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0 there exists N E N such that
I D� Uk l
(8.1 7) Note that
(.� "��' v,')
I
iB1 r (O)
E
K and every
( M · y , \l p (y)) =
d x d matrix M, we have
I
This concludes the proof.
o
o
References [1] G.ALBERTI: Rank-one properties for derivatives of functions with bounded variation. Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 239-274. [2] G .ALBERTI & L .AMBROSIO: A geometrical approach to monotone functions in Rn . Math. Z . , 230 (1999) , 259-316. [3] L.AMBROSIO & G .DAL MASO: A general chain rule for distributional derivatives. Proc. Amer. Math. Soc., 108 (1990) , 691-702. [4] L .AMBROSIO, A.COSCIA & G.DAL MAso: Fine properties of functions in BD. Arch. Rat. Mech. Anal. , 139 (1997), 201-238. [5] L.AMBROSIO, N.Fusco & D.PALLARA: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, 2000. [6] L.AMBROSIO: Transport equation and Cauchy problem for BV vector fields. Inventiones Mathematicae, 158 (2004), 227-260. [7] L. AMBROSIO: Lecture notes on transport equation and Cauchy problem for BV vector fields and applications. To be published in the proceedings of the School on Geometric Measure Theory, Luminy, October 2003, available at http: // cvgmt.sns.it . [8] L. AMBROSIO & C . DE LELLIS: Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions. IMRN, 41 (2003) , 2205-2220. [9] L .AMBROSIO & C.DE LELLIS: A note on admissible solutions of ld scalar conservation laws and 2d Hamilton-Jacobi equations Journal of Hyperbolic Differential Equations, 1 (4) (2004), 813-826. [10] L .AMBROSIO, F .BouCHUT & C.DE LELLIS: Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Comm. Partial Differential Equations, 29 (2004), 1635-1651. [11] L .AMBROSIO, G.CRIPPA & S .MANIGLIA: Traces and fine properties of a B D class of vector fields and applications. Preprint, 2004. [12] G.ANZELLOTTI: Pairings between measures and bounded functions and compensated com pactness. Ann. Mat. Pura App., 135 (1983) , 293-318. [13] G . ANZELLOTTI: The Euler equation for functionals with linear growth. Trans. Amer. Mat. Soc., 290 (1985), 483-501. [14] G.ANZELLOTTI: Traces of bounded vectorfields and the divergence theorem. Unpublished preprint, 1983. [15] F .BouCHUT & F .JAMES: One dimensional transport equation with discontinuous coefficients. Nonlinear Analysis, 32 (1998) , 891-933. [16] F .BouCHUT: Renormalized solutions to the Vlasov equation with coefficients of bounded variation. Arch. Rational Mech. Anal. , 157 (2001), 75-90. [17] F .BouCHUT, F . JAMES & S . MANCINI: Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficients. Preprint, 2004.
THE CHAIN RULE FOR THE DIVERGENCE OF VECTOR FIELDS
67
[18] A.BRESSAN: An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend. Sem. Mat. Univ. Padova, 110 (2003) , 103-1 17. [19] r.CAPUZZO DOLCETTA & B.PERTHAME: On some analogy between different approaches to first order PDE's with nonsmooth coefficients. Adv. Math. Sci Appl., 6 ( 1996), 689-703. [20] G.-Q. CHEN & H.FRID: Divergence-measure fields and conservation laws. Arch. Rational Mech. Anal., 147 ( 1999) , 89-118. [21] G.-Q. CHEN & H . FRID: Extended divergence-measure fields and the Euler equation of gas dynamics. Comm. Math. Phys., 236 (2003) , 251-280. [22] F. COLOMBINI & N .LERNER: Uniqueness of continuous solutions for BV vector fields. Duke Math. J . , 111 (2002) , 357-384. [23] F.COLOMBINI & N.LERNER: Uniqueness of LOO solutions for a class of conormal B V vector fields. Preprint, 2003. [24] C.M.DAFERMOS: Hyperbolic conservation laws in continuum physics. Springer-Verlag, Berlin, 1999. [25] R.J.DI PERNA & P.L.LIONS: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. , 98 (1989) , 511-547. [26] L.C.EvANS & R.F. GARIEPY: Lecture notes on measure theory and fine properties of func tions, CRC Press, 1992. [27] H .FEDERER: Geometric measure theory, Springer, 1969. [28] B.L.KEYFITZ & H . C .KRANZER: A system of nonstrictly hyperbolic conservation laws arising in elasticity theory. Arch. Rational Mech. Anal., 72 ( 1980) , 219-241. [29] S.N. KRUZHKOV First order quasilinear equations in several independent variables. Math USSR-Sbornik, 10 (1970) , 2 1 7-243. [30] P.L.LIONS: Generalized Solutions of Hamilton-Jacobi Equations, volume 69 of Research Notes in Mathematics. Pitman Advanced Publishing Program, London, 1982. [31] P.L. LIONS: Sur les equations differentielles ordinaires et les equations de transport. C. R. Acad. Sci. Paris Ser. I, 326 ( 1998), 833-838. [32] G.PETROVA & B .Popov: Linear transport equation with discontinuous coefficients. Comm. PDE, 24 ( 1999) , 1849-1873. [33] F.POPAUD & M . RASCLE: Measure solutions to the liner multidimensional transport equation with non-smooth coefficients. Comm. Partial Differential Equations, 22 ( 1997) , 337-358. [34] R. TEMAM: Problemes mathematiques en plasticiU. Gauthier-Villars, Paris, 1983. [35] A.VAss EuR: Strong traces for solutions of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal., 160 (2001), 181-193. [36] A.r.VOL'PERT: Spaces BV and quasilinear equations. Mat. Sb. (N.S.) 73 (115) ( 1967) , 255302. SCUOLA NORMALE SUPERIORE, PIAZZA DEI CAVALIERI, 56126 PISA, ITALY E-mail address: 1 . ambrosio@sns . it WINTERTHURERSTRASSE 190, CH-8057 ZURICH, SWITZERLAND E-mail address: camillo . delellis@math . unizh . ch FACULTY OF MATHEMATICS AND PHYSICS, CHARLES UNIVERSITY, SOKOLOVSKA 83, 186 75, PRAHA 8, CZECH REPUBLIC and FACULTY OF SCIENCE, J . E. PURKYNE UNIVERSITY, C ESKE MLADEZE 8, 400 96 U STf NAD LABEM, CZECH REPUBLIC E-mail address: maly@karlin . mff . cuni . cz
Contemporary Mathematics Volume 446, 2007
Compactness A.
Bahri
Dedicated to Haim Brezis on his sixtieth birthday
Contact Form Geometry has developed over the two last decades into a major field, connected to Symplectic Geometry and Topology but independent, with its specific aims and goals. Central to the field have been two major questions which are on one hand the study of periodic orbits of Reeb vector-fields, the so-called Weinstein conjecture [8] and on the other hand the classification of Contact Structures, on three dimensional manifolds in particular, see for example [12] . Legendrian curves have been useful in this classification. A definite progress in the study of these questions has come from Symplectic Topology. For example, Helmut Hofer [10] has been able to (almost) solve the Weinstein conjecture in dimension 3. On another hand, Eliashberg, Givental and Hofer [11] have defined a contact homology which might be related to the one defined here. However, there are clear shortcomings and unsolved issues in all these tech niques and the results obtained thus far. For example, as far as periodic orbits are concerned, one would expect to find infinitely many of them if they are non degenerate. The results and the method of proof of Helmut Hofer [10] do not yet reach this aim. The computation of the Contact Homology of [11] is also complicated because the definition involves elaborate spaces of pseudo-holomorphic curves of arbitrary genus; several non compactness problems do arise on one hand. On the other hand, one finds oneself confronted with the same problem as in Degree Theory once it is proven to be well defined: one has still to compute the degree and prove that it is non zero. For all these reasons, a multiplicity of approaches is better than a single one. Our Legendrian approach has started earlier than Symplectic Topology [1] . We have studied in details in over three books [1] , [2] , [3] a variational problem J over a Legendrian space of curves CfJ associated to the problem of finding periodic orbits of the Reeb vector-field � of a contact form 0: on a three dimensional contact man ifold without boundary M3. The generality of this framework has been discussed elsewhere [3] , [7] . Slowly, we have reached in [7] the definition of a homology in this Legendrian framework using the flow-lines originating at the periodic orbits of �. ©2007 American Mathematical Society
69
70
A. BARR!
In this work, we describe an important improvement in this approach, namely: Let (M3 , a) be a three-dimensional compact orient able manifold without bound ary and let a be a contact form on M3 . Assume that there is a non-singular vector field v in ker a such that f3 = da( v, . ) is a contact form with the same orientation as a. We introduce the action functional J(x) = J; ax (x)dt on the space of Legendrian curves 0(3 = {x E Hl(8l, M) s.t. f3x(x) == 0, a(x) = positive constant} . The various hypotheses involved in this construction are discussed i n other papers [3] , [Section IV of [5ll. The critical points of J are the periodic orbits of � which is the Reeb field of a. We refer t o [2] for the construction of a special decreasing pseudo-gradient which we built for this variational problem. We prove in this work that, under two suitable hypotheses, denoted Hypothesis (A) and Hypothesis (B) described below, compactness holds for some flow-lines ( the general result is in [6] ) on the unstable manifold of a periodic orbit. This result is potentially far reaching because it indicates the existence of an invariant sub-Morse complex made of flow-lines connecting periodic orbits, with no additional asymptots involved. We have defined in [3] a homology related to the periodic orbits using J or 0(3 . The existence of this invariant sub-Morse complex can be equivalently rephrased into the more technical statement that this homology has only periodic orbits as intermediate critical points (at infinity). We believe that this homology is equal to the homology of PCoo for the standard contact form on 83 . Some of the tools required in order to compute this homology in this particular case extend to the general case, see [7] . The paper is organized as follows: In Section 1 , we recall (without proof) some basic facts about a, v, J and 0(3 . Our proof uses little of the analytical framework. In Section 2 we consider a periodic orbit of index m in 0(3 and we build a model for its unstable manifold. In Section 3, we introduce hypothesis (A) and we state our results. In Sections 4 and 5, we proceed with the proofs. In Section 6, we extend the result to flow-lines from X2k+ l to xu, (In Sections 4 and 5 we had only considered the case X2k - XU, l ) ' Some Fredholm and transversality issues are left aside here for the sake of conciseness. These issues are resolved in [7] . 1. Some basic facts
a is a contact form, v is a vector-field in ker a. We are assuming that f3 = da(v, ·) is a contact form with the same orientation than a. We rescale v so that f3 1\ df3 = a 1\ da. The following results are established in [2] , [3] .
PROPOSITION A.
i) da(v, [�, v]) - 1 , ii) [�, [�, vll = TV, iii) The contact vector-field of f3 is w = -[�, v]+p� where p = da( v, [v, [�, [�, v]]) . =
-
71
COMPACTNESS
Let
H 1 (Sl , M) S.t. A tangent vector z t o M reads Cj3 = {x
E
f3x (i;)
==
0, ax (x) = a positive constant. }
z = A� + J-lV + TJW. If x belongs to Cj3, then x = a � + bv, with a being a positive constant. A tangent vector z at x to Cj3 reads z = A� + J-lV + TJW with:
{
A -+ pTJ = bTJ - fo1 bTJ
r, = J-la - Ab.
Let J(x) =
A , J-l , TJ
1 - periodic
fo1 ax (x)dt.
We have established in [2] , [3] : PROPOSITION B . There exists a decreasing pseudo-gradient Z for J on Cj3 such that i) The number of zeros of b does not increase on the decreasing flow-lines of Z. ii) On each flow-line, fo1 I b l (s) � C + fo1 I bl (O). iii) At the blow-up time, b( s, t) converges weakly to L:: 1 Cibt" with ICi I :::: Co > O. We also recall, see [1] , [2] , [3] , that if a vector z is transported by v, then its components verify (derivatives are taken with respect to the time along v):
On the other hand, if a vector (derivatives are taken along �):
v
{
A -+ Pl1 = TJ
r, = - A . is transported by �, then its components satisfy
2. A model for Wu (xm ) , the unstable manifold in Cj3 of a periodic orbit of index m Let X m be a periodic orbit of index m. Let r 2m be the set of curves made of m pieces of �-orbits alternated with m pieces of ±v-orbits. We recall, see [2] , [3] ,
that the second derivative f (xm ) . z . z reads as fo1 i]2 - a 2 TJ2z . a is the �-length of Xm , TJ is the w-component. We provide in what follows a model for the unstable manifold of X m in Cj3 . PROPOSITION 1 .
mo (mo
:::: 4) .
The unstable manifold of X m can be achieved in r2m for m ::::
72
A. BAHRI
PROOF. Let us assume in a first step that X m is a simple periodic orbit. We indicate at the end of the proof how to extend the result to iterates. We start with the case where m = 2k + 1 . Xm can be elliptic or hyperbolic. Assume that it is hyperbolic to start with. Then, if i! is the Poincare-return map of the periodic orbit and u is a real eigenvector of di! in ker (x, then u is transported by �, while v rotates ( considerably: m :2: ma ) . Setting xm(O) at a point where v and u coincide, setting 7) to be the -[�, v] component of u , we have
7)(0)
=
7)(1)
=
0
7) can easily be seen to have at least 2k genuine zeros, also 1j(1)1j(0) < 0 because the periodic orbit is hyperbolic of odd Monse index. The zeros of 'f) are t 1 = 0, t2 , . . . , t2k+ 1 , t2k+2 = 1 . Setting 7) = 7)i l [ti , t i+ l l , i = 1 , . . . , 2 k + 1 we derive 2 k + I-functions which are pairwise orthogonal and are orthogonal to themselves. Each 7)i defines a tangent vector Zi with .. + 2 7) = s; + s; 7)i a iT CiUti Ci+1Uti+l ' Thus, Span {Zl , " " Z2k+d is achieved in r2m. I f X2k+ 1 i s of odd Morse index 2k + 1 but elliptic, the [C v]- component 7) of any transported vector u has always exactly 2k, 2k + 1 or 2k + 2 zeros. Indeed, it needs to have at least 2k, at most 2k + 2, zeros and if the number of zeros were to change, there would be a point where u(v) would be mapped onto "Yuh-v) with 'Y < 0, yielding 2k + 1 zeros, 2k of which are genuine i.e. a hyperbolic orbit. Taking the base point at one of the nodes of the most oscillating eigenfunction, 7) must have 2k + 1 zeros to the least. This yields 2k function 7)1 , . . . , 7)2k which are zero at h = 0, t2 , . . . , t2k+1 and yield 2k vectors Zl , . . . , Z2k which are pairwise orthogonal and orthogonal to themselves. For each of these vectors, we have .. + a2 7)i = 7)i T CiUti + Ci+ 1 Uti+l so that Span {Zl ' . . . , Z2k } is achieved in r 4k + 2. We still need an additional vector Z2k+1 ' At xm(O) , we introduce the solution s;
of
s;
di!(u) - u = v .
This yields a new vector Z2k+ 1 ' 7) of Z2k+ 1 , which we denote fj2k+1 , is not zero at x(O) or at h = 0 since the orbit is not hyperbolic. If we compute the second derivative f (xm) . Z2k+1 . Z (the - [�, v]- component of Z is 7)) , we find
f (xm) . Z2k+1 . Z =
-1 1 (�2k+1
+ a 2 fj2k+1 T )7) = 7)(0) .
Thus, Z2k+1 is if (xm ) -orthogonal to Zl , . . . , z2k . We claim that we can take, after deformation of the contact form, fj2k+ 1 (0) to be negative and Z2k+1 in the negative eigenspace of if (xm ) . Then, if (xm) is non positive on Span {Zl ' . . . , Z2k, Z2k+1} which is again achieved in r4k+2 . In order to prove our claim, we complete a deformation of the contact form so that X m , without degenerating, changes from elliptic to hyperbolic with eigenvalues equal to - 1 and back to elliptic i.e. X m iterated twice degenerates but not X m which stays of odd Morse index. At the switch, di!l ker is -Id and we can set x(t d = x(O)
a
73
COMPACTNESS
at any point. We may solve continuously, through the degeneracy, the equation d£(u) - u = v. ii2 k +l (0) is zero at the switch and changes sign through it. We consider the side of the switch where it is negative. Zl , . . . , Z2 k still exist but Z2 k + 1 might have disappeared (when the Poincare-return map has an eigenvalue equal to -1 and 0 is at the node, Z2 k+l exists) . The negativity of ii2 k + l (0) means that i' (x m ) . Z2 k + 1 . Z2 k + 1 < o. We thus will have produced a space of dimension 2k + 1, Span {Zl , . . . , Z2 k l Z2 k+ l } where J" (xm ) is non positive. If X m = X 2k is of Morse index 2k, the most negative eigenfunction of (ii + a2'r}T ) has 2k zeros. An elliptic orbit is, as we will see, of odd Morse index. Thus X 2 k is hyberbolic. We pick an eigenvector u . Since m � 4, u coincides with v at some points on the periodic orbit. We set x (O) at such a time. We transport v along the periodic orbit. This yields an 'r}-component of the transported vector. Using Sturm-Liouville arguments, 'r} vanishes at 2k + 1 times, tl = 0, tl , . . . , t k +1 = O. 2 This yields 2k functions 'r}l , . . . , 'r}2 k with -
and 2k vectors Zl , . . . , Z2 k which are pairwise J" (xm )-orthogonal and orthogonal to themselves. Again, Span {Zl ' . . . , Z2 k } is achieved in f 2m . " In each occurrence, J (xm ) is non positive on the vector-space which we build and we therefore can take these directions to achieve the unstable manifold of X m Let us enter into some more details and check that our representation of Wu (x m ) in f 2m works. We will consider for simplicity first the case of a hyperbolic orbit of index 2k. We then have 2k nodes tl , t2 , . . . , t2 k where we locate the v-jumps. The basic equation is 2k 2 = a Ci6fi , 'r } T + ii 'r} reads as L: ;! l ciiii where each iii solves
L i= l
iii is 1 - periodic. We may consider the above equation when the ti 's are in the vicinity of the ti 'S. We find in this way 2k functions 'r}l , 'r}2 , . . . , 'r}2k . The second derivative J; iJ2 - a2'r}2 T in Span{'r}l , · · · , 'r}2 d reads as dAc = L: iopj CiCj ('r}i (tj) + 'r}j (ti) ) + L: ct'r}i (ti) . Clearly after integration by parts, 'r}i (tj ) = 'r}j (ti). When all the ti'S are located at the ti 's, this quadratic form is identically zero.
We claim that
74
A. BAHRI
when c is a fixed positive constant. Thus, by moving slightly for each direction (C1 , ' " , C2k) the location of the Dirac masses, we may achieve that J " (X2k) is negative in this space. Assume that the above inequality does not hold. Then, for each i such that I Ci l ::::: C1 vI: c% , where C1 is a small constant ( C1 = ..jC for example) ,
Getting rid in the remainder of the other c� s ( a�i T/j (ti) is bounded as we will see) we derive that the matrix (off diagonal terms should be multiplied by 2)
or some non trivial sub-matrix of the above one should have a zero eigenvalue. For the sake of simplicity, we will assume that the number of points (t1 , ' " , t2k) has not been reduced in the above process. We can compute this matrix as follows: we set a section to � at h , tangent to kera at X 2k (h ) . Let 'Ij; be the Poincare-return map. 'TIl is obtained by solving the equation
D'Ij;(z) - Z = v, Z is in kerax2k(tl ) and reads Z =
/11 V - 'TId�, v] .
Let 'lj;o be the Poincare-return map at t1 . X2k is hyperbolic so that D'Ij;o (v) = "(v. D'Ij; reads as (Y 0 D'Ij;o 0 (y- 1 and our equation above becomes
Let 6.t = -t 1 + t1 which we can assume to be small,. We then have, using 1 the transport equations (j.L = -'TIT, � = /1) : (Y- (z) = ( /1 1 + O ('TIl 6.t))V - ('TIl + /1 1 6.t) [�, V] + o(6.t). Observe that ih (t1 ) = 0 so that 'TI1 (t 1 ) = 0(1) and (Y- 1 (z) = /1 1 V - ('TIl + /11 6.t) [�, v] + o(6.t) . The matrix of D'Ij;o on kera reads
D'Ij;o - Id =
(
"(
�1
�� 1
)
(6" %)
in the (v, - [�, v]) basis so that
.
Our equations on z re-read then:
h - 1)/1 1
+
a'TI1
=
1 + o(6.t)
1 - "( -('TIl + /1 1 6.t) = 6.t(l + 0(1 ) ) .
"( Thus, since 6.t = -h + t1 and 'i/1 (t1 ) = 0, we find:
and this extends to give
75
COMPACTNESS
If we want to compute now a�i r/j (ti) I fi for i -I- j , we use that fact that rjj = ILj so that a�i r/j (ti) I fi = ILj (ti ) . Between tj and ti , along �, v is mapped onto ej v so that
Our matrix thus reads
2e�
The e{ satisfy the relations (we go from x(tj ) to X(ti) along +0 : e{ e; = ')' ; e{ eJ = ef if X2k (tj ) is in between X2k (ti) and X2k(t£) , e{eJ = ,),ef otherwise. We have to compute the determinant of this matrix. We multiply the matrix by ')' 1 and each line i by eI ( ei = 1) . We find then the following determinant (after a manipulation and a transposition with the above notations) :
1
-
... 2e�k +1 20J 2')' b + 1 )e� 2e�k 2e� 2e�k �� 2�� ')' b ��� e� . . 2')' 2eh 2eh . . . b + 1)e�k
.
�
Ol
2 2 2 +1 2')' b + 1) 2 2')' ,),+ 1 Olk
1
:�
2')'
2 2 2
2')' " , ,), + 1
which is not zero for ,), -I- 1 and ')' -I- -1 . The proof extends verbatim for X2k+l hyperbolic. We turn now to the case of elliptic orbits to prove our claims and prove also that we can produce for Wu(xm) an m-dimensional manifold - with ±v-jumps which can be tracked down - in r2 m where J" (xm) is negative. Using a scheme similar to the one used in Lemma 1 1 of [3] , we can see that an elliptic orbit must be of odd Morse index. Indeed, deforming the contact form as in Lemma 1 1 of [3] , we can change this elliptic orbit into a hyperbolic orbit: we rotate Cv(O) on Av(l), A < 0; C is the Poincare -return map. We cannot complete the other move allowed by Lemma 1 1 of [3] with a critical point of infinity i.e. bring Cv(O) onto Av(l), A > 0 since the Poincare-return map C would then degenerate for a periodic orbit, while it does not for a critical point at infinity. We thus have changed our elliptic orbit into a hyperbolic orbit with eigenvalue ')' = - 1 . Let us follow the proof of Lemma 1 1 of [3] : -1 is of course a double eigenvalue at the switch; C =-Id in fact. Beyond the switch, see the proof of Lemma 11 of [3] , Xm is still an elliptic orbit of index m ( I trC I remains less than 2 since C remains conjugate to a rotation) . But da(Cv(O) , v(O)) has changed sign. The analysis which we completed above for hyperbolic orbits applies; but it does not lead to the conclusion which we desire since ,), = -1 (observe that a = 0) . We consider a base point O. Its location is not important since D'l/Jo = I d. Taking the solution of i7 + a 2 77Y = 0 , 77(0) = 0, i}(0) = 1 , we claim that we can produce 2k oscillations, thus produce 2k functions T}i in the null eigenvalue of the 2 2 2 quadratic form 'lj - a 77 y using 2k + 1 ± v-jumps. We still need an additional -
f01
76
A. BARRI
direction Z2k + l which we create by solving D'I/J(z) = Z + v at O. Since 1]i (O) = 0 for i = 1 , . . . , 2k, Z2k+ l and the z/s associated to the 1]/s are orthogonal for if (xm). We claim that l' (xm) Z2k +l . Z2k +l < 0
on one side of the switch: indeed, ihk + l (0) is zero at the switch. Solving the equation for il2k +l continuously through C = -Id, il2k +l (0) becomes negative on one side or the other since da( Cv(O) , v(O)) changes sign (in the argument of Lemma 1 1 of [3] , Cv(O) crosses J.Lv(1 ) , J.L E lR - here J.L = - 1 - through the deformation) . Since
our claim is established. We now claim - this will conclude the proof of Proposition 1 after a perturbation argument bringing us back to the elliptic case - that, at the switch,
for Z E Span {Zl ' . . . , z2d . Zi is built using 1]i . The ill 's are not here the 1]l ' s of the zl 's. They are the ill 's corresponding to the various solutions of D'I/J(z) - Z v at the various nodes. Computing as in the case when Xm was hyperbolic (here Xm is elliptic turned to hyperbolic with 'Y = - 1 ) , we find that such an inequality holds for all vectors (C l . . . , C2k) which are not in the vicinity of unit vectors such that an odd number ' of Ci ' S are non zero. In fact, the zero eigenvectors of the matrices A - these matrices may be p x p, for any p ::; m if some Ci 'S are o ( vI: c;) - are equal to (1, - 1 , 1 , - 1 , . . . ) with an odd number of components (some intermediate c/s might be zero) . The 1]-component of all these eigenvectors read I: Cj �j . The �j reads as �ilj J1 2k and are equal to I: + (-1) i - j 1]i with 1]i > 0; such a combination I: cAj with the =
;
cl ' s as above (an odd number of them is non zero and equal to 1 in absolute value) is far from Span {Zl , " " Z2k } : a vector in Span{zl , . . . , Z2k } has no component on 1]2k +l ' The conclusion follows. Another line of proof uses Proposition 29, page 198 of [2] : once C -Id, we can perturb slightly C and bring C to have real eigenvalues very close to - 1 but different from -1. Xm has become hyperbolic and the framework developed for hyperbolic orbits applies. This method has the definite advantage that the representation of Wu(Xm) extend to iterates since the Morse index of iterates grows then linearly. Turning all simple periodic orbits into hyperbolic orbits might be the D best choice for this homology. =
m
Observe that the unstable manifold of Xm, as we start near ± v-jumps which have a location than can be tracked down. * will designate below the location of such a ±v-jump
Xm, is provided by
COMPACTNESS
3.
77
Hypothesis (A) , Hypothesis (B) , Statement of the result
J on C(3 has critical points at infinity which are curves of Ukr2k . Each r2k is the space of curves made of k pieces of �-orbits alternated with k pieces of ±v orbits. Denoting ai the time spent along � on the i-th �-piece of a curve of r2k , we introduce the functional Joo equal to L ai. The critical points at infinity of J on C(3 are the critical points o f Joo on Ukr2k . They are described in [3] , there is a vast zoology among them. Let us recall the two following definitions: DEFINITION 1 . A v-jump between two points Xo and X l = X S I is a v-jump between conjugate points if, denoting CPs , the one-parameter group of v, we have:
( cp: I 0:) Xl
= O:XI .
Such points liv.e generically on a hypersurface L of M. DEFINITION 2. A �-piece [Yo, Yl ] of an orbit is characteristic if v has completed exactly k (k E Z) half revolutions from Yo to Yl .
The description of the critical points at infinity x oo of J goes as follows ([3]): • If X oo has no characteristic �-piece, all its v-jumps are v-jumps between conjugate points. • If x oo has some characteristic �-pieces, some additional conditions have to be satisfied, see [3] . Any v-jump between non characteristic pieces is a v-jump between conjugate points. To each x oo is associated a suitahle Poincare-return map C (see [3]) which preserves area. We have shown in [3] , Proposition 16 and Lemma 1 1 , how, without modifying C, we can redistribute the v-rotation from a non-characteristic �-piece of orbit to another one at the expense of creating additional critical points at infinity with more characteristic �-pieces than X oo . We may thus redistribute all the v-rotation from all the characteristic �-pieces on a single one (up to E > 0; a small amount of rotation is to be left on each �-piece). We then introduce: Hypothesis (A) .
Assume that x oo has at most one characteristic � -piece. As the number of � pieces of xoo tends to infinity, the amount of rotation of v derived after rescaling the rotation from all the non-characteristic � -pieces onto a single one of them is at least 27r (the � -piece where the rotation is relocated is of our choice). We also introduce: Hypothesis (B) .
On a given characteristic piece, all the ±v-jumps belonging to the same family which are a little bit inside (counting v-rotation) this characteristic piece are of comparable relative sizes. Furthermore, the time span, measured in terms of v rotation along the � -piece between the left and the right extreme ±v-jumps of a given family, is less than 7r . We prove the following result:
78
A. BAHR!
THEOREM 1 . Let Y:;: l be a critical point at infinity of index m - 1 having at least one characteristic piece of strict HJ -index larger than or equal to 3 . Assume that the maximal number of zeros of b on its unstable manifold is 2 [�] . Assume that, if m is odd, m is large, the HJ -index (strict if degenerate}of each � -piece is at least 3 and that the number of ±v-jumps of Y:;:- l is also large. Let Ym be a periodic orbit of � of index m. Then, the intersection number i(Ym , Y:;:- l ) for the flow of [3] is zero. Observation: The statement of Theorem 1 holds without the use of Hypothesis (B) and without the restriction on the strict HJ-index (see [3] ) if the flow-lines from Ym to a neighborhood of Y:;: l do not involve companions, see 5.2 for the definition of this notion. Flow-lines from Ym to Y:;: l with Y:;: l having only non characteristic �-pieces have already been studied and ruled out for m large, under Hypothesis (A) in [3] . Hypothesis (A) and the first part of Hypothesis (B) are very natural. The second part of Hypothesis (B) is removed in [6] , [7] and Theorem 1 is generalized under somewhat different assumptions (the assumptions in the case when m is odd are then also required when m is even and the HJ-index of the �-pieces, instead of being lower-bounded by 3, is then assumed to be larger than a constant mo depending only on the geometry of 0:, v ) 4. The hole flow 4.1. Combinatorics. We start with an abstract result which may be viewed as an (elementary) observation in combinatorics. We consider a sequence of 2k points each bearing a sign, + or -. Such a sequence, together with the assigned distribution of signs, is called in the sequel a configuration. A configuration contains at most 2k sign changes. A configuration with 2k sign changes is called maximal. Given a configuration, we consider two consecutive points. We assume k ;:::: 2. We are given a sign of rotation (the same for all configurations, which are placed on a curve). Then, one of these points is the first one (using the positive rotation) and the other one is the second one. Let us assume that the sign + is assigned to the first one: +
1 2 3 The hole flow assigns signs to the intervals between the points as follows: Starting from the + assigned to 1 and independently of the configuration which we are facing, we assign signs to the remaining 2k - 1 points so that there is an alternance each shift between the + and - signs. Thus, we have +
+
1
2
+
3
4
signs of the configuration
79
COMPACTNESS
1
2
+
3
4
+
alternating distribution We then introduce on each interval distribution.
[i, i + 1 ]
the sign of
i
in the alternating
We claim
PROPOSITION 2 . The original configuration of signs with the additional inter mediate signs has at most 2k sign changes. PROOF. We reverse the process and start with the alternated configuration. We add in between its jump the signs of the original configuration, inserting between 1 and 2, the sign of 2 for the original configuration and so forth. The result is the 0 same. Viewed in this way, the claim of the Proposition 2 is obvious. Next, we discuss the choice of the starting point 1 . We assume that we have some freedom of choice on 1 but the sign assigned to the choice should be + (i.e. the same, it could also be -) for all possible choices in the configuration. We then claim:
PROPOSITION 3 . Let us consider two distinct choices 1 and I' for 1 extracted from the same configuration. Then either the configuration has a sign repetition between 1 and I' or the two hole flows, the one corresponding to 1 and the one corresponding to 1', coincide. PROOF. The alternating distributions for 1 and I' coincide if and only if there is an odd number of points between 1 and I'. Then ( and only then) , the hole flows coincide. If there is an even number of points between 1 and 1 ' , then there is a forced 0 sign repetition in the configuration since 1 and I' bear the same sign. An obvious observation which we will be using below states: Observation 1 Given a configuration, assume that between two identical signs points there is an even number of points or that between two reverse signs points there is an odd number of points, then the configuration is forced to contain a sign repetition. Next, we define:
DEFINITION 3 . Given a configuration, an elementary operation on this con figuration is the reversal of the sign of one and only one point in the configuration. PROPOSITION 4. Given a configuration with a choice of 1 and the addition of the intermediate signs of the hole flow, any elementary operation completed on a point of the configuration distinct from 1 does not increase the number of sign changes of the configuration beyond 2k.
80
A. BAHRI
P ROOF. The hole flow is unperturbed. We are simply modifying the original configuration outside of 1. 0 Let us assume that we are considering several configurations (J which all have the sign + on point j and have a forced repetition between positive signs in [i, j ] , i < j. Assume that two distinct alternating distributions starting from a + can be defined on (J. We then have PROPOSITION 5 . Choose a hole flow using one of the alternating distributions. Introduce the corresponding intermediate signs in all intervals [k, k + 1] except those contained in [i, j] . Introduce furthermore a sign - between j and j + 1. The distribution of signs derived in this way has at most 2k sign changes. Another more obvious Proposition states: P ROPOSITION 6 . Given all configurations which have a sign repetition, we can introduce once an arbitrary additional sign on the interval of our choice without increasing the number of sign changes beyond 2k. (J
Observation 2 For Proposition 6, since we are not assuming that we proceed as for Proposition 2, we need to be careful and not use the hole flow as we are introducing the additional sign. We may use, however, elementary operations on the configurations (J as long as these elementary operations do not destroy one sign repetition at least. P ROOF OF PROPOSITION 5 . In the case where the hole flow used is not the one provided by the use of j as initial point, then this hole flow gives a - between j and j + 1 (since the one of j gives a + in this interval) . Thus this hole flow is compatible with the additional introduction of a - between j and j + 1 and the number of sign changes does not increase beyond 2k. We consider now the hole flow provided by the use of j as initial point. We introduce a - between j and j + 1. This might increase the number of zeros beyond
2k.
From i to j , we then have one sign repetition to the least. The full use of the hole flow associated to j (before the introduction of the - between j and j + 1) will result in an additional sign change between i and j with respect to (J . This is obvious if this sign repetition occurs between j and j - 1. Then the hole flow would introduce a - between j - 1 and j which are both positive. Otherwise, starting from the + sign introduced by the hole flow between j and j + 1, we evolve left towards i, alternating the signs. Before hitting a repetition of sign, the sign introduced by the hole flow agrees with the sign of the left edge of the interval [k, k + 1] considered; otherwise an additional sign change occurs and the claim follows:
+ j-2 etc.
+
+ + j-l
J
j+l
COMPACTNESS
81
Accordingly, the sign introduced has to disagree with the sign of the right edge of the interval. But then, we arrive at the repetition, a sign change has to occur. We thus see that if we had used the hole flow between i and j, we would have introduced a sign change, i.e., two additional zeros. The total number of sign changes would be 2k. However, we have not used this hole flow between and j. The introduction of the sign - between j and j + 1 would only provide for one sign change. Proposition 5 follows.
i
Observation 3 We could have introduced the + of the hole flow of j or [j, j + 1] , only that we would need to complete it close to j and introduce the - after it is the interval. With this provision, the statement of Proposition 5 can be modified into "Introduce the corresponding intermediate signs-except those contained in [i, j]." 4.2. Normals. Given a critical point as infinity xco, with a �-piece [xo , x ci ] which is characteris tic, i.e., along which v has completed exactly a certain number m of half-revolutions in the �-transport, we may define m + 1 nodes,
Yo = Xo < Y1 < . . . < Ym
=
xci ·
These nodes are the points Yi along the characteristic �-piece at which v, starting from Yo = xci , has completed exactly i half-revolutions. To each interval (Yi, Yi+ 1 ), are associated a decreasing normal to the right and a decreasing normal to the left, in fact continuously varying families of these as follows. We choose a point z in (Yi, Yi + d. We consider the vector v at z and we transport it to Yo = Xo for the normal to the right and to Ym = xci for the normal to the left. We derive vectors which have components on [�, v] since z is not a node. On the other hand, this characteristic �-piece is preceded and is followed by �-pieces:
l' ,' "0
We take � at x �- , x�+ and we transport along v to Yo, Ym " We derive vectors at Yo, Ym, equal respectively to (1 + Al)� + Bl [�, v] + lev, (1 + At)� + Bt [�, v] + f1+ v . We scale these vectors so that they compensate exactly the [C v]-component of the transport of v(z) at Yo, Ym:
82
A. BAHRI
We adjust the � and v-components by modifying the lengths of the � and v pieces. We derive in this way a normal to the right and a normal to the left,
±Ni- (z) , ±Nt (z) .
We assign to each of them the orientation which corresponds to decrease J. Indeed, A1 , At are not zero and
We derive our normals N! (z) . To each of them is associated on orientation of the ± v-jump which is introduced at z. Observe that there is a way to combine ±Ni- (z) with -:tNi+ (Z) in order to build a tangent vector into x oo ; we simply assign the same orientation to the v-jump at z. Because XOO is critical in its stratification,
Thus, if u is chosen with Ni- on the left side, it was -Nt on the right side. Niand Nt corresponding to ± v-jumps having reverse orientations. Accordingly, if we think of the curves X OO + ENi- , X OO + ENt , E > 0, these two curves will contain at z a ± v-jump with the same orientation. This orientation is the same throughout (Yi , Ui+ l ) . We will refer to it in the remainder of this work as the preferred orientation of the normals in (Yi , Yi+ l ) . DEFINITION 4 . X OO will be labeled false if the preferred orientation i n (Yo, Yl) (or (Ym - l , Ym) ) corresponds to the orientation of the left edge of the �-piece (respec tively the orientation of the right edge of this �-piece). Otherwise, XOO is sign-true. We then have:
If X OO is sign-true, then the edge orientations are reversed; is odd, they agree.
P ROPOSITION 7.
otherwise, if
m
COMPACTNESS
83
PROOF. Assume the left edge is positively oriented. Since XOO is sign true, No- ( z ) has - as preferred orientation. The preferred orientation switches from a nodal zone to the next nodal zone. If m is even, N;;" _ l ( z ) has + as preferred orientation. Since XOO is sign-true, the right edge has the negative orientation. Proposition 7 follows.
0
4.3. Hole flow and Normal ( H) -flow on curves of r4k near xoo . r4k is the space of curves made of 2k �-pieces alternating with 2k ± v-pieces. The 2k ± v-pieces build a configuration. Some of these ±v-jumps might be zero. We are assuming throughout 4 that we can keep track of all the 2k± v-pieces on our deformation classes. Let us consider such curves near xoo . Since they are close in graph to x oo , they must have a nearly �-piece (a "�-piece" broken maybe with ±v-jumps) close to the characteristic piece of XOO and they must have nearly ±v-jumps corresponding to the two edges of this characteristic piece. Let us assume, in a first step, that we can associate without ambiguity a ±v jump on our curves which corresponds to the left edge. We will discuss this point later. We have "nodes" also, Xo, . . . , Xm which correspond to Yo , . . . , Ym and can be roughly defined using the v-rotation in the transport along the nearly �-piece, up to 0(1 ) . We thus have preferred orientations on each (Xi, Xi + ! ) and also normals Ni± defined on our curves (the definition of preferred orientation, normals on XOO extends easily). Let us consider such a curve and the sequence of its ± v-jumps on the nearby �-piece. We denote the nodes (up to 0(1» by vertical lines, the sequences of ± v-jumps by *'s. We will also have a line indicating the preferred orientation of each nodal zone, which will be labeled "Area of + " or "Area of -" . We will assume for the sake of simplicity that the left edge is positively oriented and we will introduce the hole flow associated to this left edge. Its distribution of signs will be indicated at the bottom of our drawings: Preferred orientation'\., +
/'
(Area of ) Hole flow
A. BARRI
84
Observe that there is a * with the positive orientation at Xo or nearby, corre sponding to the left edge. These should be also another one at Xm or nearby (up to ambiguity) corresponding to the right edge. We then have: PROPOSITION 8 . If at any point between two consecutive * 's, not close 0(1) to the nodes, there is agreement between the sign of the hole flow and the preferred orientation, a decreasing normal can be defined which will bring the curve below the level of J(XOO) without increasing the number of sign changes beyond 2k. If there are several points of this type, these decreasing normals can be convex combined and the claims remain unchanged. P ROOF . The point lies between two nodes (Xi , Xi+1 ) ' It is not close 0(1) to the edges of this interval. We may use Ni± (�) ' This use decreases J at a negative rate bounded away from zero. Along this displacement, the nodes may move. We need to decrease J only by an amount equal to 0(1) so that the ± v-jump introduced by the normal is small and the nodes move little. The points when Ni- was introduced was not 0(1) close to the nodes so that the argument proceeds above J(XOO) E . -
Since the sign(s) introduced by this (these) normals agree with the sign(s) of the hole flow associated to the left edge, the number of sign changes does not increase beyond 2k. C OROLLARY 1 .
introduced.
If there is a node with no * close to it, such a normal can be
P ROOF. Indeed, there are then two consecutive *'s with points inbetween them not close 0(1) to nodes and having reversed preferred orientations. One of them 0 agrees with the sign of the hole flow on the interval. PROPOSITION 9 . Assume that a * is in between nodes, not close 0(1) to any of them. Then, a normal can be introduced at the location of this * with the associated preferred orientation. This normal will decrease the curve below J(XOO) - E without increasing the number of sign changes beyond 2k. The use of several such normals together, also combined with the hole flow, has the same properties. PROOF. This corresponds to elementary operations on a configuration. Since the * corresponding to the the left edge is not in between nodes, we are 0 not perturbing it. The conclusion then follows from Proposition 4 above. Definition of the * of an edge. We consider the case of the left edge and an approaching configuration (j. We can define an "average left edge" e which varies continuously with (j . We consider a fixed small number b > 0 and all the ± v-jumps of (j which are contained in a b-neighborhood of e, V6 . J is fixed as (j approaches x oo . b is as small as we may wish. These ± v-jumps are ordered according to the time-parameter on the
85
COMPACTNESS
corresponding curve. Co is a fixed small parameter. We consider the last of the ± v-jumps in Va of size at least 2co having the orientation of the left edge. If this last ± v-jump is defined unambiguously, the corresponding * is the * of the left edge. As this ± v-jump becomes of size 2co or starts to move out of Va , we may have an overlap of various choices for this ± v-jump and therefore of various choices for the * defining the left edge. A problem of definition for our deformation arises when these various *'s define different hole flows; we will address this issue in our arguments. As we point out below, in the proof of Proposition 12, when the last of these large ± v-jumps becomes small (of size T < c < co ) , we can move it away from this Va-neighborhood while decreasing J. This uses Proposition 18 (below) . 4.4. Forced repetition. The construction completed in 4.3 rests upon the definition of the hole flow which, in turn, rests upon the choice of a representative for the left edge If there is a clear representative for this left edge-as when the left edge of the curve neighboring X OO is large and isolated from the other ± v-jumps-this hole flow is well defined. But if there is an ambiguity-as when the left "edge" of the neighboring curve is made of several ± v-jumps having the same orientation-then the hole flow might not be well defined. In view of the proof of Proposition 5, the configuration has then to contain a forced repetition corresponding to the orientation of the left edge (positive). Thus, PROPOSITION 1 0 . If the construction of 4.3 cannot be carried out at a curve x near x oo , then the configuration of x contains a forced repetition (with the positive orientation) near the left edge of Xoo . Outside of the curves x having such configu rations, the constructions of 4.3 can be carried out continuously. Furthermore, PROPOSITION 1 1 . Let x be a curve neighboring X OO with a configuration for which the hole flow of the left edge is defined without ambiguity. If the construction of 4 · 3 cannot be carried on a whole neighborhood of x, then contains a forced repetition between the representatives of the left edge and the right edge of X OO (J"
(J"
(J"
•
PROOF. In view of 4.3, we must then have at least a * at each node Yl , · · · , Ym - l . Two very close *'s can be separated, see Proposition 18, below. Thus, there is ex actly one * at each node (close 0(1 ) ) . In view of 4.3, again, we cannot have then a * between nodes. Finally, before the family of *'s at Xl, · · · , Xm - l , there must be a * at Xo (close 0(1 ) ) ) with the orientation of the left edge. A similar statement can be made for the right edge. We are assuming that XOO is sign true. Thus, if m is odd, the edge orientations agree and we have a sign repetition in between. The argument repeats for m even 0 since the edge orientations are then reversed.
We continue our study of such configurations and observe: PROPOSITION 1 2 . A neighborhood U of such configurations may be chosen so that the construction of 4.3. on aU reduces to the use of the normal (II) flow on * 's which do not represent the left edge, while all configurations in U contain a forced repetition.
86
A. BARRI
P ROOF. If U is taken small enough, all the configurations will have exactly one * in a neighborhood of a node. On aU, one of these * ' s has to be "away" from its corresponding node and the normal (II) flow can then be applied to this *. Since this * might represent the left edge, we have to modify U. We observe that if any of the *'s which are located near X l , · · · , Xm - l moves more than 0(1) from the related node, the configuration has crossed in a region where the use of the normal (II) flow on a * which does not represent the left edge is available. We then consider the first * (*0) before the * at Xl having the same orientation than the left edge and the first *(* m ) after the * at Xm - l having the same orientation than the right edge. If one of these *'s advances (for Xl ) or recedes (for xm ) towards the neighboring * on the characteristic piece, then this * has to move out of the node and the configuration crosses into a region with a good normal (II) flow. One of these * ' s could also exit the characteristic piece and reverse sign or become small after leaving the edge (forcing thereby all preceding *'s on the left, all following *'s for the right to leave also) . But since we need a representative for the edge, a * at the node Xl of Xm - l has to move out of the node, with the same conclusion. Finally, one of these * ' s, *0 or Xm, could become smaller and smaller, reaching a size c, l' < C < Co, Co small and fixed. Since all our configurations are near x oo , one or several large ± v-jumps build the edges and none of them is then *0 or * m " whichever has become small. We are assuming, for example, that *0 is becoming small. There is a *, denoted * 1 , within 0(1) of Xl and no * between *0 and * 1 . Using Proposition 18 (see below) on such configurations, we can move *0 away from the large ± v-jumps of the edge towards Xo, while decreasing J. Once *0 is a little bit away from the edge, since it does not represent it anymore (it is small and slightly away from the "average edge" ), the normal (II) flow can be used on it in order to decrease below X OO . This covers in particular the situation, near aU, when the jump *0 defining the left edge (or *m for the right edge) is becoming small and is being replaced by another jump, another * for the left edge. In the transition, the initial *0 can be moved inside the characteristic piece. As the definition of the * associated to the left edge changes, the normal (II) flow can be used on this initial *0 to decrease below x oo . We thus see that we can track down all these configurations and that they contain, throughout, a forced repetition inside the characteristic piece, i.e., between *0 and * m and that the exit set is through the frontier aU of a domain U where a normal (II) flow is available. This includes the case when the jump at *0 or * m becomes of a size c, l' < C < Co. Then we are changing the definition of *0 or * m for the left or right edge (respectively). In the transition region, the normal (II) flow (without any use of another hole flow or the introduction of an additional± v-jump) can be used on the jump *0 or * m which has become small. D
Observation 4 Some further thinking shows that the above reasoning on *0 and * m is not needed. The basic argument runs as follows: U is defined to be the set on Wu ( X2 k ) near X� _ l where there is a * near each node Xl , . . . , xm . Would there be other *'s in between nodes, the normal II flow can be used on them. This is in particular the case on aU. Assuming that there is a * near each node and none in between, in Ul C U 1 C U, we can introduce in each zone a decreasing normal corresponding to the preferred orientation. This corresponds to a distribution of
COMPACTNESS
87
signs which behaves exactly as a hole flow, only that the initial 1 is not defined for this distribution of signs. However, since we are introducing this alternating distribution once, we do not increase the number of signs changes beyond 2k. The definition of this alternating distribution is clear and unique. The use of this distribution is limited to Ul , shielded from Uc . It convex - combines naturally with the normal II flow-. The advantage of this argument is that it bypasses the use of Proposition 1 1 and the forced repetition in U, thus the definition of *0 and * m as well as of the *'s of the edges as this point. 4.5. The Global picture, the degree is zero. Two or more *'s can be assumed not to be too close. Otherwise, using Propo sition 18 below we can bring them apart (in a global deformation) while decreasing
J.
We have now four regions and the transition between them; there is first U and U, a smaller version of U. On Ul we use the normal No (z) and extensions of it (alternated sign distribution starting with No (z) between nodes) related to (xo, Xl ) and we convex-combine it with the normal (II) flow as above on U Ul . Using Proposition 6 and Observation 2, we have not raised the number of zeros beyond 2k. Outside of U, we use the construction 3. when available. It provides with combinations of hole flow and normal (II) flows in the region where the representative of the left edge is defined properly. In the remainder, we have several possible representatives. We may assume that the hole flows corresponding to these representatives do not coincide. Otherwise, we find no problem in order to define our decreasing deformation. We then must have a forced repetition. In order to decrease J in this region, we introduce again a normal No (z) related to (xo , xI ) . This is completed ato followto: we may atotoume that there ito no * "between nodes" (i.e. not in the immediate vicinity of nodes) . Otherwise, we use the normal (II) flow maybe on several nodes at the same time. This flow can easily be seen, after the construction of No- (z), convex-combined through "sliding" with No- (z) (we can use several consecutive copies of No- ( z ) in the same interval between two * 's to complete the convex-combination) . Under this assumption, there is a last * "before the node at Xl " (i.e. before Xl and not in the vicinity of xI ) . This last * (*0) is close to the left edge. No (z) is introduced between this * and X l . It is introduced in the "middle" of the first nodal zone. The representatives for the left edge are chosen among the v-jumps of size � Co having the orientation of the left edge and close to the average left edge. Thus No (z) is introduced after any of these representatives. It is also introduced before (on the characteristic piece) the ±v-jump corresponding to the hole flow (whatever it is) in the interval starting with *0. In fact, since the only ±v-jumps needed for the hole flow can be located in the vicinity of the nodes and the other ones taken to be zero, we may assume that No (z) is introduced before any ±v-jumps used by any of the hole flows on the configuration. Thus we do not introduce v-jumps "in the edge" i.e. before *0. Since the competing *'s for the definition of our hole flows are before *0 (*0 possibly included) , we do not introduce ±v-jumps between the starting 1 's of our hole flows. We have to worry about the convex-combination of No (z) and one of the competing hole-flows, namely the one such that the ±v-jump introduced between *0 and * 1 (the next *, to the right of *0) is not negative. If * 1 is not beyond X l , no positive v-jump is introduced before * 1 since this v-jump would not decrease J.
Ul
C
-
88
A. BARR!
We need to worry only if *0 is a positive v-jump. Otherwise if *0 is negative or zero, No (z) and *0 are compatible (including if *0 is zero) . If *0 is positive, it generates a hole flow which may be viewed as one of the competing hole flows. We thus may view *0 as a positive j generating one hole flow and we have another i, i < j , with a * at i, positive again, in the left edge, defining the competing hole flow. We thus have replaced our two competing * 's with the * at i and *0 ' We then invoke Proposition 5: The use of No- (z) is allowed without increase of the number of sign changes beyond 2k. Along the decreasing deformation, *0 is untouched and therefore its sign unchanged, unless it moves inside the characteristic piece. But, then, No (z) locates precisely at *0 and becomes the normal II flow which is compatible with all hole flows. At the boundary of the region where there is a hole but not a well-defined representative for the left edge, No (z) will convex-combine with each of the hole flows (used on disjoint closed regions) and by Proposition 5, this combination will not raise the number of zeros beyond 2k. With the support of a drawing:
))
� No
\Nol1nal (II) flow
and
(Hole flow I
Nonnal (II)
flow
;;
convex
combinations
+
of
No ( ) and the variom flows
Using now a standard deformation lemma, we may assert that a given compact K in Jcoo +e' with Coo = J(Xoo) , can be deformed into Jc oo -c' U V, where V is contained in a neighborhood, as small as we please, of x oo . In this neighborhood, the nodes are defined up to o( 1 ) . If K is in r4k and if its ± v-jumps can be tracked down, as is the case of K's contained in Wu (xu,) , X2 k a periodic orbit of index 2k , then after the use of the deformation lemma, we may apply our construction to the part of the deformed set in V. The result is that all of K is moved below Coo . Thus - and this is the form under which we will use this conclusion.
-
PROPOSITION 1 3 . Let X� _ be a true critical point at infinity of index 2k 1 (x�_ l could be a cycle at infinity) with at least one characteristic piece. Assume that all the oscillations on Wu (X 2k ) near X 2k -l are small (i. e. no Fredholm issue). Then, the intersection number of Wu (X2k ) with Ws (x� _ ) is zero if there is no intermediate false critical point at infinity between X 2k and x�_ l ' l
l
COMPACTNESS
89
5. Companions 5 . 1 . Their definition, births and deaths. When we defined normals, we defined also what a "preferred orientation" was. If the preferred orientation on (Yo, yd or (Ym - l , Ym) coincides with the orientation of the closest edge, XOO is labeled false. False x oo 's can be bypassed downwards, without increase in the number of sign changes but at the expense of introducing a normal, i.e. , a new ± v-jump which is the immediate neighbor of an edge ± v-jump bearing the same orientation. Bypassing such x oo 's modifies our configurations in one regard: individual jumps are replaced by families of companions around a single original jump. If we wish to extend Proposition 13 to this new situation, we find out very soon that the reasoning has to be different since a family can control several nodes through its various companions. Let us observe though that as long as there are two companions or more in a family,the original companion is among them and it survives their death, with the same orientation, i.e., as an added companion goes away, its original companion survives its death, with the same orientation. Elsewhere, this original jump might switch signs, but it then has no companion. This observation is important as can be seen when we try to understand how a true critical point at infinity Y= dominates another true critical point at infinity x= . If x= has several characteristic pieces, then XOO is in fact a cluster of several critical points x';' , x� 1 , . . . , x�R. x�R is the critical point at infinity which has the same graph as XOO but its associated cycle does not use any full (half)-unstable manifold associated to a characteristic �-piece. x�R+ 1 builds a cycle through a combination of various characteristic �-pieces, one at a time, as pieces of a puzzle. x�R+ 2 using two of them at a time, etc. As soon as Yoo dominates x,;" it has to dominate X� l ' · · · , x�R and if we try to cancel Wu(Y=) n W (x';'), it is in s fact Wu(Y=) n Ws (x�R)' a stratified space of top dimension £ which we need to move down past the level of Xoo. Along this stratified space, configurations vary, companions are given birth to, then die, etc,. It is a full story which we are facing. The following Proposition which we will prove later reduces considerably this otherwise quite complicated scenery: PROPOSITION 1 4 . When a companion in a family which is not a family of an edge, and such that the v-rotation between its extreme companions, on the left and on the right, is less than 0, 0 > c, becomes extremely small, it can be brought back close to its closest surviving companion (towards the original companion). 7r
-
We will prove this Proposition later. Its proof involves slight modifications of the flow which we will indicate later. It is of relevant importance in that no companion can appear suddenly or disappear somewhere. All these births and deaths are within the family. 5.2. Families and nodes
a. Critical Configurations of families
The basic observation which we make here is that a family of two companions or more cannot overlap a node which is not Xo or xm, i.e. , an edge node; or if it does, the associated configuration, in fact all the configurations containing such an overlap, can be moved continuously down, past J (XOC).
90
A. BAHRI
Indeed, the family covers then at least two distinct preferred orientations; one of them coincides with the orientation of the ± v-jumps of the family or by cre ating a new companion (obviously with the same orientation) in between existing companions, we can move all these configurations down. Each time a deformation (obviously a J-downwards deformation) is defined, we study the configurations which are "at the boundary" of such a deformation, i.e., the transitions to other configurations where such a deformation cannot be defined. For the deformations defined above, their definition is directly related to the overlap. As the overlap fades away and the family recedes (up to 0 ( 1 )) on one side of the node, this deformation cannot be defined anymore. Thus, we have to find another way of decreasing these configurations once the overlap on a node is one-sided up to a small constant c. We will then scale the growth of the companion which we grew or introduced, putting it to a smaller and smaller growth. It does not suffice anymore to move the configuration down. We have to find other, compatible ways to decrease J over such configurations (by deforming them). The configurations which we consider below are therefore configurations such that any family is within 0 ( 1 ) of at most two consecutive nodes. Indeed, otherwise, there is a definite overlap. On the other hand, we will assume that the hole flow of the left edge is well defined, which amounts to say that the family associated to the left edge is defined without ambiguity, then its hole flow can be used in between families, which implies that continuously defined and decreasing deformations can be defined on configurations such that one node among X l , ' " , Xm - l is not within 0 ( 1 ) reach of a family. The hole flow convex-combines in a natural way with the previous flow related to the overlaps. They coexist without increasing the number of sign changes in the configuration beyond 2k. Lastly, we observe that if a family lies between two consecutive nodes, within c > 0, c small and fixed, of each node, then we can think of the whole family as reduced to its original companion. Such a family is obviously not associated to an edge of the characteristic piece and the use of the normal (II) flow, increasing or decreasing all members of the family at once, whatever suitable, even reversing, all together, their orientation, is available for it. Again this portion of the flow convex-combines with the other portions defined above and there is no family which cannot have at least a node with 0 ( 1 ) reach. We thus define: DEFINITION 5 . A critical configuration on a characteristic ,-piece of XOO is a configuration such that every node X l , ' " , Xm - l is within o(l)-reach of one family, every family with support on the characteristic piece is within o(l)-reach of a node at least and every such family is within o(l)-reach of at most two consecutive nodes.
We then have: PROPOSITION 1 5 . Consider all critical configurations such that one family is within o(l)-reach of two consecutive non-edge nodes. Assume that the hole flow of the left edge is well defined. Then all these configurations can be deformed below J ( X OO ) using the hole flow of the left edge.
COMPACTNESS
91
PROOF. On each side of the area where the family sits (up to 0(1 ) ) , the pre ferred orientations agree. There is one side where the preferred orientation agrees 0 with the sign of the ± v-jump introduced by the hole flow as we use it.
We extend Proposition 15 as follows: PROPOSITION 1 6 . Assume that a critical configuration a does not contain a forced sign repetition, with the sign of the left edge, between the family defining this left edge and Xo + c, c > 0 small. Then, Proposition 1 5 extends to all such a 's which contain a family within 0(1) of Xo and Xl . REMARK.
Xo + c stands for a point close to Xo inside the characteristic piece.
PROOF. This family has to have the orientation of the edge, otherwise we could separate it away from the edge (a family with the wrong orientation cannot leave the characteristic piece through the left edge unless it is followed by a family having the orientation of the edge which will take the place of the family of the edge. Such a family would be after Xl and we would have ample room to separate the edge and the family with the wrong orientation.) The hole flow starting at this family has to agree, under our assumption (no sign repetition near the left edge, with the sign of the left edge), with the hole flow defined by the family of the edge. This hole flow provides a sign + after Xl and this agrees with the preferred orientation of (Xl , X2) q.e.d.
COROLLARY 2. If a critical configuration a contains two or more families within o(I)-reach of two consecutive nodes, then either a can be decreased below J(XOO) in a continuously defined deformation or a contains a forced sign repetition, with the sign of the left edge, between the family defining this left edge and Xo + c, c > 0 small. PROOF. At least one of the families does not control the nodes controls another pair of nodes q.e.d.
x m - l , Xm but
b. Critical configurations with one family within o(I) -reach of (Xm - l , xm).
We thus see that the only critical configurations which we have not been able to decrease below J(XOO) are those such a family is within o(I)-reach of (Xm - l , x m). If an additional family is within o(I)-reach of two other nodes, we have defined such a deformation. However, once we define a decreasing deformation for the configurations with only one family within o(I)-reach of (Xm - l , x m), we need to check that the flows convex-combine and generate a global decrease below J(XOO), without increase of the number of zeros beyond 2k. Let us consider such a configuration
92
A. BAHRI
+
II 111111111111
right edge
We use in the sequel the hole flow of the right edge, between Xm - 3 and Xm - l . It is well defined if there is no repetition in the configuration related to this edge. We make this assumption in a first step. Since we are using the hole flow of the left edge in other parts, we need to convex-combine in some regions and this needs careful checking. This is the reason why we are not starting from a critical configuration and we are simply assuming that a family controls Xm - l and Xm . The normal (II) flow on intermediate families is compatible with both flows. Thus, we assume that every family has to be within o ( l ) -reach of a node and that it does not overlap over a node. We cannot yet assert that every node is controlled by a family. We focus on Xm - 2. PROPOSITION 1 7 . If no family is within o(l)-reach of Xm- 2 , then the two hole flows can be convex-combined near the configuration on the interval preced ing ( Xm - l , xm ) .
PROOF. Since XOO is sign-true, the preferred orientation of ( xm - 2 , x m - d is + and the preferred orientation of ( Xm - l , Xm - 2 ) is -. If the two hole flows disagree, then one requires the use of + between the family controlling Xm - l , Xm and the previous family, while the other one requires the use of -. The introduction of + follows the introduction of - and is compatible with the orientation of the family controlling ( Xm - l , x m ) . The convex-combination can proceed. It makes use of a single interval in the transition but, on each side, the use of the full hole flow is warranted q.e.d.
Thus a family must be within o ( l )-reach of Xm - 2 , on one side or on the other side, with no overlap. Let Uo be a small neighborhood of the set of all configurations containing such a behavior, with no repetitions related to the right as well as to the left edges.
Let Fl be the family within o ( l )-reach of (Xm-l , Xm ) and F2 be the family within o ( l ) -reach of Xm -2
93
COMPACTNESS
+
IIII
III1II11
Xm-2
or
+
IIIWi
11111111
J
+
We claim: We consider in what follows three distinct consecutive families of companions F1 , F2 , F3 . F2 follows F3 and Fl follows F2 . The support of F2 is inside a charac teristic piece. We define the thickness T of F2 to be the �-length between the first and the last companion of H . We assume throughout this part that the maximal size of a companion of F2 is o(T) and that T :s; � . We then claim:
PROPOSITION 1 8 . We can rearrange Fl , F2 , F3 along a J-decreasing deforma tion so that i) The v-rotation between the right edge of F2 and the left edge of Fl is at least 8, where 8 is a fixed positive number. ii) If T > 0, the v-rotation between the left edge of F2 and the right edge of F3 is also at least 8 . It is also at least 8 if the v-rotation between Fl and F2 is more than + 8. The first part of ii) has to be understood as a statement involving T after the deformation. iii) Assume that F2 had initially a thickness T � c, a fixed positive number c :s; � . Assume that F2 can be separated in two distinct families of consecutive ± v-jumps F2- , Fi and that the maximum size of the jumps of Fi , s+ is small 0 of the maximum size of the jump of F2- , L . Then, after decreasing deformation, all of Fi is at a � -distance equal to 19O at most of F2- . Furthermore, s+ changes into s�, with �s+ :s; s� :s; cs+ and L into L ( 1 + 0(1) ) . 7r
-
7r
7r
-
-
7r
PROOF. PROOF OF I ) . see Section 8, Proposition 20 of [7] . Given two consecutive ± v jumps, the �-piece between them has HJ-index zero (is minimal) if the v-rotation along this �-piece is at most 7r 8. If it is less than 7r 8, then we can "widen" the -
-
94
A. BARRI
intermediate �-piece between them by inserting v-verticals to the left of left jump or to the right of the right jump. We take the side which corresponds to the smallest jump. In this way, no v-jump disappears. We need to take this cautionary step only if the jumps have opposite orientations. This widening process never stops as long as the v-rotation is less than Jr. There are indeed no "small" rectangles made of two small ± v-jumps and two large �-pieces as long as the v-rotation along these �-pieces is less than Jr - 8. See Section 8 for further details.
PROOF OF II) . Once the distance between Fl and F2 is Jr - 8 or more, we turn to F2 and F3 and complete the same process. If the thickness of F2 does not fade away in this process, then it goes to the end and the distance between F2 and F3 is at least Jr - 8. It can also happen that the thickness of F2 is zero or becomes very close to zero while the distance between Fl and F2 is more than Jr .
At the end of the process, if either T > 0 or the v-rotation between Fl and F2 is more than Jr , the v-rotation between F2 and F3 is at least Jr - 8. PROOF OF III) . The claim can be derived from the case when F2- is reduced to a single ± v-jump of size c - and Fi is reduced to a single ± v-jump of size c+ , with c+ = o ( c - ) assume that both ± v-jumps have the positive orientation for example.
+
x
� T
Our deformation is the composition of two deformations. The first one involves a recession of the right v-jump towards the left. This one might not be (in fact is not) J-decreasing. We thus have to compose this deformation with a second one which decreases J. To define the first deformation, we consider the �-orbit through x+ and we consider a distance T+ along this �-orbit before x + ; we find a point x - :
COMPACTNESS
95
x-
We introduce the v-vertical at X - . There is a unique piece of �-orbit between the v-vertical corresponding to the v-jump of size c- and the v-vertical at x - :
When T+ = T , the distance between the v-verticals through x- and x - is O ( c+ ) and so is the length of the �-piece between them. We need, however, to check that the length c- has not been consumed along this process. The condition is easy to see: Let J be the size of the v-jump at X - . This v-jump transported T - T+ backwards along � and the v-jump transported backwards T along � should have the same [�, v]-component. This gives us the equation:
Then,
96
A. BAHRI
and the size of the v-jump removed from o ( c+
We thus want to have: i.e.,
is
0(8 + c+), i.e. ,
+ Tc+ TT+ ) .
T T - T+
For example, this works if
x-
T - T+ T
_
= =
c 0( _ ) c+
19± . Vc
As pointed out above, this deformation is not J-decreasing, it is in fact J-increasing. To compute the increase, we compare J at two nearby curves along the deformation:
The change in J is c - If:>. Q . � is obtained from c� by v-transport during a time
{ �+�� == �
c.
Thus, since
- ,\
c
-
and
0 (82 ) . The size 8 of the v-jump has been computed above. Thus, 0(C�T2 ) dJ dT+ (T - T+ ) 2
If:>.
Q
=
I...l. A J
_ -
O ( c2+ T )
It is
� ;-::-r:r; et=" 0 ( c3/2 + V c T) . V
Tc�J+ (1 + 0(1 ) ) .
_ -
In order to define a decreasing deformation, we need to find another deformation of the same curves which would be decreasing and which would convex-combine with the previous one.
COMPACTNESS
97
We recall for this that F2 has support in a characteristic piece and that its thickness T is less than � and larger than c > O. Thus, one of these v-jumps (we are assuming for the sake of simplicity that F2 is made of two jumps. There is no loss of generality in this assumption) is c/2 to the least away from a node. A decreasing normal can be defined at such a jump and it will warrant a decrease of J at a rate � Ac, A > 0 fixed. Consuming a size c+ /100 of the related v-jump (this is possible since c+ = o( c )) which we set aside and do not touch during the first deformation if the normal has to be taken at x+, we warrant a constant decrease of J. This works if the normal at x+ or £- decreases the size of the v-jump. It also works in the other case at £ - . If the normal at x+ increases the size of the v-jump, then we engineer the decrease of J by increasing the size of the v-jump as it travels along the first deformation. We then stop-to avoid entering into technical difficulties-when the right v-jump is at 60 of the left v-jump since 1 then the distance to a node might have shrinked to 260 ' We could of course have used other smaller fractions of c. The statement about s and L follows from the estimate on 6.J and on 8 with + T - T+ � 160 ' This proof contains the proof of Proposition 14 as well. 0
PROPOSITION 19. If Xm-2 is not xo, all such configurations in Uo can be de formed below J(x oo ) and the deformation can be convex-combined, with global de crease and no increase in the number of sign changes beyond 2k, with the hole flow of the left edge. PROOF. On all the configurations which we are considering, there are three consecutive families. F1 and F2 are as above. F3 is the first family before F2. We -3 pick up a small constant 8 = 2cOO and we reorder these families, along a J-decreasing deformation as described in Proposition 20, Section 8 below.
are now away by 7r - 2cOO to the least. If they are away by more than 7r + c or if F2 has some thickness, then F2 and F3 are also away by 7r - 2c�0 to the least. Finally, the two-edge jumps of F2 are of comparable size. Otherwise, the thickness of F2 is less than 160 ' We assume in a first step that Xm - 3 is not Xo. We distinguish five distinct situations according to the behaviour of the right and left edge of F2, denoted L.E.F2 and R.E.F2 with respect to Xm - 2. 1st Case: The distance (all distances are along �) o f R.E.F2 t o Xm-2 is more than C. Since the v-rotation between R.E.F2 and L.E.F1 is at least 7r - � , there is a hole at Xm - 1 . We use the hole flow of the left edge in this hole to decrease J. Observe that we need to prove Proposition 19 if the hole flows of the right edge and hole flow of the left edge do not coincide. This implies at once that the hole flow of the left edge should yield a - between F1 and F2. If it yields a +, Proposition 19 is a straightforward statement since this + coincides with the orientation of F1 , and is provided by the hole flow which we are using outside of Uo. 2nd Case: The distance of R.E.F2 to Xm - 2 is less than c but more than � and the distance of L.E.F2 to Xm - 2 is less than 1c�0 ' F2 has then a thickness larger than 160 ' This implies that its right edge jump and its left edge jump (we may assume that there are only two of them) are of comparable size. If we come back to our computation, we find:
F1
and
F2
-3
c+ c-
-> - Cc
98
A. BARR!
where C is a universal constant. This uses Hypothesis (B) . The rate of decrease of a normal at the location of the right edge jump is lowerbound by C1 c since the distance of RE.F2 to Xm - 2 is more than � . The rate of variation of a normal at -3 the location of the left edge jump is upperbounded by C 1cOO since the distance of 3 L.E.F2 to Xm-2 is at most 1�- 0 ' We can therefore use the two normals together to decrease J and since the size of the jump is comparable, we can get all the ± v-jumps of F2 to switch signs together along this unhindered J-decreasing deformation. This acts as an normal (II) flow. This flow can be convex-combined with the flow defined in the first case without increase of the number of zeros beyond 2k since this would yield a combination of normal (II) flow (generalized) with a hole flow (used partially near xm- d (see [7] for more details) . -3 3rd Case: The R.E.F2 is within � of Xm - 2 while L.E.F2 is within 1cOO of Xm - 2 ' We then introduce a normal between R.E.F2 and Xm - 1 which yields, since this is the preferred orientation of this area, a positive v-jump. (If + was the preferred orientation of (Xm - 1 , xm ) , we would grow Fl ' Proposition 19 would be a straight forward statement. ) This can be considered to be a local use of the other hole flow. While it provides a decrease of J, it is not compatible with the hole flow of the left edge. However, this flow which is used locally and does not require the hole flow of the right edge to be defined, is compatible with the normal (II) flow of the second step. It is incompatible with the local use of the hole flow of the left edge as in the 1st step and the uses of these two flows are shielded one from the other by the flow of the second step. This flow, because it introduces a + after F2 and before F1 and because F1 is positively oriented, is also compatible with the use of the hole flow before F2. It is also compatible with any normal (II) flow (under the form of overlap as well) . 4th Case: L.E.F2 is more than 1"';0 to the left of Xm-2. R.E.F2 is less than 1"';0 to the left of Xm-2 and less than c to its right. It lies in between. We use i) of Proposition 18 (5 2"';0 ) ' Either F2 overlap Xm - 3 or there is a hole at Xm- 3 ' We can use the overlap (a normal (II)-type) flow or the hole flow of the left edge. They are compatible, thus can be convex-combined for transition, with all the flows we defined above. We made a special point about this in the third case. -3 5th Case: R.E.F2 is more then too to the left of Xm - 2. We use the hole flow of the left edge near Xm-2. It introduces a negative v-jump before Xm - 2 and is compatible with all previous flows. The above arguments assume in fact that F2 and F1 have opposite orientation so that as we use the widening process of Proposition 19 between them, pushing away F1 , the left edge of F2 does not grow. If the orientation of F1 and the orientation of F2 agree, the argument requires an additional construction, see [6] for more details. =
We summarized these steps in a chart with L.E.F2 as ordinate y, R.E.F2 as abcissa x. We have y ::; x. We have introduced the various regions with the essential flows used in each of them. At the frontiers, convex-combinations are used. We have discussed above their compatibility.
99
COMPACTNESS
H.F.L:
hole flow left edge
Proposition 19 follows under the assumption that the hole flow of the left edge is well defined. If there is a repetition barring the use of the left edge hole flow, we need to introduce a decreasing normal and we are allowed to introduce it with an additional sign change if the repetition is not already exhausted by the use of one of the non compatible hole flows We have gone over this argument before: The repetition occurs "in the left edge" in that, in our construction, we may require that the family of the left edge should be at a distance 0(1) from the average left edge (which can be defined easily) . It follows that no normal is used in between the repetition. In the interval where it is used, this normal identifies then with the orientation of one of the hole flows. With the other one, it might induce an additional sign change but this would correspond exactly to the single additional sign change allowed. q.e.d. 6. Flow-lines for
X2k + 1
to xu,
We sketch in the following how we can rule out most of the flow-lines from a periodic orbit of index 2k + 1 to an xu, when the number of ±v-jumps of x u, is large. The cases which are left open are similar to the ones related to X2k - XU,_l ; they involve characteristic pieces of strict index zero or of strict index 1 and two families (families of each edge, with non zero companions at least for one of the families) living on them. Let us first consider the case of a critical point at infinity x u, having at least two characteristic pieces of strict index different from zero or 1 . We know that Wu(X2k+ l) can be achieved in r4k+ 2 and that the 2k + 1 (families of) ±v-jumps can be tracked down along Wu(X2k + 1 )' Completing the same analysis as in the
100
A. BAHRI
case of X2 k - x� _ l ' we conclude that, on each of these characteristic pieces, either there is a hole or there is a repetition, at least in the case where the families are reduced to single ±v-jumps. If there are two holes on one characteristic piece or a hole over two nodes or one hole on two different characteristic pieces, we can build a decreasing deformation which does not raise the number of zeros beyond 2k. We are left with configurations with no holes or at most one hole. In this latter case, we can use the hole flow on this characteristic piece unless a repetition develops in the left edge. If two characteristic pieces at least are "symmetric" (this means that there are as many *'s mod 2 between *0 and * m as between the left and right edges of the characteristic piece) , we have the freedom of one additional repetition and we can build a decreasing deformation. Assuming all characteristic pieces bear no hole and are, but one maybe, all not symmetric, we count the *'s over the configuration. We find using the results of [3] that if n is the number of characteristic pieces, then the number £ of full characteristic pieces used jointly in the definition of x� (Le. £ is the number of full half unstable manifolds used in the definition of x�) is equal to n - 1 . We also find by a further argument that the number of non-degenerate �-pieces with "Ij =I- 0, see [2] , [3] is at least n31 . If n is large, we use then Hypothesis (A), conclude that there is a large amount of v-rotation and using Proposition 1 1 of [3] , we can modify a in the vicinity of x� and get rid of it in our homology. Thus n is bounded, £ is bounded and as k tends to 00 there is a characteristic �-piece with a large HJ-index. This concludes this outline of the proof. References 1. Bahri, A. , Pseudo-Orbits of Contact Forms, Pitman Research Notes in Mathematics Series No. 173, Longman Scientific and Technical, Longman, London, 1988. 2. Bahri, A., Classical and Quantic periodic motions of multiply polarized spin-manifolds., Pit man Research Notes in Mathematics Series No. 378, Longman and Addison - Wesley, London and Reading, MA, 1998. 3. Bahri, A., Flow-lines and Algebraic invariants in Contact Form Geometry PNLDE 53 (2003), Birkhauser, Boston. 4. Bahri, A., Recent Progress in Conformal Geometry, Adv. Nonlinear Stud. 3 (2003), 65-150. 5. Bahri, A., Xu, Y, Recent Progress in Conformal Geometry, Imperial College Press (to appear) . 6. Bahri, A. (to appear) . 7. Bahri, A., Compactness, preprint Rutgers 2006. 8. Weinstein, A., On the hypotheses of Rabinowitz's periodic orbits theorems, J. Diff. Equ. 133 ( 1979), 353-358. 9. Bennequin, D., Asterisque (1983) , 106-107. 10. Hofer, H., Pseudo-holomorphic curves in symplectization with applications to the three di mensional Weinstein conjecture, Inventiones Math 114 (1993) , 515-565. 1 1 . Eliashberg, Y., Givental, A., Hofer, H . , Introduction to Symplectic Field Theory, Geom. Funet. Anal. 2000, 560-673. 12. Eliashberg, Y., Classification of overlwisted contact structures on three manifolds, Invent. Math. (1989) , 623-637. RUTGERS, THE STATE UNIVERSITY OF N EW JERSEY DEPARTMENT FRELINGHUYSEN ROAD P ISCATAWAY, N J 08854-8019 E-mail address: abahriCDmath . rutgers . edu
OF
MATHEMATICS 1 1 0
Contemporary Mathematics Volume 446, 2007
Generalized travelling waves for reaction-diffusion equations Henri Berestycki and Franc;ois Hamel
A Haim Brezis, en temoignage d 'admiration et d 'amitie ABSTRACT. In this paper, we introduce a generalization of travelling waves for evolution equations. We are especially interested in reaction-diffusion equa tions and systems in heterogeneous media (with general operators and general geometry) . Our goal is threefold. First we give several definitions, for fronts, pulses, speed of propagation, etc. Next, we discuss the meaning of these defi nitions in various contexts. Then, we report on several results of [4J (of which this is a companion paper) about these notions. We further establish here several new properties. For this definition to be meaningful we need to show two things. First, that the definition covers and unifies all classical cases (and does not introduce spurious objects) . Second, that it allows one to understand propagation fronts in completely new situations. In particular we report here on a result about travelling fronts passing an obstacle.
1 . Classical notions of travelling fronts 1.1. Planar fronts. Travelling fronts form a specially important class of time global solutions of reaction-diffusion equations. They arise and play an important role in various fields such as biology, population dynamics, ecology, physics, com bustion... In many situations, they describe the transition between two different states. Let us start with recalling the notion of classical travelling fronts in the homo geneous case, for the equation (1.1) U t = �u + f(u) i n ]RN , For basic properties of the linear heat equations, which allow one to derive existence and uniqueness of the Cauchy problem associated with ( 1 . 1 ) , we refer to the classical text of H. Brezis [14J . In the case of ( 1 . 1 ) , a planar travelling front connecting the uniform steady states 0 and 1 ( assuming f(O) = f(l) = 0) is a solution which propagates in a given unit direction e with a speed c, and which can then be written as u(t, x) = ¢(x·e�ct) with ¢( - (0 ) = 1 and ¢( +(0) = O. Two properties characterize such fronts: their 1991 Mathematics Subject Classification. Primary 35K55; Secondary 35B10, 35B40, 35K57. Key words and phrases. Front propagation, generalized waves, qualitative properties.
101
HENRI BERESTYCKI AND FRANQOIS HAMEL
102
level sets are parallel hyperplanes which are orthogonal to the direction e , and the solution is invariant in the moving frame with speed c in the direction e. The profile ¢ of a planar front ¢(x . e - ct) satisfies the ordinary differential equation ¢" + c¢' + I(¢) = 0 in R Existence and possible uniqueness of such fronts, formulre for the speed ( s ) of propagation are well-known [1, 2, 16, 23] and depend upon the profile of the function 1 on [0, 1] .
1.2. Curved travelling fronts. Before introducing our general definition, let us recall the known extensions in non homogeneous cases. The first such extension is still one with classical travelling fronts but which are not planar anymore. Assume that the domain is a straight infinite cylinder of the type n = ]R. x w, where W is a bounded smooth domain of ]R.N - l . Denote x = (Xl , Y) , with y E w, the variables in n and consider the reaction-diffusion-advection equation (1.2)
Ut
AU = I(y, u) - �u + a(y) � UXl
with, say, Neumann boundary conditions on on. The functions a and 1 are given and may depend on the cross variables y. Assume that l(y, O) = I(y, 1) = 0 for all y E w. In this context, a travelling front connecting 0 and 1 and propagating with speed c in the direction e l = (1, 0, . . . , 0) is a solution of the type u(t, Xl , y) = ¢(Xl - ct, y) such that ¢( -00, y) = 1 and ¢( +00 , y) = 0 uniformly in y E w. These fronts are still invariant ( in the moving frame with speed c in the direction e l ) and have a constant speed, but the profile ¢ is in general not planar anymore. It is a function of both variables s = Xl - ct E ]R. and y E w, and it satisfies the elliptic partial differential equation
��
-�¢ + (a(y) - c) with Neumann boundary conditions on on.
=
I(y, ¢) in n
Most of the known results which had been obtained on planar fronts for the homogeneous equation (1.1) have been ex tended, with PDE methods, to the case (1.2), see [2, 9, 10, 11, 27] . The case when w is periodic in the variables y can also be treated similarly, see [3] .
1.3.
Curved fronts for (1.1) in ]R. N . Non-planar fronts which arise in het erogeneous problems of the type (1.2) were recently shown to also exist even in the homogeneous case. Consider for instance the homogeneous equation (1.1) in ]R. N and call r = ( xi + . . . + X � _ l ) l / 2 . Assume that 1(0) = 1(1) = O. For the main three classical classes of reaction terms 1 ( combustion, bistable, monostable) and for any given angle a E (0, 1r /2) equation (1.1) admits "conical-shaped" non-planar fronts of the type
u(t, x ) = ¢(r, X N - ct), such that ¢(r, s ) ---t 1 ( resp. 0) uniformly as s - 'lj;(r) ---t -00 ( resp. +(0), where 'lj; satisfies: 'lj;(r)/r cot a as r +00 (see [13, 15, 17, 19, 20, 25]). The ---t
---t
profiles are still invariant in a moving frame with constant speed, but the level sets are not hyperplanes anymore. Conical-shaped fronts are also known to exist for systems of reaction-diffusion equations and for aperture angles a close to 1r /2 under some stability assumptions (see [21]). In the case when 1 is concave and positive on (0, 1), then, many more non-planar travelling fronts also exist, which are not conical-shaped, see [20].
GENERALIZED TRAVELLING WAVES FOR REACTION-DIFFUSION EQUATIONS
1 03
1.4. Pulsating travelling fronts, periodic media. Another important ex ample of travelling fronts is for heterogeneous equations of the type ( 1 .3)
Ut = V' . (A(x)V'u) + q(x) . V'u + f(x, u) in ]RN ,
where the uniformly elliptic matrix field A, the vector field q and the function are smooth and periodic in ]RN . That is, there are L 1 , . . . , L N > 0 such that
f
A(x + k) = A(x), q(x + k) = q(x), f(x + k, ') = f(x, ') for all x E ]RN and k = (k1 , . . . , kN ) E L 1 Z X . . . x L N Z. Unlike all aforementioned ( 1 .4)
cases, these equations in general are not invariant by translation in any direction. Assume that, say, f(x, l) = f(x, O) = 0 for all x E ]RN . Given a unit vector e E §N - 1 , a pulsating travelling front connecting 0 and 1 , and propagating with speed c -I- 0 in the direction e is a solution u(t, x) of (2.1) such that ( 1.5)
(
)
u t + k � e , x = u(t, x - k)
(t, x) E ]R X ]RN and k E L 1 Z X x L N Z, and u(t, x) 1 (resp. 0) as x . e � -00 (resp. X · e +(0) uniformly in t and in the variables which are orthogonal to e (see [3, 30, 31] ) . These fronts can be written as u(t, x) = ¢(x · e - ct, x) where the function (s, x) ¢(s, x) is periodic in x in the sense of (1.4), and ¢(-oo, x) = 1 , ¢( + oo, x) = 0 uniformly in x. The function ¢ satisfies a degenerate elliptic equation in the variables s and x. In the moving frame with speed c in the direction e, the profile of the front is not invariant anymore, but it is in general quasi-periodic in time. Observe that at each time t, each level set of u is trapped between two parallel hyperplanes which are orthogonal to e, but in general it is not for all
. • •
�
�
f----+
planar. Existence results and formulre for the speeds of propagation are given in [3, 6, 7, 31J . The case where the domain n satisfies ( 1 .6)
(namely n has the same periodicity (L 1 , . . . , L N ) in the variables (X l " ' " X N ) as the coefficients) has also been investigated, see [3J . For reaction-diffusion equations with time-dependent coefficients, pulsating fronts (which are defined in a similar way) are also known to exist (see [18, 26] ) . Moreover, the limiting states p± (t, x) may also depend on x or on t for space or time-periodic equations (see [8, 22, 28J for some examples) . 1.5. Almost periodic case. Our last case deals with the almost-periodic framework. Consider the case where all coefficients of ( 1 .3) are almost periodic. To make notations simpler, assume that (2. 1) reduces to (1. 7)
Ut = Uxx + b(x)f(u) in n = R
Assume that the closure H, with respect to the uniform norm on ]R, of the set of all translations uyb (with uyb(x) = b(x + y)) of the coefficient b is compact. Assume moreover that f(l) = f(O) = O. In this case, a new definition was introduced by H. Matano. Namely, a travelling wave (as defined in [24]) is a solution u for which
IIENRI
1 04
BERESTYCKJ AND FRAN 0) such that inf {f.1 (t, x) . v(x); (t, x) E JR x 80} > O. 1. Assume that u is a wave connecting p- and p+, that there is > 0 such that sup {do (x, f t- r); t E JR, x E ft} < +00, (2.3) T
106
HENRI BERES'IYCKI AND FRANQOIS HAMEL
and that (2.4) sup { dn(y, ft) ; y E nf SeX, r) } +00 unif. in t JR, x E ft . Then, for all A (p- , p+), (2.5) sup {dn (x, rt ); u(t, x) = ).} < + 00, and, for all C > 0, (2. 6) p- < inf {u(t, x) ; dn (x , ft) < C} < sup {u (t, x); dn (x , ft) < C} < p+ . 2. Conversely, if (2.5) and (2.6) hold for some choices of sets (nt, fdtEIR satisfying (2.2) and if there is do > 0 such that the sets ± {(t, X) E lR x n, x E nt , dn(x, ft » d} are connected for all d > do, then u is a wave connecting p- and p+, or p+ and p-. Roughly speaking the assumption (2.3) means that f and f are ot too far from each other. For instance, if all f are parallel hyp rplanes in n = JRN , then the assumption means that the dist ance between ft and ft-r is bounded independently e sense of t , for some T > O. The property (2.4) means that the sets nt are in s wide enough, ni formly with respect to t. Proposition 2.3 means that, under nome assumptions, the boundedness of the distance between the setn ft and the level sets of a wave is thus an intrinsic notion . n
E
l
r_+oo
E
t
e
t
t- r
om
u
It tUTllD
out that
n
the mean speed,
if any,
is also
intrinsic.
2.4. Let p± be two limiting states solving (2.1) and satisfying inf { Ip- Ct, x) - p+ (t,x) l; (t, x) E JR x 'O} > O. Let u be a wave connecting p- and p+ with a choice of sets nt satisfying (2.2) and (2.4) . If u has the mean speed c, then, for any other choice of sets nt satisfying (2.2) and (2.4), u has a mean speed and this mean speed is equal to c. PROPOSITION
2.3. Further specifications. More specific notions of fronts, pulses, inva sions or travelling waves) , almost planar waves can now be defined. These notions
(
p±
the limiting Htates or of the sets nt , and are listed in the following definitions. Here u denotes a wave connecting p and in the sense of Definition . 1 are related to some properties of
p+
2 .
2.5. (Fronts and Pllh;es) Let p± (pt, . . . , p�). We say that the wave u is a front if either pi(t, x) < pt(t, x) for all (t, x) E JR x n and 1 < i < or pi(t, x) > pt(t, x) for all (t, x) E JR x n and 1 < i < The wave u is a pulse if p-(t, x) = p+(t, x) fo all (t,x) E lR x n. 2.6. (Invasions, or t ravellin g waves) We say that p+ invades p ( resp. p- invades p+) if nt n;- (resp. nt n:;-) for all t > s and do(rt,r ) +00 as It - 8 1 +00.- Therefore, u(t, x) -p± ( t, x) 0 t ±oo (resp. t 'foo) locally uniformly in n with respect to the dist ance dn . 2.7. (Almost planar waves in the direction e) We say that the wave u is almost planar in the direction E §N if, for all t E JR, nf can be chosen so that rt = {x E n, x · e = �d DEFINITION
=
ffi ,
ffi .
r
DEFINITION
-+
:::l
:::l
DEFINITION
e
for
some
�t E lR.
-1
-+
as
.•
->
-+
-+
GENERALIZED TRAVELLING WAVES FOR REACTION-DIFFUSION EQUATIONS
107
2.4. The classical examples. Let us now come back to the usual notions which were listed in Section l. We shall see that they are all covered by the general definitions of waves and that they may correspond to some of the specific cases mentioned above. For instance, for the homogeneous equation (1.1) in JRN, if f(O) = f(l) = 0, the solutions u(t, x) = ¢(x · e - ct), with ¢(-oo) = 1 and ¢(+oo) = 0 are (almost) planar fronts connecting 1 and 0, with (mean) speed l ei - The uniform stationary state p - = 1 (resp. p+ = 0) invades the uniform stationary state p+ = 0 (resp. p- = 1) if c > 0 (resp. c < 0). The sets nt can for instance be defined as
nt = {x E JRN , ±(x · e - ct) > O} For equation (l.2) in an infinite cylinder n = JR x the solutions u(t, X l , y) = ¢(XI - ct, y) such that ¢( -00, y) = 1 and ¢( + 00, y) = 0 uniformly in y E w are almost planar fronts connecting 1 and 0, and the sets nt can be chosen as nt = { (X l , y) E JR x w, ±(X I - ct) > O}. The curved fronts u( t, x) = ¢( r, X N - ct) exhibited in Section l.3 for equation ( l . 1) can also be covered by Definition 2.1 with p - = 1 , p+ = 0 and, say, nt {x E JRN , ±(XN - ct - 1jJ(r)) > O}. They are not almost planar as soon as 1jJ(r)/r -f+ 0 as r XI + . . . + xJv - 1 � +00. The pulsating fronts which were mentioned in Section l.4 also fall within the general definition of travelling fronts with (p- , p+ ) = (1, 0) and, say, nt given by (2.7) if n = JRN. But, in a general periodic domain satisfying (l.6), the mean speed (as defined in Definition 2.2) of a pulsating front solving (l.5) is equal to 1' lcl, where l' 1'(e) ::::: 1 is such that - dn (x ' y) � 1'(e) as I x - y � (2.8) l +00, (x, y) E n x n and x - y is parallel to e. Ix - yI The constant 1'(e) is by definition larger than or equal to l . It measures the as ymptotic ratio of the geodesic and Euclidean distances along the direction e. If the domain n is invariant in the direction e, that is n = n + se for all s E JR, then 1'(e) = l. (2.7)
w,
=
=
J
=
Lastly, the almost-periodic case described in Section l . 5 is also a particular case of the general definitions. For instance, in the one-dimensional case (l. 7) with f(O) = f(l) = 0, the solutions u(t, x) satisfying ( l . 8 ) are generalized waves with (p- , p+ ) = (1, 0), n t = ( - oo, �(t)) and nt = (�(t), +oo).
To sum up, we have just seen that the general definitions given in this section generalize all the usual notions. Furthermore, what is also very important is that the new notions are both strong and wide. Indeed, first, we show in the following two sections that there is no abusive generalization since, under some assumptions, the generalized waves can be reduced to the usual notions in some particular cases. Second, we will see that the generalized waves can take into account other cases which cannot be covered by the classical definitions.
3.
Applications of the definitions to the classical cases
In this section, we see how the general definitions can reduce to the usual no tions in some particular cases. As an example of such results, we start in Section 3.1
1 08
HENRI RBREf)TYCKI AND FRANQOIS HAMEL
with the proof of a one-dimensional symmetry property for almost planar bistable type fronts in JR.N. A more general result is given in [4] . But we include the proof herf� because it is simple and explains clearly why this result is true. In Sections 3.2 to we report on some results of [4] for generalized bistable-type fronts. Finally, we prove in Section 3.5 a new classification result for generalized monostable-type fronts which are trapped between two planar fronts.
3.4
3.1. Almost planar bistable waves. We consider classical time-global boun ded real-valued solutions of (3.1)
We assume here that the function f : JR.
-->
JR. is locally Lipschitz-continuous and
1(0) = /(1) = 0, :J 0" > 0, 1 is non-increasing in (-00, 8] and in [1 - 8, +00) .
(3.2)
An example of such a function is the cubic nonlinearity 1(8) ;= 8 - 83 which arises in scalar Ginzburg-Landau equations (see [12] ) .
3.1. 1,
THEOREM Let u be a bounded almost planar wave solving (3. 1 ) , connecting p- = 0 to p+ = and assume that there exi8t e E §N- l , c > 0, M > 0 and a map JR. ':'l t �t 8uch that
V t E JR., 0; = {x E ]RN , ± (x · e - �t ) < OJ , V (t, s) E ]R2 , M < I �t � l c l t s l < M
I-->
Then
thcrc
-
-
s
-
-
exist E E { - 1 , I} and a decreasing function t/> : 1::/
(t, x) E JR.
X
JR.
-->
JR.N , u(t, x) = r/J(x , e - CEt) .
.
(0, 1 ) such that
R.oughly speaking, this result means that any almost planar wave is actually planar and invariant in the moving frame which propagates with speed c in the direction Ee (actually, if c 0, then u is stationary) . =
PROOF. Up to rotation of the frame, one can assume that e = (1, 0, . . . , 0). Denote x' = (X2, . . , XN ) and x = (X l , X') . The assumption made on et provides the existence of E E {-I, I } such that the map .
t
is bounded. Call
I-->
(t := �t - ClOt
v (t, x) = u(t, x + cete) = u(t, Xl + cet, x')
for all (t , x) E JR. X ]RN . Our goal is to prove that v depends on Xl only and that it is decreasing in X l . The function v is a generalized wave connecting p- 0 and p+ = 1 , for the equation =
(3.3)
and
nt = {x E JR.N ,
(3.4) v(t, x)
-->
1
±(Xl
Vt = llv + CEe · V'v + 1(v) ,
-
(t )
(resp. 0) as Xl
Thus, there exists A
>
0 such that
I-->
(t is bounded, it follows that
-00 (resp. + 00) unif. in (t, X')
E
JR.
v(t, x) > 1 - 0" for all Xl < - A and (t, x') E lR X lRN - t , (3.5) for all Xl � A and (t, x') E JR. x ]RN -I. v(t , x) ::; 8 Notice that one can assume without loss of generality that 8 E (0, 1/2] .
X
]RN - I .
GENERALIZED TRAVELLING WAVES FOR REACTION-DIFFUSION EQUATIONS
1 09
Choose now any T E JR and p E JRN - 1 . For all S E JR and (t, x ) E JR X JRN , denote WS (t, x ) = v(t + T, X l + s, x' + p) , and call E = { ( t , x ) E JR x JR N , X l < -A}. Fix any (J � 2A. Since v and w are globally bounded (because u is) , one has v + E � w 0, V + E � w -Bz in E,
for some constant B (remember that f is Locally Lipschitz-continuous and that Voo is bounded. The strong parabolic maximum principle implies that z(t, x ) = ° for all t ::; 0, X l ::; -A and x' E JR N- 1 . But z � E* > ° on BE, which leads to a contradiction. Thus, 10* = 0, whence v � w 0. In particular, ° such that
(-oo, p- (x) + 0]
and
[p+(x) - 0, +00)
3.2. [4] If u is an invasion of p- by p+ with
K := inf {p+ (x) - p- (x) ; X E O } > 0, I N and if there exist e E § - , C > ° and a map JR. :3 t I-> �t such that (3.9)
(3. 10 )
where
rt = {x
(3. 1 1 )
E
0,
X · e
- �t = O} and ot = {x E 0, ± (x , e - �t) < O},
then u is a pulsating front. That is u t+
I
k·e c
where I = ,,(e)
,x >
= u(t, x - k) for all (t, x) E lB'. x O and k E LI Z x . . . x LN Z, 1 is given in (2.8) . Furthermore, u is unique up to shifts in t .
GENERALIZED TRAVELLING WAVES FOR REACTION-DIFFUSION EQUATIONS
111
Remember that ,(e) measures the asymptotic ratio of the geodesic distance and the Euclidean distance in the direction e, and ,(e) is then automatically larger than or equal to 1. The speed c/,(e) is the "Euclidean" speed in the direction e, as if there were no obstacles, whereas c is the intrinsic geodesic speed which takes into account the geometry of the domain. Notice that ,(e) = 1 if 0= ]RN , or if 0 is invariant in the direction e. Theorem 3.2 says that, under the above assumptions, our general definitions do not introduce new objects in the periodic framework : almost planar travelling fronts reduce to pulsating travelling fronts in the sense of Section 1.4. 3.3. Invariance in a moving frame. In Section 1.2, we mentioned several explicit examples of usual travelling fronts which are invariant in their direction of propagation. We gave in the previous subsections some conditions under which almost planar fronts are truly planar or pulsating in homogeneous or periodic frame works. We here give a general characterization of fronts which are invariant in their moving frame, without assuming any periodicity in the medium. We assume here that 0is invariant in a direction e E SN - l , that u is a gen eralized wave connecting p - and p+ for equation (2.1), that u and p± are globally bounded, that A, q, It and p± depend only on the variables x' which are orthogonal to e, that f = f(x', u) and that (3.8) and (3.9) hold. �t PROPOSITION 3.3. [4] If there exist e E SN - 1 , C 2: 0 and a map ]R '3 t satisfying (3.10) and (3.11), then there exists c E { - 1, 1} such that u(t, x) = ¢(x e - cet, x' ) for some function ¢. Moreover, ¢ is decreasing in its first variable. If one further assumes that c = 0, then the conclusion holds good even if f and p± also depend on x . e, provided that they are nonincreasing in x . e. In particular, if u is quasi stationary in the sense of Definition 2.2, then u is stationary. As a consequence of Theorem 3.2 and Proposition 3.3, it follows that, if 0= ]RN and if A, q, f, p± are independent of t and x, then u is a truly planar travelling f---t
·
front, that is :
u(t, x) = ¢(x · e - ct), (p - , p+) i s decreasing and ¢(=Foo) = p± .
where ¢ : ]R ----t an immediate generalization of Theorem 3. 1 .
This result corresponds
3.4. Invariance in the cross-directions. In the previous subsections, we gave some conditions under which the fronts reduce to planar, pulsating or usual travelling fronts. The fronts were assumed to have a mean speed. The following result is concerned with the case of almost planar fronts which may not have any mean speed and which may not be invasion fronts. It gives some conditions under which almost planar fronts actually reduce to one-dimensional fronts. We assume here that 0= ]RN , that u is a generalized wave connecting p - and p+ for equation (2.1), that u and p± are globally bounded, that A and q depend only on t, that the limiting states p± depend only on t and x . e and are nonincreasing in X · e, that f = f(t, X · e, u) is nonincreasing in X · e, and that inf (p+ - p - ) > o. Assume also that there is J > 0 such that, for all (t, x) E ]R X ]RN , S f---t
f(t, x . e, s) is nonincreasing in (-oo, p- (t, x . e) + J]
and
[p+ (t, x . e) - J, +(0).
HENRI BERESTYCKI AND FRANQOIS HAMEL
3.4. [4J II u is almost planar in the direction e E §N- I with some sets rt and nt satisfy'ing (3. 11) and such that THEOH.EM
'v'
(J
sup { I�I CT - �t l ; t E R} < + x , +
E JR,
then u only depends on t and x for some junction ¢; ( 3 . 12)
\f (t, x)
:
e,
that is :
u ( t , x) = ¢>(t, x · e ) .
JR 2 --+ R Fur·thermore, E JR
X
and u is decreasing in x . e .
JRN ,
p- (t , x · e ) < u(t , x) < p+ (t , x · e)
Notice that the assumption that Hllp {1�t+CT - �t I; t E R} < + 00 for every (J E JR is clearly [-Munger than the property (2.3). But the map t ...... �t is not needed to be monotone and u may not be an invasion front. Actually, if the inequalities (:1.12) are assumed to hold a priori and if is assumed tu be nonincreasing in s for s in (p- (t, x e), p ( t , x e ) + 8J and [p+ (t , x . e ) - 6, p+ (t, x e)J only, instead of ( -00, p - ( t , x . e ) + 8J and [p+ (t , x . e) - 6, +(0 ) , then the strict monotonicity uf '/1, in the variable x e holds good. As a consequence of Proposition 3.3 (with c = 0), the following property holds : in Theorem 3.4, if one further assumes that the function t �t is bounded and that A , q, I and p± do not depend on t, then u depends on x . e only, that is 'u is a stationary one-dimensional front. Roughly speaking, this means that any quasi-stationary front is truly stationary. This la�t result, also corresponds to a generalization of Theorem :3 . 1 with c = O .
f
.
.
.
'
.......
3.5. Monostable waves which are trapped between two fronts. Sub
sections 3,1 to 3.4 were concerned with "bistable-type" waves, in the sense that the reaction term / was assumed to be nonincreasing in some neighbourhoods of the limiting states p± (t, x). Here, we give a clasification result for monostable fronts which are trapped between two given planar fronts. Namely, we assume that the function / : [0, 1] --+ JR is of cla.�s C 1 and that (3 . 13)
/ (0) = [ (I) = 0, / > 0 on (0, 1 ), f'(0) > 0, f'( I ) < O.
(3 1 4 )
Ut = t:.u + feu) , x E JR N
It, is known that the equation
admits planar travelling front.s of the type u(t, x) = 'Pc(x . e - et), such that 'Pc : JR -> (0, 1 ) with 'Pe( - 00) = 1 and 'Pe ( +(0) = 0, for all e E §N- 1 and for all c > c* , where the minimal speed c' is positive and does not depend on e (it is known that c* > 2 //,(0» . Therefore, for a prescribed direction e, we cannot expect any uniqueness up to shifts. However, uniqueness (up to shifts) still holds for the waves which are trapped between two shifts of the same planar front . .
THEOREM 3.5. Assume that / satisfies (3. 13) . Let tL bp. (I bo'unded almost planar wave solving (3.14), connp.ding p- = 0 and p+ = 1, and satisfying (3.15)
\f (t, x)
E JR X JRN , 'Pe(x , e - ct) < u (t , x) < 'Pc (x , e - ct - a ) ,
for some c 2:: c' , e E §N - 1 and a 2:: 0, where 'Pc : R --> (0 , 1 ) solves 'P� + 'P� + / ('Pc) = o in R with 'Pe ( -oo) = 1 and 'Pc (+oo) = O . Then there exists b E [0, (.1,] !;uch that
V (t , x)
E
JR x R N , u(t , x) = 'Pc(x ,
e
- ct - b) .
GENERALIZED TRAVELLING WAVES FOR REACTION-DIFFUSION EQUATIONS
Call
PROOF. As in the proof of Theorem 3.1, one can assume that
for all
(t, x)
113
e = ( 1 , 0, . . . , 0) .
v(t, x) u(t, x + ete) = u(t, Xl + et, x') v solves Vt = � v + ee . V'v + f ( v) =
E ]R X ]RN . The function
(3.16)
and (3. 17)
Our goal is to prove that v is a shift of the planar front 'Pc(Xl) . First, remember that 'Pc is decreasing in R It is also well-known that
-00 if e > e* , -00 if e = e* , where a > 0 and A c = (e - Je2 - 41' (0))/2 if e > e* . If e = e* , then a > 0, or a = 0 and {3 > 0; furthermore, A c = (e* + J(e* ) 2 - 41' (0)) /2. * Choose 0 E (0, 1) such that f is decreasing in [1 - 0, 1 ] , and let A > 0 such that 'Pc(Xl ) ::::: 1 - 0 for all Xl :::; -A. Call F = {(t, x) E ]R x RN , Xl > -A}. Fix any T E ]R* and p E ]R N- l . For all 0" E ]R and (t, x) E ]R x R N , denote w"(t, x) = v(t + T, Xl + O", x' + p) . Since 'P c is decreasing, (3.17) yields w" :::; v in as as
(3. 18)
s ----+
s ----+
]R x ]RN for all 0" ::::: a. Call now 0" *
= inf {O" E ]R, w"t :::; v in ]R x ]RN for all 0"' ::::: O" }
The real number 0" * is well-defined because 'Pc(-oo) = 1 has w"* :::; v in ]R x ]RN .
.
> 0 = 'Pc (+oo), and one
Assume that 0" * > O. Notice first that there exists no TJo > 0 such that w"* - 'fJ :::; v in F for all TJ E [0, TJo ] : otherwise, if such a TJo exists, there would also hold that w"* -'fJ :::; v in {Xl :::; -A} (as in the proof of Theorem 3. 1 , using that v ::::: 'Pc (Xl) ::::: 1 - 0 for all Xl :::; -A), whence w"* - 'fJ :::; v in ]R x ]RN for all TJ E [0, TJo ] , This contradicts the minimality of 0" * . Therefore, there exist two sequences (O"n ) n EN in ( 0" * - 1 , 0" * ) and (tn ' Xn ) n EN = (tn , Xl ,n , X� ) n EN in F such that
ern ----+ 0" * as n + 00 and w"n (tn l xn ) ::::: v(tn , Xn ) for all n E N. Since X l , n ::::: -A for all n E N, two cases may occur, up to extraction of a sub sequence : either Xl , n +00, or Xl ,n Xl , oo E [-A, +(0) as n +00. Let us first deal with the case when Xl , n +00 as n +00. From standard parabolic estimates and Harnack inequality, there are two positive constants Cl and C2 such ----+
----+
----+
----+
----+
----+
that, for all n E N, 0 :::;
V(tn , Xl ,n , x� ) - v(tn + T, Xl ,n + 0"* , x� + p) :::; w"n (tn , Xl , n , X� ) - w"* (tn , Xl ,n , X� ) w"* (t, x) :::; Cl X (er* - O"n ) X l max tn - StStn , lx-xn I S 2 :::; Cl C2 X (0" * - O"n ) X wU* (tn + 1 , Xl, n , x� ) :::; Cl C2 X (er* - ern ) 'Pc( Xl . n a + 0" * ) . x
-
114
HENRI BERESTYCI« AND FRAN O. Then there also exists a constant C3 > 0 such that (v - w"* ) (t - T, XI - cr* , x' - p) < C3 X (v - w"* ) (t , x r , x') for all (t, x} , x') E JR x JRN . Thus.
v(tn - kT, XI,n - kcr * , x� - kp) - v(tn - (k - l)T, XI,n - (k - 1 ) cr* , x� - (k ::; CIC2 C; x ( cr" - crn) x 'Pc(Xl,n - a + cr*) for all k E N and n E N, whence
l)p)
v(tn - kT, X I,n - kcr* , x� - kp) - v(tn + T, XI,n + cr' , x� + p) < CIC2(1 + C3 + . . . + C; ) (cr* - crn)'Pc (X I,n - a + cr' ) .
From (3. 17), it follows that
(3. 19) 'Pc(XI,n - ku * ) < [1 + CI C2 ( 1 + C3 + . . . + Cf) (cr * - crn)] 'Pc(XI,n - a + u ' )
for all k and n in N. Fix now k E N such that - ku* < -a+u* (this is possible since u* > 0). Because of (3. 18), there is c: > a such that 'Pc ( s-ku*) > ( l +C:)'Pc (s-a+u*) for all s large enough. Since 'Pc > 0 and X I,n � + 00, Un u* as n +00, the inequalities (3.19) are impossible for n large enough. In the case T < 0, similarly, there exist two positive constants C� and C� such that --;
-+
'Pc(X I,n ) - 'Pc(X l ,n - a + ku *) < v(tn' XI,n, x� ) - v(tn + kT, XI, n + kcr* , x' + kp) ::; CI CW + C� + . . . + (q ) k-l ) (cr* - crn) 'Pc (X I ,n) for all n E N and k E N, k > 1 . Choose k such that -a + ku' > 0 and, from (3.18), c: > 0 such that ( 1 + C:)'Pc ( s - a + kcr*) ::; 'Pc(s) for s large enough. Once again, one +00 in the above inequalities. gets a contradiction as n Therefore, the sequence (XI ,n)nEN has to be bounded. Up to extraction of a +00, and that subsequence, one can assume that XI,n --; X I, oo E [-A, + 00) as n the functions vn(t, x) = vet + tn , Xl , X' + x�) converge locally uniformly in JR x RN to a solution v"" of (3.3) such that z(t, x) = voc (t, x) - v",, (t + T, XI + u' , x' + p) > 0 in JR x lRN , --;
--;
with equality at (0, XI,oo , 0) . The strong maximum principle and the uniqueness of the Cauchy problem for (3.3) imply that z = 0 in JR x RN , that is Voo (t, x) = voc (t + T, Xl + u·, x' + p) for all (t, x) E lR X JRN . But v"" still satisfies (3.17) . Since u· > 0 and 'Pc ( -oo ) = 1 > 0 = 'Pc(+oo), one has reached a contradiction. As a conclusion, one has proved that cr· < O. Thus,
vet + T, Xl + cr, X' + p) for all cr > 0 and (t, x) E jR X jR N . Since T 1= 0 and p E JRN- I were arbitrary, it follows that v can be written as a nonincreasing function ¢(XI) which depends on Xl only. Because of (3.17), 0 = ¢( +00) < ¢(s) < ¢( - 00) = 1 for all s E lR and ¢ is decreasing from the strong maximum principle. Furthermore, the function ¢ satisfies ¢" + c¢' + J(¢) = 0 in R Thus, by uniqueness, ¢ = rp ( . - b) for some b E JR . Lastly, 0 < b < a because of (3. 1 7) and 'Pc is decreasing. This completes the proof of Theorem 3.5. vet, x)
>
w" (t, x)
=
3.6. 1. It is immediate to see that the conclusion of Theorem 3.5 still holds if, instead of J > 0 on (0 , 1) and 1'(0) > 0 in (3.13) , it is only assumed that all (0, 1) of ¢" +c¢' + 1(4)) = 0 in JR with ¢( -00 ) = 1 , ¢( +00) = 0 are solutions ¢ : JR eqnal to 'Pc up to shifts, and that 'Pc is decreasing and lim infs�+oo 'Pc(S - T)/rpc(S) > 1 for some T > O. REMARK
-+
11 5
GENERALIZED TRAVELLING WAVES FOR REACTION-DIFFUSION EQUATIONS 2. The assumptions (3. 15) imply that (3.20) 0
c* (c* = 2 y'1' (0) in this case) , then the assumptions (3.20) are sufficient to ensure that u is of the type u(t, x) = 'Pc(x . e - ct - b) for some b E lR. (see [20]). However, this last result is open for general monostable nonlinearities f. 4. Further qualitative properties
We now proceed to further general qualitative properties of the generalized waves. Throughout this section, m = 1 and u denotes wave connecting p - and p+, for equation (2.1). We assume that u and p ± are globally bounded in lR. x 0 and that properties (2.3), (2.4), (3.6) and (3.7) are satisfied. First, the following general monotonicity property holds.
[4] Assume that A and q do not depend on t, that f and p± are nondecreasing in t and that there is 0 > 0 such that, for all (t, x) E lR. x 0, s f(t, x, s) is nonincreasing in (-oo,p - (t, x) + 0] and (p+ (t, x) - 0, +(0) . If u is an invasion of p - by p+ with infJR x IT (p+ - p - ) > 0, then ( 4.1) V (t, x) E lR. x 0, p - (t, x) < u(t, x) < p+ (t, x). and u is increasing in time t. Notice that if (4.1) holds a priori and if f is assumed to be nonincreasing in s for s in (p- (t, x), p - (t, x)+ o] and (p+ (t, x)- o, p+(t, x)] only, instead of (-oo, p - (t, x) + o] and (p+ (t, x) - 0, +(0), then the strict monotonicity of u in t holds good. THEOREM 4. 1 . �
Actually, Theorem 4.1 plays a crucial role in the uniqueness results of Sec tions 3.2 to 3.4. It says that the "bistable-type" invasion fronts are monotone in time. In the case of almost planar fronts, one can be more precise, that is one can compare any two fronts up to shifts in time.
THEOREM 4.2. [4] Under the same conditions as in Theorem 4. 1, assume fur thermore that f and p± are independent of t, and that there exist e E §N - 1 , ;::: 0 and a map lR. 3 t �t such that (3. 10) and (3. 1 1) are satisfied. Let u be another globally bounded invasion front of p - by p+ for equation (2.1) with the boundary condition (3.6) , associated with i\ = {x E n, X · e eft = O} and fit = {x E n, ±(x · e - eft) < O} and having a mean speed c ;::: 0 such that sup { 1 dn (i\ , rs ) - C1t - sl l; (t, s) E lR.2 } < +00. Then c = c and there is (the smallest) T E lR. such that u(t + T, x) ;::: u(t, x) for all (t, x) E lR. x O. Furthermore, there exists a sequence (tn ' Xn) n EN in lR. x 0 such that (dn(xn , ftJ ) n EN is bounded and u(tn + T, xn ) - u(tn ' xn ) 0 as n +00. Lastly, either u(t + T, x) > u(t, x) for all (t, x) E lR. x 0 or u(t + T, x) = u(t, x) for all (t, x) E lR. x O. C
�
-
-+
-+
HENRJ BERESTYCKI AND FRAN -n, x E JR, ( un) t (5.3) un(-n, x ) un,o (x) : = max ('Pc_ (x + c_ n), 'Pq (x + c+n)) . where the function
'
,
--+
->
From the maximum principle, it follows that
( x - c_ t), 'Pc+ ( x - c+ t) ) < un (t , x) < 1 E N, t > -n and x E JR. These estimates imply that un(-m, ') > for all Thus, the maximum principle implies that each Um (-m, . ) in JR as soon as n > max ('Pc .
n
m.
sequence (un (t, x))n>ltl is nondecrea. 0 and T2 . VI > O. The above estimates for U ± imply that u± (x + TTl ) ---+ cpct (x . VI ) and U± (x + TT2) ---+ cpc; (x . V2) as T ---+ +00 , locally uniformly in x E JR2 . Since cl < ct and c;- < ci , the limiting profiles of U± is the directions Tl and T2 are different, and the functions U± are not equal up to shifts and rotations.
1 20
HR'\1Rl BERESTYCKI AND FRANQOIS HAMEL
For each t
E
( -00, OJ, call
n t = x E ]R2, X · VI n t = :c E ]R2 , x · VI and, for each t E [0, +(0), call
< clt >
elt
n
u
nt = X E ]R2 , x · VI < cT t U x E ]R2 , X . V2 < ct t } , nt = x E ]R2 , X · VI > eTt n x E 1l�2 , X · Vz > ctt} . With these choices of nt, it follows immediately from (6.2) that u is a generalized
wave connecting 1 and 0, and that 1 invades 0. Lastly, it is straightforward to check from the estimateH (6.2) that u(t, x ) ° as t -00, and u(t, x) - u+ (t, x) u- (t, x) () as t +00, uniformly in x E R2 . In ol;her words, --.
This
--->
--->
u(t, x + c±v±t) ---> U ± (x)
as t
--+
--->
±oo, uniformly in x
E
]R2 .
means that the solution u converges uniformly in ]R2 to two different pro files U± as t ---> ±oo in two different moving frames, which propagate with two different speeds c± into two different directions v± . This completes the proof of Proposition 6 . 1 .
7. Exterior domains and further examples
In this section , we describe another application of the general notions of trav elling waves. We deal here with propagation around an obstacle. More precisely, we assume that the domain n is a connected smooth open i:iubset of RN such that where the obstacle K is non-empty and compact. I've consider the following ques tions : given a planar front which is tmvelling in the direction of the obstable, can it propagate around the obstacle and, if the answer is positive, is iLs shape perturbed behind the obstacle, and is its profile shifted ? We give answers to these questiolli:i [or the reaction-diffusion problem Ut -
Llu = f (u) in n, V . \7u ° on an,
(7.1 )
where V - v(x) denotes the outward unit normal to n at a point x E an. We assume that the nonlinearity f is of the bistable type on [0, 1 ] , that is 1 is of class Cl ( [O, 1]), /(0) = f(l) 1(0) = 0, 1' (0) < 0, 1' ( 1 ) < 0, 1 < ° on (0, 0), f > ° on (0, 1), where 0 E (0, 1) is given . We also assume that J: f > o. It is well-known that equation (7. 1 ) admits a unique planar front profile when n = ]RN : there (0, 1) such exist a unique speed c and a unique (up to shifts) function rfJ : R that rfJ(-oo) = 1 , ¢(+ oo) = ° and 0. When n i= ]R N , the�e planar fronts
GENERALIZED TRAVELLING WAVES FOR REACTION-DIFFUSION EQUATIONS
121
THEOREM 7. 1 . [5] Assume that K is strictly star-shaped. Given any direction e E SN - l , there exists a solution u(t, x) of (7.1) defined for all (t, x) E lR x II, and such that u(t, x) - ¢(x · e - ct) ----+ 0 as t ----+ ±oo, uniformly in x E II and u(t, x) - ¢(x · e - ct) ----+ 0 as I x l ----+ +00, uniformly in t E R The proof of this theorem can be divided into three main steps, which cor respond to the behavior of the front at very negative times, when it reaches the obstacle, and lastly when it recovers its shape at large times after passing the obstacle. Let us give a few words about each main step : • firstly, the existence of a non trivial time-global solution u( t, x) being close to the front ¢(x . e - ct) when t ----+ -00 is obtained as a limit as n ----+ +00 of a sequence of Cauchy problems starting at times -n; • secondly, it is proved that u(t, x) ----+ 1 locally in x as t ----+ +00. Here, we use the fact that the obstacle K is strictly star-shaped, that is there exists Xo E K such that [xo, x] E K and v(x) · (xo - x) > 0 for all x E oK = an. We prove a useful result of independant interest, which says that any solution 0 ::::; U (x) ::::; 1 of the stationary problem associated to (7.1) such that U(x) ----+ 1 as I xl ----+ +00, has to be identically equal to 1 if K is strictly star-shaped; • lastly, we prove that the local deformations of the level sets of the front which are induced by the obstacle become negligible at large times. We use sub- and super-solutions with strong or weak diffusion in the directions which are orthogonal to e . In equation (7.1), the obstacle K can then be viewed as a local perturbation of the uniform homogeneous medium. Other problems which are similar in nature can also be investigated. For instance, in equation (2.1), some coefficients may be locally perturbed. This is the case for instance in (1.7) when b(x) - boo has a compact support, or b(x) ----+ boo as I x l ----+ +00, for some constant boo , with b(x) -=t boo · This situation is not almost periodic. These problems are the purpose of current research. Lastly, we mention that the generalized travelling waves are the good tools to describe propagation in more complex geometrical situations, like spirals, curved cylinders with two different unbounded axes . . . We also point out that these general definitions could of course be still given for other types of equations which are not of the parabolic type. 8. Open problems There are natural questions that arise from this new notion of travelling front in several contexts. In each of these, the types of questions are: existence and uniqueness of fronts, range of front velocities -is there an interval of speeds in KPP-type equations and a unique speed for bistable problems?-, stability of the fronts, etc. We mention here a non-exhaustive list of specific or more general open problems: • For an equation like (1.7) where b is equal to the sum of a constant b oo and a compactly supported function, are there bistable or KPP generalized
122
HBNR.I BERESTYCKI AND FRANCOIS HAMEL
invasion fronts ? If the answer is po�itive, what is the possible phase shift which is induced by the local perturbation in the medium ? More general The same questions can be asked when, say in dimension 1 , the function equations with locally perturbed coefficients can also be
•
b
has
two different
limits when x
->
±oo
considered.
? Obviously, all these questions
can be extended to general parabolic operators, higher dimensions and •
•
•
more general geometries. Thc t;tudy of fronts for equations with time and space-dependent coeffi cients has just started and many fundamental
questions
about existence
and dynamical propertic8 of generalized waves in this context remain open.
Time-dependent
domains can also be considered. The general definitions
can indeed be easily extended to this framework. In Section 7, we
reported on the existence of bistable almost-planar fronts
passing an obstacle.
Can the geometrical condition on the obstacle be
removed ? •
Can the solutions of by
generalized
the
Cauchy problem associated to
(2.1)
be described
waves or fronts at large times ?
References
•
[I] D.G, Aronson, H,F, Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math, 30 (1978), 33-76, [2] H, Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, In: Noulinear PUEs in Condensed Matter and Reactive Flows, NATO Science Series C, 569, H, Berestycki and Y. Pomeau eds, Kluwcr, Doordrecht, 2003. [3] H. Berestycki, F, Hamel, Front propagation in periodic excitable media, Comm . Pure AppL Math. 55 (2002), 949-1032. [4] H. Berestycki, F. Hamel, On a general definition of travelling waves and their proper'ties, preprint , [5] H. Berestycki, F. H amel, H. Matano, Travelling waves in the presence of an obstacle, preprint. [6] H . Berestycki, F. Hamel, N. Nadirashvili, The principal eigenvalue of elliptic operators with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys. 253 (2005), 451-480. [7] H. Bcrestycki, F. Hamel, N. Nadirashvili, The speed of propagation for KPP type problems. I - Periodic framework, J. European Math. Soc. 7 (2005), 173-213. [8] H. Berestycki, F. Hamel L. Roques, Analysis of the periodically fragmented environment model : II Biological invasions and pulsating travelling fronts, J, Math. Pures Appl. 84
[9J
-
,
(2005), 1101-1140.
Berestycki, B. Larrouturou, A semilinear elliptic equation in a strip a,ising in a two dimensional flame propagation model, J. Reine Angew. Math. 396 (1989), 14-40. [101 H. Bcrestycki, B. Larrouturou, P.-L. Lions, A1ulhdimensional traveling-wave solutions of a flame propagation mode� Arch, Rat, Meeh. Anal. 1 1 1 (1990), 33-49. [llJ H, Berestycki, L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. H. Poincare , Anal. non Lin. 9 (1992), 497-572. [12] F. BHhuel, H. Brezis, F. Helein, Ginzburg-Landau vortices, Birkhiiuser, 1993. [131 A. Bonnet, F. Hamel, Existe nce of non-planar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal. 31 (1999), 80-1 18. [14] H. Brezis, Analyse Fonctionnelle, Th{orie et Applications, Masson, 1 987. [15] X. Chen, J.-S. Guo, F, Hamel, H, Ninomiya, J.-M. R.oquejoffre, Traveling waves with parab oloid like interface.s for balanced bistable dynamics, Ann. lnst. H . Poincare, Anal. Non Lin., to appear. [16] p, C. Fife, Mathematical aspects of reacting and diffusing systems, Springer Verlag, 1979. [17] P.C, Fife, Dynamics of internal layers and diffusive interfaces, CBMS-NSF Regional Con ference, Series in Applied Mathematics 53, 1988. H.
GENERALIZED TRAVELLING WAVES FOR REACTION-DIFFUSION EQUATIONS
123
[18] G. Frejacques, Travelling waves in infinite cylinders with time-periodic coefficients, Ph.D. Dissertation, Universite Aix-Marseille III, 2005. [19] F. Hamel, R. Monneau, J.-M. Roquejoffre, Existence and qualitative properties of conical multidimensional bistable fronts, Disc. Cont. Dyn. Systems 13 (2005) , 1069-1096. [20] F. Hamel, N. Nadirashvili, Travelling waves and entire solutions of the Fisher-KPP equation in ]RN , Arch. Ration. Mech. Anal. 157 (2001) , 91-163. [21] M. Haragus, A. Scheel, Corner defects in almost planar interface propagation, Ann. Inst. H. Poincare, Anal. Non Lin. 23 (2006) , 283-329. [22] W. Hudson, B. Zinner, Existence of travelling waves for reaction-diffusion equations of Fisher type in periodic media, Boundary Value Problems for Functional-Differential Equations, J. Henderson (ed.) , World Scientific, 1995, pp. 187-199. [23] A.N. Kolmogorov, I.G. Petrovsky, N.S. Piskunov, Etude de l 'equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique, Bull. Univ. d'Etat it Moscou, Ser. Intern. A 1 (1937) , 1-26. [24] H. Matano, Oral communication. [25] H. Ninomiya, M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Diff. Eqs. 213 (2005) , 204-233. [26] J. Nolen, J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Disc. Cont . Dyn. Syst. 13 (2005) , 1217-1234. [27] J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solu tions of parabolic equations in cylinders, Ann. Inst. H. Poincare, Anal. Non Lin. 14 (1997) , 499-552. [28] B. Sandstede, A. Scheel, Defects in oscillatory media - towards a classification, SIAM J. Appl. Dyn. Sys. 3 (2004) , 1-68. [29] W. Shen, Dynamical systems and traveling waves in almost periodic structures, J. Diff. Eqs. 169 (2001), 493-548. [30] N. Shigesada, K. Kawasaki, E. Teramoto, Spatial segregation of interacting species, J. Theo ret. BioI. 79 (1979) , 83-99. [31] X. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Ration. Mech. Anal. 121 ( 1992) , 205-233. EHESS, CAMS, 54 BOULEVARD RASPAIL, F-75006 PARIS, FRANCE E-mail address: hb ° if t i= 1 and JR+ V (t) +00 if l -> +00 (so that 1 is the minimum) . The configuration space for the order parameter is no longer rn:1P'2 = §,2/{± 1 } but instead � JRd /{±l } . It possesses in particular the positive cone property and is singular at the origin. Moreover the metric is fiat except at zero. Notice that a related model was proposed by J. Ericksen, where � is replaced by the cone {(s, 71) E JR X IR3 , l s i = lui}. Several important mathematical itlsues [or this slightly different model were treated by F.H. Lin and R. Hardt ( [20] , [14) ) . An interesting aspect of the previous setting iH that the minimization problem is now well-defined for any given 9 E H� (al1, �). Indeed, we have (see [9)) where V
---t
---t
PROPOSITION 1 . Let
9
E
H � (8B3, �), for �
IR3 / {±1}. Then,
H; (B3 , � ) "1 0.
Moreover, th"re exists a; E (0, 1) such that any minimizer of (2) in H� (B3 , �) belongs to the Holder space C�;� (B3 �). ,
In view of the conical singularity of � at the origin, the highest possible reg ularity one might expect and actually define is lipsehit>l regularity. However, it is not known whether minimizers for (2) are lipschitz continuous maps.
129
SOME QUESTIONS
If, in the definition of E, one replaces V by -i-x V, where E is a small parameter, then in the limit E � 0, minimizers will take value in ]R1P'2 and have possibly singularities of codimension two, i. e. lines in dimension three: such singularities are indeed experimentally observed (see [12] , [18] , [24] ) and are called disclinations for nematic liquid crystals.
3.
Some results and open questions for Problem 1
In order to emphasize some of the main features of Problem 1 , we start with the Sobolev space W l ,P(O, N) for 1 :s; p :s; 00. In order to define this space, we consider an isometric embedding of N into some ]Rq (this is possible by Nash Moser's Theorem) and set
W l ,P (O, N)
==
{u E W l ,P (O, ]Rq), u(x) E N a.e.}.
It can be checked that this definition is independent of the chosen embedding. We define similarly the space W l ,P(O, E). In the sequel, ° C ]RN is a smooth bounded, simply connected domain in dimension N 2: 2 (for instance, 0 = EN C ]RN is the unit ball). We first have the classical observation LEMMA 1 . If 1 :s; p < 2 and 1f l (N) i- {O}, then there exists some u E W l ,p (O, N) such that there does not exist 'P E W l ,P(O, E) such that u 1f 0 'P. PROOF. It suffices to consider a smooth map "( : § l N such that "( is non trivial in 1f l (N) (it exists by assumption). Up to a translation, we may assume o E 0. Then, the map u defined for x = (X l , ... , XN ) by (X , X ) u(x) = "( ( l 2 ) (3) I (X l , x 2 ) 1 is smooth in ° \ {Xl = X 2 = O}, uniformly bounded, and satisfies =
�
_
u E W l ,P(O, N). However, if there exists 'P E W l , l (O, E) u = 1f 0 'P, then by Fubini's theorem, there exists r > 0 such that uisr = 1r 0 'P I Sr and 'Pisr E WI, 1 (STl E) c CO (SrJ ) , where Sr denotes the circle in the (X l , x2 )-plane of center 0 and radius r. This last fact can not be true since § l :3 uls)rw) = "((w) is not homotopic to a constant. D
for 1 :s; p such that
1, which is very similar to the case s = 1 considered before.
THEOREM 2. Let be given 1 < s < 00 and 1 < p < 00. i) If sp < 2 and 1Tl (N) =I- {O}, then there exists some u E WS,P(O, N) such that there does not exist rp E WS,P (O, E) such that u 1T 0 rp. ii) If sp :2: 2, then for any u E WS,P (O, N) , there exists rp E w s ,p n W1,SP(O, E) such that u 1T 0 rp. The proof of Theorem 2 is given in the Appendix (section A. l ) . We have defined, as for the case s = 1 , WS ,P (O, £ ) == {rp E WS,P( f! , lRq ) , rp(x) E E a e . } . =
=
.
The case 0 < s < 1 is more delicate, and the question is not completely settled. A first difficulty is related to the very definition of WS,P(O, E). In what follows, our choice is (4) WS,P(O, E)
==
{ rp
E U (O, lRq) ,
rp(x) E E a.e. and de (rp(x), rp�)) I x y l s+p _
E U (O
x
}
O) ,
where de denotes the geodesic distance i n E. Notice i n particular that this definition is independent of the embedding5 . One may choose, instead of the geodesic distance de( rp(x), rp(y)), the euclidean distance I l rp(x) rp ( y ) l l . The two definitions obviously coincide when E is compact; in the case E is not compact, it is not clear whether this alternate definition is equivalent to (4) and independent of the choice of the embedding. We have -
THEOREM 3. Let be given 0 < s < 1 and 1 < p < 00. i) If 1 ::; sp < 2 and 1Tl (N) =I- {O}, then there exists some u E WS,P (O, N) such that there does not exist rp E WS ,P(O, E) such that u = 1T 0 rp. ii) If sp < 1 or sp :2: N, then for any u E WS,P (O , N) , there exists rp E WS,P(O, E) such that u 1T 0 rp. =
The proof of Theorem 3 follows essentially the lines of [5] and is given in the Appendix ( section A.2 ) . It is worthwhile noticing that the simple connectedness of ° is not necessary if sp < 1.
)
(
5 In particular, if cp belongs t o w" p(n, E), then necessarily de(cp(.), z E LP(n, ffi.) for any or one, since n is bounded z E E. Indeed, for a.e. y E n, de(cp(.), cp(y» E LP(n, ffi.) by Fubini's theorem. The same property holds for s = 1 , since W1 ,p C W" p for 0 < s < 1 .
)
FABRICE BETHUEL AND DAVID CHIRON
132
2. In case the lifting exists for 0 < s < 00 , 1 < P < 00 and sp > 1 , it is unique up to the action of an element of 7r1 (N) (!;ee Lemma A.4 in the Appendix with s = (T and p = q). REMARK
3. In the statements of the Theorems, we have assumed N > 2. For N = 1 and n = ( a, b) , the lifting property always holds. This follows easily from Lemma 2 and the proofs of Theorems 2 ii) and 3 ii) . REMARK
In the special case N = §1, J. Bourgain, H. Brezis and P. Mironcscu ( [5]) were able to extend Theorem 3 i) to the whole range 1 < sp < N with an explicit counter example. A similar counter-example can still be constructed in the case where E is non compact. A first observation is that since N is compact, E is compact if and only if 7r1 (N) is finite. Therefore, E non compact means 7r1 (N) infinite.
2 . Let ° < s < 1, 1 < sp < N . If E is non compact (i. e. 7r1 (N) is infinite), then there exists 1k E WS,p (n, N) such that u can not be written as U = 7r 0
+00.
Consider next a minimizing geodesic In : [0, in] E with unit speed from e to �n (since E is complete, such geodesics exist) . From the minimality of In, we infer that for any t, 8 E [0, in], --->
(5)
Since bn) is a sequence of l-lipschitzian maps satisfying 1,, (0) = �, we deduce by the Ascoli-Arzela compactness theorem that for a subsequence, still denoted In , there exists some function f : IR+ -+ E such that f(O) = � and In
--.
f
uniformly on compact subsets of IR+.
The function f has the desired property passing to the limit in (5).
Step 2: Construction of a map without lifting. COIlsider 0 that
< 8 < 1 , 1 < p < 00 such that 1 < sp < N, and let N - sp
'--
N - sp 2,
vE
(6)
< 2, then, for any [ compact, the answer to (OQ 1 ) is also negative. PROPOSITION
4.
Given a base point b E N, we denote h] E 1l'1 (N) = 1l') (N, b) the homotopy class of the loop i : §1 N at b. We also denote 1 the trivial homotopy class, "." the group operation in 11'1 (N, b), and [il k - hl · . . · h] with factors. We easily check that the loop ik : §1 -+ N defined by ik(Z) == i(zk) for z §1 belongs if and only if to h]". A loop 'Y is said to be of order PROO F .
-+
.
h] k =
1
k E N*
and
hP ¥ 1
which implies in particular that ik : §1
-+
for
1 < j < k,
k E
E, i. e.
N has a smooth lifting rk : §l -+ k 'liz § 1 . 1l' 0 rk (z) = 'Yk (Z) = i(z ) (7) Recall that [ is assumed to be compact, that is 7fl (N) is finite, and therefore every element of 1l'1 (N, b) is of finite order. The proof is completed in two steps. Step 1.' There exists an embedding i : §l
k E N, k > 2.
-+
E
N such that the loop I has finite order
135
SOME QUESTIONS
By assumption, there exists a non trivial element 9 E 7f (N, b). By standard l results, there exists a minimal geodesic , in the homotopy class of 9 based at the base point b, i. e. a loop , : §l --+ N such that 9 = b] and is a geodesic with minimal length in g. It can be checked that , has at most a finite number of self-intersections. We wish , to have no self-intersection. If it is not the case, we proceed as follows. We first see , as a path defined on [0, 1] with ,(0) = ,(1) = b. If ,(0) = ,(t) for a t E (0, 1) for instance, then one of the loops 'HO ,t] or 'I [t , ] is l non trivial ( for otherwise , would be trivial) , say 'I [O ,t] , thus we can replace , by 'I [O ,t] · In a finite number of steps, we obtain a closed geodesic , : §l --+ N with no self-intersection, that is , is an embedding. Moreover, by construction, , is a non-trivial loop, and its order is finite since E is compact (i. e. 7f (N) is finite) . l As already seen, the loop 'k has a (smooth) lifting rk : §l --+ E and (7) holds. The map rk is also an embedding. Indeed, if rk (eit ) = rk (eis) (0 ::; s < t < 27f) , k then ,(ei t ) = ,(eik s). Since , is an embedding, then kt - ks = 27fn for some n E Z. Necessarily, 0 < n < k since 0 < t - s < 27f and we then infer that [,] n = 1 with 0 < n < k, a contradiction with the fact that , has order k.
Step 2: Construction of a map which has no lifting. By hypothesis, (Pb 2)k has a negative answer for the integer k given in Step 1 , thus there exists v E WS,P (O, §l ) such that v ¢. w k in ° for any w E WS ,P (O, §l) . Let now u == , O V : ° --+ N. Since , is lipschitzian, u E WS,P (O, N) . Suppose u has a lifting rp E WS ,P (O, E). Then, rp has values a.e. in rk (§l ) = 7f - l (r(§l ) ) . Denoting by r; l : rk (§l ) --+ §l the inverse of the embedding rk, r; l is also lipschitzian, hence w == r;l 0 rp belongs to WS ,P (O, §l) and verifiel:>, by (7), , 0 v = u = 7f 0 rp = k k 7f O rk O W = ,k O W = ,(w ) . Since , is an embedding Of §l , we infer v = w for some w E WS ,p (O, §l ) , a contradiction. Therefore, u has no lifting in WS,P (O, E) . 0
5. Lifting in fiber bundles The lifting in fiber bundles in the Sobolev context, i. e. (Pb 4), has been little studied so far. We will see in this section that the problem is presumably more complex than for the covering case, and at this stage, it is not clear what the expected results might be. In view of subtle problems related to the topology7 of the domain ° for (Pb 4), we may restrict ourselves to the case In this setting, if u : BN --+ N is continuous, then there always exists a continuous lifting rp : BN --+ E such that u = 7f 0 rp. As expected, we have PROPOSITION 5. Let be given 0 < s ::; 1 and 1 ::; p ::; 00 such that sp > N. Then, for every u E WS ,P (BN , N), there exists rp E WS ,P (BN, E) such that u = 7fOrp. 7In the case o f a covering, only the first fundamental group o f the domain plays a role. For fiberings, other aspects of the topology of the domain are of importance, even in the case of continuous maps. In particular, let us point out that if 1l' : E N is a covering, then 7rk(E) =: 7rk (N) for k 2: 2, whereas it may not be the case for a fibering. ---+
136
FABRICE BETHUEL AND DAVID CHIRON
The proof follows essentially the same arguments as in the continuous case, we therefore omit it. It is however not completely clear if this construction extend to the case s > (in view of the higher regul arity required) . Similarly, we conjecture that this lifting property holds in the critical case s < I and sp = N (see (OQ
1
0
§3 such that u = 7r 0 'P. Moreover, two given continuous maps Ul and U2 from §3 into §2 are in the same homotopy class if and only if corresponding liftings 'P 1 and 'P2 are also homotopic from §3 into §3 . It follows that 7r3 (§2 ) = Z and one may therefore define the Hopf invariant of u as H{u) REMARK 7. Let w : §2 can be shown that
---->
==
deg('P) .
§2 and u : §3
---->
§2 be continuous maps. Then, it
H(w 0 u) = (deg w) 2 H(u) , whereas for every continuous 9 : §3 ----> §3 , then it can be deduced from the discussion above that H(u 0 g) = (deg g)H(u) . REMARK 8. It is worthwhile (in particular in view of the proof of case d) in Theorem 4 below) comparing the 3-Dirichlet energy of a map u : §3 ----> §2 with the 3-Dirichlet energy of a lifting 'P. In view of the integral definition of the degree d == H(u) = deg('P) = so that (11)
; 13 J'P d1i3 ,
1 31
1 38
FABRICE BETHUEL AND DAVID CHIRON
On the other hand, it is shown in [25] , using Remark 7, that for some smooth map Ud : §3 -+ §2 such that H (Ud ) = d and
r
iS3
d E Z, there exists
IV Udl 3 d'H3 < Gl dl � ,
so that, by (ll), the 3-energy of Ud becomes negligible compared9 to the 3-energy of any lifting 'Pd of Ud as d -+ +00. The previous argument can be adapted to prove for a given d E Z the existence of smooth maps Wd : B3 -+ §2 such that Wd = 0"1 §2 on 8B3, H(Wd ) = d and
E
83
IV'Wdl 3 dx < C1 dl ! .
Here, in the definition of the Hopf invariant H(Wd), we compactify the boundary 8B3 to a point. As above, one proves that for any liftingl O 'Pd of Wd on B3 , we have for some G 0
>
( 12) Choosing a good gauge. \Ve have seen in the case of coverings that the choice of
a lifting is very reduced. In particular, in many cases, the imposed regularity yields the uniqueness of the lifting, up to the action of an clement of 7fl (N) (see Lemma AA in the Appendix) . By contrast, when the fiber is a continuum, if the lifting exi�t�, then there is also a continuum l l of possible liftings, so that, if one imposes some regularity to the lifting, then one has to make a suitable choice, usually called a gauge fixing. In the case of the Hopf fibration, with abelian fiber § l = U (l) , we will describe next the Coulomb gauge. exp((J(x)O"l )g(X), so Let (J : BN --+ R be a scalar function, and set go(x) that U = IT 0 90 and explicit computations show that g;Idgo = g- I dg + (dB)O"I. In particular, _
We claim that there exi�ts Bo- such that the 1-form Al identify with the vector field AI , satisfies divAt = d*-Al = 0 (13) At n = (A d N = 0 -
.
Indeed, it suffices to choose
(Jo such that
= - div(Al) = -AI ' n
In particular, if A l the functional
gIn view of (9),
in BN , on 8BN.
E L2 (BN ), then (Jo can be obtained by minimizing on Hl (BN )
this means that the 3-energy of 'f'd is "concentrated" in A I , which is not seen
by IV'Udl .
lOThe lifting is not necessarily constant on 8B3: however, the image 'Pd(8B3) is of measure
zero in S3 .
llmuch larger than the fib er!
1 39
SOME QUESTIONS
The curvature equation. The term A l (g) is related to the projection by the following classical equation 12
(14)
dA 1 = 2u ( *
w
u = II 0 9
),
where w denotes the standard volume form on §2 , and u* (w) its pull-back on BN . In coordinates, " ." standing for the scalar product in su( 2 ) , u (w ) writes as
u
*
(w )
=
L I �J<j-::;'
N
aU U · (ax ' '
x
)
*
au dXi 1\ dX . j ax ·
J
Combining (13) with (14), we obtain an elliptic system for the Coulomb gauge A I . More precisely, by Poincare lemma, there exists a 2-form such that d = 0, T = 0 on aB N and
(15) Inserting (16)
(15) into (14), we are led to the elliptic equation 13 -� = u ( ) *
w
Lifting maps in W1,P ( BN, §2 ) . The lifting problem for the Hopf fibration II : SU(2) � §3 � §2 for s = 1 can be settled in a number of cases. (i)
THEOREM
If either
4. Let 1 :s; p :s; 00 and N 2 2 . (b) p 2 N 2 3,
(a) l :S; p < 2 :S; N,
then for every u E W 1 ,p (BN , §2 ) , there exists c.p E W1 ,p (BN , §3) such that u = IIoc.p. (ii) If either (c) 2 :S; p < 3 :S; N, then there exists a map u
W 1 ,P (BN , §3) .
This last Theorem (OQ 3) p = H l (B2 , §3 ) ?
N =
2,
E
W 1 ,P(BN , §2 )
(d) p = 3 < N, which does not admit any lifting c.p
E
4 leaves two cases open i. e.
does every map in H I (B2 , §2 ) admit a lifting c.p E
(OQ 4) Is the lifting property true for
3 3, we can consider the map u : EN (X l , X2 , X3 ) _ u (x ) =
->
§2 defined by
I (X l , X2 , X 3 ) 1 '
U(nN) for every 1 < P < 3, hence u E We have l V'ul (x) < I (X l , X2 , X3 )I-l Wl'P ( EN , §2) for 1 < p < 3. We are going to show that u can not be lifted to a map 'P E Wl,p(nN, §2) if 2 < P < 3. We argue by contradiction: assume that 'P E WI,p(nN, §2) satisfies u = II 0 'P. By the usual averaging argument we have used in the proof of Lemma 1, this yields a radius r E (0, 1 ) and a E EN-3 such that 'u ls� has a lifting 'P l s� E Wl,p(S; , §3), where S; is the sphere S; = {x = (X I , X2 , X3 , X') E nN, xI + x� + x§ = r2, x' = a}. Changing coordinates, this means that I d§2 has a lifting lP WI ,p (§2 , §3) . If p > 2, then lP is a continuous lifting of Ids2 , a contradiction with the property seen earlier. In the limiting case p = 2, wc argue by approximation. There exists a sequence 1/Jn E C = (§2 , §3) such that lPn -> lP in HI (§2, §3) . Therefore, II 0 lP", --> II 0 lP = Id§2 in HI (§2, §2) , which yields, by continuity of the degree in H I (§2 , §2) , deg(II 0 lPn) -> deg(Id§2 ) = 1 as n -> +00. However, since 'IjJn : §2 -> §3 is smooth and '7r2 (§3) = {O}, then §2 also, hence lPn : §2 -> §3 is homotopic to a constant, thus II 0 lPn : §2 deg(II 0 lPn) = 0, a contradiction.
E
E
-+
REMARK 9. An alternate proof of case ( c) relies on density properties of smooth maps in Sobolev spaces between manifolds. Recall that since 7r2 (§3) = {O}, then c= (nN , §3) is dense in Wl,p(nN, §3) for 2 < P < :� ::; N (see [1] ) . In particular, for any 'P E WI,P(EN, §3) , II 0 'P belongs to Coo (EN, §2 ) i- Wl,P(EN, §2 ) since 7r2 (§2) =I {O} (see [1] ) .
141
SOME QUESTIONS
This Remark raises the following question:
(OQ 5) For 2 ::; p < 3 ::; N, does any map in coo (B N , §2 ) has a lifting in W 1 ,P(B N , §3) ? In other words, do we have II(W 1 ,P(B N , §3 ) ) = coo (B N , §2 ) -=I W 1 ,P(B N , §2 ) ? REMARK 1 0 . The argument of the proof of case (c) shows that Id§2 has no lifting in VMO(§2 , §3 ) , and therefore, I d§2 has no lifting in WS,P (B2 , §3) for every ° < s < 00 and 1 < p < 00 such that sp 2 2 . We use in this case the continuity of the degree in VMO(§2 , §2 ) (see [7] ) .
Proof of case (d). I t suffices t o consider the case N = 4 (for N > 4, one adds suitably many dimensions) . The central argument in the proof is a dipole construction introduced in [4] . Consider, for A > 0, d E Z the function Wdip (A, d) : jR4 ----+ §2 defined for X l E jR, X' E jR3 by Wdip(A, d) (X 1 ' x, ) Wd ==
( 1 2 1 xAX'1 - 1 1 )
1 if X E B4 , and Wdip(A, d) is extended by 0'1 outside B4. Here, Wd is the map constructed in [25] and described in Remark 8. Notice that Wdip is locally lip schitz continuous away from (± � , 0, 0, 0) , where it has singularities of Hopf in variant ±d. Moreover, Wdip = 0'1 outside some neighborhood L: A of the segment L: oo == [(- � , O, O, O) , ( � , O, O, O)] which shrinks to L:oo as A ----+ +00. Finally,
r
W , d dx C d i . Jr.). IV dip(C W ::; 1 l The main point is that, as in Remark 8, any lifting of Wdip has a 3-energy larger than C 1 d l . We consider finally the scaled and translated versions of Wdip Wdip (A, C, d, a) WdiP ( ' d) � a , so that Wdip(A, C, d, a) = 0'1 outside L:i a a + CL:� and ==
,
==
� C ) £
( 1 7) Our counter-example writes
u L: Xr.n Wdi ( An , Cn , an , dn ) + NXB4\(Un24r. n ) , n 2:4 A n L:an , � then
>1
1 11> l l w2 .!f < Cpl l u * (w) l l d < Cp l lV'u l l ip · Therefore, by (15) and Sobolev embedding, we have 1> E W1,p and (19) follows, that is 2 ' I I A l l b = l i d 1> I ILP < Cpl l V'uI ILP -
*
To complete thc proof, let 1L E W 1 ,P(BN, §2), and let Vn E coo (BN , §2 ) be such that Vn -+ 1L in W 1 'P (BN, §2 ) . Let 9n E COO(BN, §3) be the lift constructed above for v = vn . It follows from ( 18) that 9n is bounded in W1'P(BN, §3) . Therefore, up to a subsequence, we may assume gn ......l. g in W1'P(BN , §3) and check that 1L = 110g. D This finishes the proof of (a ) and thus of Theorem 4.
1 1 . The previous argument does not work for N = p = 2 since then u* (w) is only bounded in L l , and the elliptic theory does not allow to conclude. Actually, it turns out that inequality (18) does not hold for N = p = 2. More precisely, let P : rc -+ §2 \ { -ad be the stereographic projection and let, for A > 0, U\(z) = P(AZ) . Then, one has fB 2 1 V'u \ 12 < 1 §2 1 but REMARK
( inf \ -++00 lim
As a matter of fact, u\ A --> +00.
......l. - 0' 1
lV'g l 2 dx, 11 0 g = u\
)
in Hl (B2, §2) and u* (w)
=
+00.
......l.
1 §2 1 bo as measure as
However, this does not exclude the fact that ( 18) might hold for p = small energies, and this raises the open question
N
=
2 for
(OQ 6) Does there exists 6 > 0 and b > 0 such that if u E HI (B2, §2) verifies fIJ2 1V'u12 dx < 6, then there exists g E H1 (B2, §3) such that 1L = II 0 g
and
{ lV'g l 2 dx < b ? JB2
As in [5] , the case s i= 1 is also of interest . However, only the case (c) of Theorem 4 extends easily for 2 < sp < 3 in view of Remark 10.
1 43
SOME QUESTIONS
Appendix A A.I. Proof of Theorem 2. Proof of i). We use the same counter-example as in Lemma 1 , and check that the argument can be transposed. Indeed, for these s's and p ' s, s < 2 and 1 ::; p < 00, u E W 1 ,p and 8u (x) = 8Xl
�r I" (( l , O)
_
X1 (X�' X 2 ) , r
)
where r == (xi + x�) 1 /2 and 1" denotes tangential derivative for I' ( and a similar au au expression holds for aX2 ) , thus aX l E ws- 1 ,p for sp < 2. As we have seen during 0 the proof of Lemma 1 , u can not have a lifting in W 1 , 1 (0, f). For the proof of ii), we will use as in [5] the Gagliardo-Nirenberg inequality. We give here a general version (see e.g. [6] , Corollary 2) .
LEMMA A.L (Gagliardo-Nirenberg inequality) Let 0 < So, Sl < 00, , PO P1 ::; 00 and ° be a smooth bounded domain in JRN . For 0 < () < 1 , define 1 ())so and - == () + 1 - () So == ()S l + ( 1 Po Po P1 l l o o (0, JR) , then u E WS9 ,P9 (0, JR) and ,P (0, JR) n WS If u E ws ,p -
-
1
2, then by Theorem 1 , there exists
Ucil that sp > 2 and every u E WS,P(l1,N) and 'P E W),SP (rt, f) such that u = 11' 0 'P, then :p E W',P (rt, f).
In the Cal:i€ S = 1 , there is nothing to prove. Let us assume next that. (Is) holds for a given s E N" , and let 1 < P < 00 verifying (s + l)p > 2 , u E ws+1,P(rt, N) and 'P E W 1 ,(sH)P(rt, £) be such that lL = 11' 0 'P. By the induction assumption, we have 'P E W',P(rt, f) . We next claim that (A , 2)
'P E ws+l,P (rt, f),
Proof of (A.2) . We will repeatedly use the fact that N is compact, hence u E Loo . We have u E wsH,p n Loo , it follows from the Gagliardo-Nirenberg inequality that U E ws,p(1+ ! ) . Applying the induction hypothesis (Is) with lip" = p(I + ! ) E ( 1 , 00) verifying sp(l + !) = sp + s > sp > 2 , we obtain 'P E w s ,p( 1+ ; ) . Moreover, we have 'P E W 1 , ( s+ 1 ) p hy hypothesis, hence 'P E W·,p(l+ ; ) n WI ( H)p . ,
.•
1 45
SOME QUESTIONS
i
This implies, by Lemma A.3, that we have, for any 1 S; S; e, ei (rp) E ws,p(l+ � ) ( fl, lRq ) , since ei is a smooth map on E, with uniformly bounded derivatives of any order since N is compact. Using the Gagliardo-Nirenberg inequality (ei(rp) E Loo ) , one . s+1 infers that for any 0 S; j S; s, j E N, ei ( rp) E W S -J ,P s-j ( fl, lRq ) , so that in particular .
s+1
DS -J (ei(rp)) E £p s-j ,
with p��� = 00 (recall l ei l = 1 on E) if j = s. Next, since U E ws+ 1 ,p n L oo C ws+ 1 ,p n W 1 ,(s+l )p by Lemma A. 1 , we deduce lli (U) E ws+ 1 ,P ( fl, lRq ) by Lemma A.3. Applying Lemma A.2 with lli (U), U E ws+ 1 ,p n Loo , we infer 9i == (\7u, lli (U»)) E ws,p. We also have 9i E L(s+l )p since lli (U) E Loo and \7u E L(s+ l )p . Hence 9i E ws,p n L(s+l)p, from which we infer by Lemma A. 1 that, . s+1 for any 0 S; j S; s, j E N, 9i E WJ ,P HI , that is (A.3)
.
s+1
DJ (9i) E £P H I . For a = ( a 1 , " " a N ) E N N , we denote l a l a 1 + . . . + a N E N and al a i aa = - axr1 ax� 2 . . . ax(lr We' differentiate ( A. 1 ) s times: for any multi-index = ( a 1 , " " aN ) E N N such that l a l = s, the Leibniz formula yields (A.4)
==
-::--::-:--=--c:----=-=_
a
aa \7 rp =
L L 2= ENN , S R
a� (9i)aa - � (ei(rp)) , � �� � ( ( ) �� ) ( ) � a
... 1� where f3 S; a means f31 S; aI , f32 S; a 2 , . . . , f3N S; a N · For 0 S; 1f31 = j S; s, j E N, . s+1 s+l . we have by (A.3) and (A.4) , DS-J (ei(rp)) E LP s-j and DJ (9i ) E £P HI , with
s-j j+1 p( s + 1) + p( s + 1 )
1 p'
thus a�(9i)aa-�(ei (rp) E £P by Holder and then Ds+ 1 rp E £P, that is rp E ws+ 1 ,P ( fl, E) as claimed. This finishes the proof by induction for integer s. We turn now to the case s rf N. Let m = [s] E N* , so that m < s < m + 1. First, as for the case s E N, U E ws,p n UXJ c W 1 ,sp implies lli (U) E ws,p n Loo . Since U E ws,p n L oo , Lemma A.2 yields gi E ws- 1 ,p. Moreover, 9i E LSP since lli (U) E Loo and \7u E Up. Hence,
9i E w s - 1 ,p n Up. Furthermore, by Lemma A . 1 , U E ws,p n Loo , thus ( m < s) U E wm, � n L oo . The result being already established for s integer, we infer rp E wm, � . Moreover, rp E W 1 ,sp by construction, thus rp E W 1 ,sp n wm, � and Lemma A.3 implies (A.6) fi ei (rp) E wm, � n L oo . From (A.5) and (A.6), we are now in position to apply Lemma A.2 with "(f, g, s, t, r, p, O)" = (A.5)
==
(fi, gi , m, � , Sp,p, s�l ), so that "Os" = s - 1 and the relation 1 0 1 s - l = 1 -r + -t = sp + -m;: p is satisfied, which yields fi 9i E ws- 1 ,p for any 1 S; i S; e. \7 rp E ws- 1 ,p, that is rp
Therefore, (A. 1 ) gives 0
E WS,P, and the proof is finished for non-integer s.
146
FABRICE BETHUEL AND DAVID CHIRON
A.2. Proof of Theorem 3. Proof of i). The case
1 < sp < 2 < N follows
once more from the map introduced in Lemma 1 . Indeed, we have U E W 1 ,p n LOO for any 1 :::; p < so that by the Gagliardo-Nirenberg inequality, u E ws ,p for every 0 < s < 1 and 1 < p < 00 verifying sp < As we have already seen by a slicing argument, U can not have a lifting in W1 /q,q for 1 < q < 00, for otherwise, I would have a lifting in VMO (St , £). 0
2,
2.
Sinee N is a compact manifold without boundary, there exists firstly a constant C > 0 such that for any u, v E N,
Iu - vi
>
:::;
d./lf(u, v) < Clu
-
i
v ,
and secondly a constant Ii 0 such that the nearest point projection p : No -+ N from the Ii-neighborhood No of N in ]Rq onto N is well-defined and smooth. Finally, since N is compact and 11' is a covering, there exists another constant T) > 0 having the following property: if B is a geodesic ball in N of radius < T), then 1T-1 (B) is a union of disjoint geodesic balls in c diffeomorphic and isometric (by 11' ) to B. In particular, inf{ds (cp, 'Ij! ) , IT ( cp) = 7r('Ij!) , 1/J =j:. cp}
(A.7)
Proof of ii) : the case
8p
> 2T) > O.
< 1 . We proceed essentially as in [5) (Appendix A).
The construction is based on a approximation of maps in WS'P, sp < 1 by maps which are piecewise constant on cubes. First, we extend u by an arbitrary constant a E N outside 0 and still have a map in W·,p, which allows to reduce to the case o = (0, l)N. We define Pj the dyadic partition of 0 into 2j N cubes of size 2-j , and let Xj be the set of maps in L l ( O , lRq ) which are constant on each cube of Pj ' For u E L1 (0, N) and j E N, we set
Ej (u)(x)
1
=
u(y) dy,
I Qj (x ) 1 Qj (x) where Qj (x) E Pj is the cube defined by x E Qj (x). Fix a E N and let, for j > 0, if Ej (u) E No, Ej (u)) p ( UJ (x) otherwise. a We have Uj E Xj and Uj -+ u a.e. as j -+ +00, since Ej (u) To define cp, we will make use of the following claim.
Claim 1 . There exists C = C (c, N) exists cp E E satisfying 7r (cp) = u and
->
u a.e. as j
->
+00.
> 0 such that for any 1/J E E and u E N, there
ds (cp, 1/J) < C l u - 7r('Ij!) I .
Proof of claim 1 . Since N is compact, there exists K c E compact such that 7r(K.) = N (cover N by finitely many closed balls trivializing the covering 7r ) . Let be given u and 'Ij!. There exists h E 7rl (N) such that14 h · 'Ij! E K. If lu - 1T(1/J) 1 > 1], then let cp' E K. be such that 7r(cp') = u, and define cp h- 1 . cp' E c. Then _
IT(cp) = 1l'(cp' )
= U,
and since h acts isometrically and h . 1/J, cp' belong to K.,
de (cp, 1/J) = ds (h - 1 cp', 1/J) = ds (cp' h 1/J) < diam(K.) :::; CT) :::; Cl u .
l1h
.
1jJ denotes the action of h
E
1l']
(.�
,
on
.
1jJ
E E.
..
-
1l'(1/J ) I
147
SOME QUESTIONS
provided C 2': diam(K)1] - I . If Iu - 1f ( 1,b) I < 1], then 1f is an isometry from Be: ('P, 1]) onto BN(U, 1]), of inverse a , and it suffices to take 'P = a(u), so that
and the proof of the claim is finished. Let 'Po E E be such that 1f( 'Po) = Uo. We then define 'Pj E Xj by induction. Assume 'Pj E Xj is constructed, and consider Qj E Pj and Qj + ! E Pj + l , Qj + ! C Qj . Claim 1 applied with 1,b = 'Pj ( Qj) (so that 1f(1,b) = 1f ('Pj ( Qj)) = Uj ( Qj)) and u = Uj + l ( Qj + d gives us a 'P E E, and we set 'Pj + ! ( Qj + l) = 'P. Then, we have 'Pj + l E Xj + ! and (A.8) Claim 2. There exists ( O, I ) N ,
C = C(E , N)
> 0 such that for any j E N* and a.e.
m
IUj - Uj - 1 1 :::; C(lu - Ej (u) 1 + lu - Ej- 1 (u)l). Proof of claim 2. If Ej (u) � No (or Ej_ 1 (u) � No) , then l u - Ej (u) 1 2': 8 (or lu - Ej_ 1 (u) 1 2': 8) , hence IUj - Uj - 1 1 :::; diam(N) :::; Clu - Ej (u) 1 (or :::; Clu Ej - 1 (u) 1) provided C 2': diam(N)8 - 1 . If Ej(u), Ej_ 1 (u) E No, then we have, since
p
is lipschitzian,
so the proof of the claim is complete. As a consequence, one has by (A.8), claim 2 and taking the 00
LP norm 1 5
00
j L 2 SP dfp ( [l , e:) ('Pj , 'Pj - l) :::; C L 2spj I IEj (u) - ull fp · j= 1 j =O From Theorem A . I in Appendix A of [5] , we have 00
L 2spj I I Ej (u) - u W �,p :::; Clul fvs,p , j =O thus I:� 1 2Spj df p ([l e: ) ('Pj , 'Pj - d converges and we infer that ('Pj) is Cauchy , LP(O, E) , so that there exists 'P E LP(O, £) such that 'Pj 'P in LP(O, £) j +00. As in [5] , we now prove that 'P E WS ,P(O, E) and (A.9)
----+
in as
----+
(A. IO)
l (n lPW5,p r
=
1 1[lx[l
de: ('P (x), 'P(Y))P dxdy < C � 2Spj l E - (u) - uI P < CluiP . l I LP Ws,p - L I x Y I sp+ N j =1 _
J
We propose a direct proof of (A.IO), based on the following claim.
15dLP (f!,e) denotes the natural distance in LP(O, £) defined by dLP(f!,e) (ip, 'IjJ)P If! de (ip(x), 'IjJ(x))P dx.
_
148
FABRICE BETHUEL AND DAVID CHIRON
Claim 3. There exists GI, G2 E Pj ,
C=
C ( s, p, N) such that for every j E N and every cubes
(A . 1 1 )
Assuming claim 3 for a moment, we complete the proof of ii) when sp < 1 . Summing (A. Il) over all cubes C] , C2 E Pj gives, using t:laim Z for the last inequality,
-- 1 £
de (CPj + I (X ) , CPj +l (Y» P d d I CPJ. +l IPW··P X Y I I ."L - Y N+.p . . fl x ll de ( cpj (x) , cPj ( Y» P dxdy + 2C2spj r I U x) - U x) IP dx < r ( .( + + N sp J In J l Jflxn Ix - yl < l'Pj lfv,.p + G'Zspj I l lu - Ej(u) IP + I u - Ej +I(U) IP dx,
J
i
where the com;tant C' depends only on N, p and infer for k E N ,
k+ l ICPk l�v" p < G L 2Spj I IEj (u) - uW;',p j =O
Since
1 ) , Un has a lifting ° uniformly for x E (0, 1 ) . Therefore, we are left with the case ° < s < 1 sp < p, for which we are in a VMO type1 6 embedding w s ,p c VMO. In I, we bave ly - z ls+ ! « 2c) � since sp = l . With -->
-->
->
->
=
s+
F (y , z) = dd¢ (y) , ¢ (z)) E P « O " 1 ) 2 ) I Y Z I 1.I> -
16However, the definition of VMO((O, 1 ) , £) should be precised, since [; is not endowed with the euclidean distance.
• ·
• •
• • ·
,
SOME QUESTIONS
we then infer by Holder inequality (with p' = OS
(2€) �
IS ( 2€) 2
s
l l x+c
X-c
x+c
X-c
(2€)2( � -1) (2€) :r
151
� E (1, 00) such that � + ? = 1)
F( y , z) dydz
(1�:c 1�:€ FP(y, z) dYdZ) � = (1�:€ 1�:€ FP (y, z) dYdZ) �
as € � ° uniformly for x E (0, 1) since FP E L 1 . Therefore, f E VMO((O, 1), JR) and it follows from [7] (Section 1.5) that the essential range f((O, 1)) of f is connected. Since 7l" 0
0. Since {x E (0, 1), hx = h} has positive measure, it follows that f == 0, that is 'IjJ = h .
3. In particular in 3D, the equation ( 1 .5 ) is sub critical for p < 6. The main ingredient in the proof of ( 1 .4) are refined versions of Strichartz' theory for the linear Schrodinger group eitf', (which are of course also essential to the Cauchy problem) . Unlike the Zakharov-Shabat ID cubic NLS
( 1.6) equations ( 1 . 1 ) , ( 1 .2) are non-integrable and it is generally believed that weak tur bulence may occur (one may in fact come up with examples of NLS with smooth nonlinearity that do exhibit the phenomenon, but no example with a local polyno mial nonlinearity seems known) . An issue closely related to weak turbulence is the question of abundance of time quasi-periodic and almost-periodic solutions in phase space. Again, since ( 1 . 1 ) , (1.2) are non-integrable, we certainly do not expect every classical solution of (1.1) or ( 1 .2) to be almost periodic. Recall that if u(t, x ) is a solution and u : lR � H8 (']fd ) is almost periodic as a function of time, then surely SUPt I lu(t) II H' < 00. Over the last decade, lots of work has been done on the construction of quasi-periodic solutions for NLS both in I D and D > 2. A quasi-periodic solution corresponds to a finite-dimensional invariant torus in phase space (its dimension is the number of frequencies) . Two methods have been used to study the problem (involving an infinite dimensional phase space). The first is an adaptation of the classical KAM theory to the PDE-setting (see in particular [10]). The second is a direct applica tion of a Nash-Moser type scheme with a control on the linearized (non-diagonal) equation through more sophisticated small divisors analysis (which is reminiscent to the theory of localization for quasi-periodic lattice Schrodinger equations, cf. [2] ) . This second method applies also beyond the Hamiltonian context and has the appearance of more general implicit function theorems involving small divisors. Among the several expository and reference works, we mention [10], [11] , [8],
[2] , [13] .
Much less has been done an almost periodic solutions ( infinite dimensional invariant tori) for Hamiltonian PDE's (putting the integrable cases such as KdV, KPII or I D cubic NLS aside) . The papers of J. Poschel (see [15]) and the author (see [3) ) , present constructions for ID NLS and I D NLW ( nonlinear wave equation) for a full set of frequencies. Both arguments are based on the idea of consecutive perturbations of finite dimem,ionaJ tori, eventually producing an invariant torus of
NORMAL FORMS AND THE NONLINEAR SCHRODINGER EQUATION
1 55
full dimension, but with very strong compactness properties (in fact the decay of the action variables is hardly explicit ) . These constructions surely do not address in any way the issue of 'abundance', even not in the real algebraic category. In recent work [4] , the author succeeded in carrying out the KAM sheme in the context of a ID NLS
(1.7) where P(t) is an arbitrary polynomial and M is a bounded Fourier multiplier pro viding an extra source of parameters. Thus
(1. 8 )
M¢(x)
=
¢ (n)Mn e27rinX L n EZ
and the Mn are parameters. In this setting, invariant tori of full dimension are constructed, which are 'abun dant' in the real analytic or even Gevrey category. We follow the 'standard' normal forms procedure as worked out in [14J for short range interactions (notice that the nonlinearity in (1.7), expressed in Fourier modes, is not short range) but the analysis here is significantly different. It involves the fine features of the model, including the behavior of the spectral asymptotics {n 2 l n E Z} and the form of the nonlinearity. We do not know at present how to carry out a similar argument in 2D. A key issue is of an arithmetical nature and may be explained as follows. The system of equations
{ nnil -- nn�2
= a i=- 0 =b
n l , n 2 in ID (i.e. a, b, n l , n2 E Z) but not in 2D. PROBLEM 1.2. Construct almost periodic solutions for the analogue of (1.7) in
determines
2D.
The problem may be more than just technical. Of course, the other issue is to dismiss the Fourier multiplier M. This forces us to extract parameters from the nonlinearity by amplitude-frequency modulation. In lD with nonlinearity of the form iUt + Uxx + mu + u l u l 2 + ( higher order) = O. (1.9) this was achieved in [12J in the construction of finite dimensional tori. In the present problem where an infinite dimensional invariant torus is pro duced, the major difficulty is that on one hand the iterative scheme requires a suffi ciently fast decay of the action variables In = I qn 1 2 (qn = ,;;(t) ( n)) ; but if the action variables are small, so is the frequency modulation and the small divisor problems become worse. PROBLEM 1.3. Construct invariant tori of full dimension for (1.9) . The paper [5J illustrates the difficulty brought up above, where it is to some extent resolved in the more modest attempt of establishing Nekhoroshev (=long time) stability properties for small solutions of (1.9) . It is shown in [5J that if Ilu (O) II H s
< C
J. BOURGAIN
156
(s large) and u(O) is 'typical' in some sense, then in particular lI u(t) II H' < 1 for I t I < To C )A, where A A(s) .�';" 00 . =
=
In the more recent papers [6] , [7] we started exploring the application of normal forms to NLS for coarser topologies. A first task here is to understand the precise mapping properties on HS-spaces of the sympleetie transformations involved in the Birkhoff normal forms reduction of the nonlinearity to a 'resonant part' and a 'small' remainder term (recall that the symplectic space of NLS expressed in Oarboux coordinates is L2) . The description 'resonanL' refers Lo the ull-IIlodulated frequencies {n2 ln E Z} (in 10). Thus a monomial q"l qn2 . . . q"2. - 1 q"2. ' where nl n2 + . + n2s - 1 n 2. = 0 is called resonant if -
.
-
.
( 1 . 10)
(removal of such terms requires modulation of the n2-frequencies, for instance us ing a Fourier-multiplier M as in (1 .7) and leads to the 'small-divi::;Uf::;' analysis encountered in the construction of 'true' invariant tori) . We cite two POE results that came out of this analysis. Both related to equation (1.1). •
THEOREM
1 .4. �6]). Let
s > 1 and u the solution of the Cauchy problem =
i.ut + Uxx u l ul4 u(O) E W (1J'). -
Then
,-,; - 1
l I u ( t ) I I H' « t · for t
The second is THEOREM
is
a
-+
00.
low regularity result.
1 .5. ([7]).
Cauchy problem (1.11)
u (x, t) = u (x + 1 , t )
0
There is
a
Sobolev exponent 0 < So
80 -
globally wellpo8ed and u(t)
E
1 2
such that the
n(:c, t) = n (x + 1 , t)
HS for all Lime.
Recall that ( 1 . 1 1 ) is locally wellposed for all 8 > 0 (see [1] ) . The claim is thus that this local solution extends to a global one for 8 > 80 (it is unknown if we may take 8 0 = 0) . The interest of the second theorem is that it provides a direct proof of the uniquely defined invariant dynamics on the support of the (formally invariant) Gibbs measure introduced in the work of Lebowitz, Rose and Speer (1.12)
where ( 1 . 1 3)
H (4))
=
. 11'
(1\74>12 + p2 1cPI2 + 1 4>16)
(see again [1] and relevant reference::;).
i ,
,
,
•
NORMAL FORMS AND THE NONLINEAR SCHRODINGER EQUATION
157
The measure ( 1 . 12) is indeed absolutely continuous with respect to the 'free' measure induced by the Gaussian process ( 1 . 14) where
g e21Cinx L Jnn(w) + p2 2 n EZ
{gn l n E Z} are independent normalized complex Gaussian random variables. References
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
J. Bourgain, Nonlinear Schrodinger equations, in Hyperbolic equations and frequency in teractions, ( Park City, UT, 1995) , 3-157, lAS/Park City Math. Ser. , 5, Amer. Math. Soc., Providence, RI, 1999. J. Bourgain, Green's function estimates for lattice Schrodinger operators and applications, Annals of Mathematics Studies, 158, Princeton University Press, 2005. J. Bourgain, Construction of approximative and almost periodic solutions of perturbed linear Schrodinger and wave equations, Geom. Funct. Ana!. 6 (1996), no 2, 201-230. J. Bourgain, On invariant tori of full dimension for lD NLS, J. of Funct. Ana!., to appear. J. Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDE, J. Ana!. Math,
80 (2000), 1-35.
J. Bourgain, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations, Ergodic Theory Dynam. Systems 24 (2004), no. 5, 1331-
1357.
J. Bourgain, A remark on normal forms and the "I-method" for periodic NLS, J. Ana!. Math.
94 (2004), 125-157.
W. Craig, Problemes de petits diviseurs dans les equations aux derivees partielles, Panoramas et Syntheses, 9, Societe Mathematique de France, Paris, 2000. H. Brezis, T. Gallouet, Nonlinear Schrodinger evolution equations, Nonlinear Ana!. 4 ( 1980) , no 4, 677-681. S. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993. S. Kuksin, Fifteen years of KAM for PDE. Geometry, topology, and mathematical physics, Amer. Math. Soc. Trans!. Ser 2, 212, Amer. Math. Soc, Providence, RI, 2004. S. Kuksin, J. Poschel, Invariant Cantor manifolds of quasi-periodic oscillations for a non linear Schrodinger equation, Ann. of Math. (2) 143 ( 1996), no 1, 149-179. T. Kappeler, J. Poschel, KAM and Kd V, Springer, 2003. J. Poschel, Small divisors with spatial structure in infinite-dimensional Hamiltonian systems, Comm. Math. Phys. 127 ( 1990) , no 2, 351-393. J. Poschel, On the construction of almost periodic solutions for a nonlinear Schrodinger equation, Ergodic Theory Dynam. Systems 22 (2002), no 5, 1537-1549. INSTITUTE FOR ADVANCED STUDY, PRINCETON, NJ E-mail address: bourgain ). * . In both the elliptic and parabolic settings, we describe recent results concerning the existence, uniqueness, regularity, and stability of solutions for the extremal problem at ).* . Two phenomena are studied in detail: an apparent "failure" of the Implicit Function Theorem at the extremal problem, and the complete blow-up (instantaneous complete blow-up in the parabolic case) of approximate solutions for )' > ). * .
1 . Introduction This article is concerned with several semilinear elliptic problems of the form g>. (x, u) in 0 (1 . 1 ) o on 80, that admit a minimal (or smallest ) solution for each A � A * , where A * is a certain extremal parameter, and that have no solution for every A > A* . The minimal solution u* corresponding to A = A* is called the extremal solution. An example of this situation is given by Gelfand's problem
{
-D.u u
Aeu 0
in 0 on 80,
and more generally by (1 . 2 )
Ag(u) o
in 0 on 80,
2000 Mathematics Subject Classification. Primary 35J60, 35K55, 35B35, 35B40, 35R25, 35D05. Key words and phrases. Nonlinear elliptic and parabolic problems, extremal solutions, regu larity of solutions, non-existence of very weak solutions, instantaneous complete blow-up. Supported by MEC Spanish grant MTM2005-07660-C02-0l , the RTN Program Front Singularities HPRN-CT-2002-00274, and by the European Science Foundation PESC Programme "Global" . ©2007 American Mathematical Society
159
XAVIER CABRE
160
whenever 9 > 0 is increasing and convex 011 ]R and superlinear at + 00 in the sense (2.2) stated helow. Under these assumptions, the extremal parameter of (1.2) satisfies 0 < ,\* < 00. Here and throughout the paper, !1 is a smooth bounded domain of ]RN . For each 0 < ,\ < >. * , the minimal solution of (1. 2) is classical. Its limit as >. r ,\* is the extremal solution u' , which may be singular (i.e., unbounded) for some dimensions, nonlinearitiet>, and domains. \Vhen g(u) = e", it is known that u · E LOO(n) if N < 9 (for every !1), while u* (x) = log(1/lxI 2 ) if ::::: 1 0 and !1 = B1 • Brezis and Vazquez [BV] raised the question of determining the regularity of u' , depending on the dimen sion N , for general nonlinearities 9 as above. Below we explain recent results of Nedev [NI , N2] and of the author and Capella [CCa2] on this delicate regularity issue. While nonexistence of elassical solutions for ,\ > ,\* follows by the definition of >" , Brc�is, Cazenave, Martel, and Ramiandrisoa [BCMR] proved that, in addition, no weak solution of (1 .2) exists for >. > >.* . L ater Brezis and Vazquez [BV] characterized singular extremal solutions of ( 1 .2) by means of a stability condition. FOT some equations in dimension N > 1 1 , they also show that an app uent "failure" of the Implicit Function Theorem occurs at ( u* , >. * ) , since the linearized problem at · u turns out to be formally invertible (for some problems) and, however, no weak solution exists for ,\ > >" . This article is a survey of several results and methods from [BC, BCMR, BV, CCa2, CMal, CMa2, Mal, NI, N2, PV] concerning the previous issues, both in the elliptic and the parabolic settings. Special attention will he made on the regularity of the extremal solution u· , as well as on how nonexistence develops for >. > >" . We will see that this last question is related to a phenomenon of complete blow-up of approximate solutions, and of instantaneous complete blow-up in the parabolic setting. Sections 2 and a are devoted to existence, nonexistence, uniqueness, stability, and boundedness of solutions of ( 1 .2). We describe some re�ults of [BCMR, BV, CCa2, CMal, Mal , NI, N2] ahout weak solutions of ( 1.2). We are particularly interested on the regularity of the extremal solution u ' and on the linearized prob lem of ( 1 .2) at the extremal solution and its connections with the Implicit Function Theorem. Section 4 is concerned with [Be], where Brezis and the author make a detailed analysis of how nonexistence develops for ,\ > ,\. , and also of the apparent "failure" of the Implicit Function Theorem, for the simple model
N
,
,
(1 .3)
-6.71,
u
U
2
Ixl 2
o
+ ,\
'
in !1 on 8!1,
in all dimensions N > 2. Here 0 E !1, u· = 0, and >. * = O. Indeed, it. is proved in [Be] that, for every >. > 0, (1 .3) has no solution in a very weak sense. Moreover, approximate solutions of ( 1 .3) blow-up everywhere in !1, i.e., there is complete blow-up in the sense established by Baras and Cohen [BaCo] for Home parabolic problems. In Section 5 we present, related nonexistence and blow-up results obtained in [Be] for the parabolic version of (1.3) . They extend, in particular, a result. of Peral
-
161
EXTREMAL SOLUTIONS AND BLOW-UP
{
and Vazquez [PV] . Namely, the problem
Ut flu u u(O) log (1/ l xj 2 )
(1.4)
-
= =
2(N - 2)
e
0
U
Uo and Uo =t u, =
in (0, T) x B1 on (0, T) x aB1 on B1 ,
with Uo 2': u := has no weak solution u(t, x) 2': u(x) even for small time: instantaneous complete blow-up occurs. To prove this result we will consider the equation satisfied by u u. It leads to the parabolic version of problem (1 . 3 ) . The potential I x l - 2 also appears when linearizing problem (1.4) at u. Hence, the study of the linear problem Ut flu = >"lxl - 2 u is also of interest. In Section 6 we discuss the classical work of Baras and Goldstein [BaG] on this equation, as well as results of the author and Martel [CMa2] on linear parabolic equations with singular potentials: -
-
{
(1.5)
Ut flu u u(O) -
a (x)u 0
= =
Uo
=
in (0, T) x D on (0, T) x aD on D,
with a 2': 0 and Uo 2': 0 in Lfoc (D) . We give conditions, based on the validity of a Hardy type inequality with weight a (x) , that ensure either the existence of global weak solutions or the instantaneous complete blow-up of approximate solutions of (1.5). Replacing a (x) by >..a (x) in (1.5), the extremal parameter >..* can be explicitly computed for some critical potentials a (x) This work extends the result proved by Baras and Goldstein [BaG] in 1984. Namely, if a (x) = >" lxl - 2 , 0 E D, and N 2': 3, then >.. * = (N - 2) 2 /4. More precisely, (1.5) has a global weak solution for every Uo E L 2 (D) and every 0 :::; >.. :::; >..* = (N - 2) 2 /4. Instead, if >.. > >.. * = (N - 2) 2 /4 then (1.5) has no positive weak solution for every T > 0 and every Uo E Lfoc(D) with Uo 2': 0 and Uo =t O. .
The paper is organized as follows: 1. Introduction. 2. The extremal solution. 3. An apparent "failure" of the Implicit Function Theorem. 4. Nonlinear elliptic problems: very weak solutions. 5. Nonlinear parabolic problems. 6. Linear parabolic problems with singular potentials. 2. The extremal solution We start considering the problem
(2.1>J
{ -fluu
= =
>..g (u) o
in D on aD,
where >.. 2': 0 is a parameter, and D is a smooth bounded domain of JRN , N 2': The nonlinearity 9 is a C 1 , increasing and convex function on [0, (0 ) such that
g(u) +00. U Since 9 > 0, we consider nonnegative solutions of (2.1>.). The cases g(u) g(u) (1 + u)P, with p > 1, are classical examples of such nonlinearities. (2.2)
g(O) > 0
=
and
lim
u-->+oo
2.
=
= e
U
and
16 2
XAVIER CABRE
These reaction-diffusion problems appear in numerous models in physics, chem
istry, and biology. The case of an exponential reaction term
g(u) = eU is a very Since the work of Gelfand [G] in the sixties,
simplified model in combustion theory.
problem (2.1,), ) has heen extensively studied.
2.1. Minimal stable solutions. It is well known that there exists a param eter A* with 0 < ..\* < 00, called the extremal parameter, such that if 0 < A < "\" then (2.1,),) has a minimal classical solution u,),. Here, by "minimal solution" we mean that u,), is smaller than every other solut.ion or supersolution of (2. 1,), ). On the other hand, if ..\ > ..\ * then (2.1,),) has no classical solution. The set { u,), ; 0 < A < A' } forms a branch of solutions increasing in A. More over, every u,), is a stable solution, in the sense that the first Dirichlet eigenvalue of the linearized problem at u,), is positive: 11'1 {-.6. - >.g' (u,),}; O} > o. In particular, for the quadratic form associated to the linearized problem we have
(2.3)
Qu,, (r.p} :=
10 1\7r.p1 2 - 10 ..\g' (u,), )r.p2 > 0
for every r.p
This condition, that we refer as the "semi-stability of
u,),
E HJ (O).
, is equivalent to /1 1
where /11 is the first eigenvalue of the linearized problem above. Recall that the first Dirichlet eigenvalue in
-.6. - a (x )
0
is defined by A /1 1 {' -u
(2.4)
-
a(
x ) ,. H "}
-
_
.
mf
O;;E 0,
of a linear operator of the form
In I \7r.p I
2
2 a(x}r.p In In r.p2 -
.
Later, we will use this expression as definition of generalized first eigenvalue when
E
L�oc (O), - 00.
a
a
> 0 a.e"
and a is singular. In this case, /11 {-.6.
- a(x}j O}
could be
{ u,), ; 0 :S ..\ < ..\*} can be proved using the Implicit FUnction Theorem (starting from ..\ = 0). The solution u,), may also be obtained by the monotone iteration procedure, with 0 < >. < >" fixed, starting from u = 0 (note that u = 0 is a strict subsolution of the problem ) . These results, among others concerning (2.1,),), are proved in [CrR] and [MP ] . The existence of the branch
2.2. Weak solutions.
The increasing limit of
tremal solution U · . It is proved in
..\ =
>. *
in the following sense:
DEFINITION U E
for all
We say that
2.1.
LI (O),
( E C2 (0)
with
to the boundary of
[BCMR]
=
0
on
ao,
as ..\ i >. * is called the ex
is a weak solution of
u*
is a weak solution of
u
g(u)8 E Ll(O), (
that
u,),
u.6.( =
and
where
(2. 1 >J
8(x)
n
= dist (x,
80)
n
(2. 1,),)
for
if
>.g(u)(
denotes the distance
0.
More generally, the definition of weak solution for problem
one: we now assume
in the above integral.
g,). (x, u(x))8(x)
E
Ll (O)
The nonexistence of classical solution for
Cazenave, Martel, and Ramiandrisoa
and we
A > A* [BCMR] :
( 1 . 1 ) is the analogous replace >.g(u)( by g,),(az, u)(
has been improved by Brezis,
EXTREMAL SOLUTIONS AND BLOW-UP
THEOREM 2.2 ( [BCMR] ) .
For every A > A* ,
(2.1>.)
1 63
admits no weak solution.
In case g(u) = eU, a similar result had been obtained by Gallouet, Mignot, and Puel [GMP] . The proof of Theorem 2.2 given in [BCMR] uses a new and interesting method. One assumes that (2.1>.) has a weak solution for some A > A*. One then considers the equation satisfied by 1>( u), where 1> is a positive, bounded, and concave function which is chosen appropriately depending on the nonlinearity g. It is possible to construct in this way a bounded supersolution (and hence a classical solution) of (6( 1 - 0)>')' for each 0 < E < 1 . But this is a contradiction if E is small enough to guarantee ( 1 - E)A > A * . This procedure could be called the "method of generalized truncations" 1>(u) of u. It is also used in [BCMR] to study the global existence and the blow-up in finite time of solutions to the evolution problem Ut - flu = Ag(U ) , and in [Ma2] to prove results on complete blow-up of solutions. A refined version of the method is also used by Martel [Mal] to show the following property of extremal solutions: THEOREM 2.3 ( [Mal] ) .
u* is the unique weak solution of (2.1;. . ) .
2.3. Regularity of the extremal solution. The extremal solution u * may be classical or singular depending on each problem. For instance, for Gelfand's problem we have: THEOREM 2.4 ( [CrR, MP, JL] ) . Let u* be the extremal solution of (2. 1>.) . (i) If g(u) = eU and N :S 9, then u* E L oo (0.) (i. e., u* is classical for all 0.) . (ii) If 0. = B1 (the unit ball), g( u) = eU, and N � 10, then u* = log ( 1/ lxI 2 ) and A* = 2(N - 2) . There is an analogous result for g(u) = ( 1 + u)P. In this case, the explicit radial solution is given by Ixl - 2 / (p- 1 ) 1 and coincides with the extremal solution u* when 0. = Bl for large values of N and p (see [BV, CCa2] for more details). Part (i) of Theorem 2.4 was proven by Crandall and Rabinowitz [CrR] and by Mignot and Puel [MP] . The proof uses the stability of the minimal solutions, as follows. One takes cp = eC>U>' - 1 in the semi-stability condition (2.3) . One then uses that u>. is a solution of (2. 1>.). This leads to an L 1 (0.) bound for ef3u>. , uniform in A, for some exponents (3. That is, -flu>. is uniformly bounded in Lf3 (0.) and, hence, u>. is uniformly bounded in W 2 ,f3(0.). If N :S 9, the exponent (3 turns out to be bigger that N/2 and, therefore, we have a uniform L oo (0.) bound for u>.. Part (ii) of the theorem had been proved by Joseph and Lundgren [JL] in their exhaustive study of the radial case using phase plane analysis. More recently, Brezis and Vazquez [BV] have introduced a simple approach to this question, based on the following characterization of singular extremal solutions by their semi-stability property: -
THEOREM 2.5 ( [BV] ) . Let u solution of (2.1>.) for some A > O.
HJ (0.), u rt L oo (0.), be an unbounded weak Then, the following are equivalent :
E
(a) The solution u satisfies (2.5) Ag'(u)cp2 � 0 IV'cpl 2
In
- In
(b) A = A* and u = u* .
for every cp E HJ (0.).
XAVIER CABRE
164
That (a) holds for u' follows immediately from the semi-stability (2.3) of miIl imal solutions, by letting .\ r '>" . The key idea behind the other implication of the theorem is that, for each .>. > 0, (2.h) has at most one semi-stable solution -a consequence uf the convexity of g. Following [BV] we can deduce part (ii) of Theorem 2.4 from Theorem 2.5. Indeed, let 0 = B 1 and u = log(1/lxI2). A direct computation shows that this function is a solution of (2.1>.) for .>. = ,\ = 2(N 2). The linearized operator at u i� given by 2(N 2) £r.p = t:J.r.p 2(N - 2) e r.p = - t:J.r.p r.p. 2 I xl H N > 10 then the first eigenvalue of L in B1 "atisfies P1 { L; Bd > O. This is a consequence of Hardy 's ine.q1tality: ,
-
-
(N - 2) 2
1
B,
u
-
r.p2 < Ix l 2 -
1V'r.p12
-
-
for every r.p
H3 (B1) ,
B, and the fact that (N - 2)2/4 > 2(N 2) if N > 10. Applying Theorem 2.5 to (u, .>.) we deduce part (ii) of Theorem 2.4. Before proceeding we recall that, by scale invariance, Hardy's inequality also holds in every domain 0 C jRN with N > �t Namely, (N - 2)2 r.p2 2 for every r.p E Hci (O). (2.6) < 1V'r.p1 4 in fl Ixl 2 Moreover, if 0 E 0 then the optimal constant is (N 2)2/4 and it is not attained in JIJ (O); see [BV] . Theorem 2.4 is a precise result on the boundedness of u' when g( u) = e U • A related regularity result from [BV] states that if the nonlinearity satisfies in addition lim inf7l->oo ug'(u)/g(u) 1 , then u· E HJ (O) for all 0 and N. However, to establish reglilarity of u' without additional assumptions on 9 a question raised by Brezis and Vazquez [BV]- is a much harder task. The best known results for general domains and nonlinearitiet:> (within the hypothese made at the beginning of Section 2) are due to Nedev [NI, N2]: 4
E
-
r
-
>
THEOREM 2.6 ( [NI, N2] ) . Let u* be the e:r,tremal solution of (2 . h ) . (i) If N < 3, then u· E £= (0) (for every 0) . (ii) If N < 5, then u· E HJ (0) (for every 0). (iii) For every dimension N , if 0 is strictly wrwex then u* E H6 (O) .
The proof of this t.heorem uses a very refined version of the method of [erR, MP] described above in relation to Theorem 2. 4 . There are still many question;; to be answered in the case of general 0 and g. For instance, it is not known if an extremal solution may be singular in dimensions 4 S N < 9, for some domain and nonlinearity. However, the radial case 0 = Bl has been recently settled by the author and Capella [CCa2] . This work does not use phase plane analysis, but instead PDE techniques. The result establishes optimal regularity results for general g. To state it, we define exponents qk fur k E {O, 1 , 2, 3} by 1
-
(2.7)
=
qk qk
=
1
2
-
-
+ 00
.IN - l k - 2 +� N N.
for N > 10 for N < 9.
-
EXTREMAL SOLUTIONS AND BLOW-UP
165
Note that all the exponents are well defined and satisfy 2 < qk :'::: +00. As in Theorem 2.6, the following result holds for every nonlinearity 9 satisfying the as sumptions at the beginning of Section 2. THEOREM 2.7 ( [CCa2] ) . Let n = B 1 and u* be the extremal solution of (2.1).,). (i) If N :'::: 9, then u* E L= (Bd . (ii) If N = 10, then u* (lxl) :'::: C log(I/lxl) in B1 for some constant C. (iii) If N ;::::: 1 1 , then
u* ( l xl) :'::: C ! X I- Nj 2+ v'N - 1 +2 Jlog(I/lx l ) in B1 for some constant C. In particular, u* E Lq(Bd for every q < % . Moreover, for every N ;::::: 11 there exists PN > 1 such that u* tI- Lqo (B1) when g( u) = (1 + U) PN (iv) u* E W k ,q(Bd for every k E { I , 2, 3} and q < qk ' In particular, u* E H3 (Bd for every N . Moreover, for all N ;::::: 10 and k E { I , 2, 3 } , 1 8k u * (lxl) 1 :'::: C !xl- Nj 2 + v'N - 1 +2 -k (1 + Jlog(I/lxl)) in B1 for some constant C. Here 8k u* denotes the k-th derivative of the radial function u* . Theorem 2.4, which deals with g( u) = eU, shows the optimality of Theo rem 2.7(i) (ii) , including the logarithmic pointwise bound of part (ii) . The Lq regu larity stated in part (iii) (q < qo ) is also optimal. This is shown considering 9 ( u) = (1 + U)PN (for an explicit PN ), in which case u* (lxl) = I x l - Nj2+ v'N - 1 +2 - 1 . This function differs from the pointwise power bound (2.8) for the factor Jlog(I/lxl). It is an open problem t o know i f this logarithmic factor i n (2.8) can b e removed. (2.8)
•
The exponents qk in the Sobolev estimates of part (iv) are optimal. This follows immediately from the optimality of qo and the fact that all qk are related among them by optimal Sobolev embeddings. The proof of Theorem 2.7 was inspired by the proof of Simons theorem on the nonexistence of singular minimal cones in ]RN for N :'::: 7. The key idea is to take cp = (r - /3 - 1 )ur , where r = l xi, and compute Qu * ((r - /3 - 1)ur) in the semi-stability property (2.3) satisfied by u*. The nonnegativeness of Qu* leads to an L 2 bound for urr - a, with 0: depending on the dimension N. This is the key point in the proof. A similar method was employed in [CCal] to study stability properties of radial solutions in all of ]R N . The method originates from certain relations between semilinear equations and minimal surfaces studied recently in connection with a conjecture of De Giorgi (see [CCa2] for more details) . Theorem 2.7 has been extended in [CCaS] to equations involving the p-Lapla cian and having general semilinear reaction terms. Previously, Boccardo, Escobedo, and Peral [BEP] had extended Theorem 2.4 on the exponential nonlinearity to the p-Laplacian case in general domains. See [CS] and references therein for recent extensions of Theorem 2.4 to more general nonlinearities in the p-Laplacian case and in general domains. 3. An apparent "failure" of the Implicit Function Theorem 3.1. Hardy'S inequality and the Implicit Function Theorem. As a con sequence of (2.3) , we know that the extremal solution is semi-stable, i.e., it satisfies ILl {-�-A*g' (u*); n} ;::::: O. If u* is classical then necessarily ILl { - � -A * g'(u*); n} =
166
XAVIER CABRE
O. This is an immediate consequence of the Implicit Function Theorem and the
>
nonexistence of classical solutions of (2.1,),) for each A A*' Brezis and Vazquez [BVI point out that the property jjd-� - A*g' (U' ) i Q} = 0 fails for some problems in which the extremal solution u' is singular. They also show that jjd -� - Ag'(u,), ) ; Q} may decrease to a positive number as A r A* . An example of this situation is given by -�u u
(3.1)
= =
Ae" 0
in Bl on OBI ,
in every dimension N > 1 1 . Indeed, we know that for this problem 'U' = log(1/lxI2) and A* = 2(N - 2). The linearized operator at u' is given by L = -� 2(N 2) l x l -2 , which has positive first eigenvalue in Bl if N > 1 1 , by Hardy's inequal ity (2.6). We conclude that, for N 1 1 , the linearized operator of (3. 1 ) at u * is bijec l tive, for example between HJ (Br) and H- (Bl ). However, the Implicit Function Theorem can not be applied to problem (3.1) at (u' , A'), since there are no weak solutions of (3. 1 ) for A > A', by Theorem 2.2. Similarly, the Inverse Function Theorem can not be applied neither, since it is proved in [BCMRI that, for each A 0, problem -�u = 2(N - 2)eU + A in BI , U = 0 on OBI admits no weak solution if N > 10. This apparent contradiction is explained by the absence of appropriate func tional spaces in which one could apply the Imp li cit or Inverse Function Theorems. -
>
>
,
3.2. Weak eigenfunctions. Motivated by this apparent "failure" of the Im plicit Function Theorem, the author and Martel [CMal] have shown that the linearized problem of (2.1,),") at 'U" always admits a positive weak eigenfunction belonging to Ll (Q) with eigenvalue 0, even for the problems in which this operator has positive first eigenvalue in HJ (Q). The fact that 0 is a weak eigenvalue is one explanation of the apparent "failure" of the Implicit Function Theorem at u· . THEOREM
3.1 ([CMal]). There exists a f'Unction 'P > 0 in Q such that in n on aQ,
-�'P = A' g' (u* )'P 'P = 0
(3.2)
in the weak sense of Definition 2.1, i.e., and
In
'P�(
=
n
A*g' (U* )'P(
for all ( C2 (Q) with ( = 0 on aQ. We recall that 8(x) = dist(x, (0) . Moreover, 'P E U(O) for every 1 < q < N/(N - 2).
E
-
We also prove the existence of a positive weak eigenfunction of the operator -� - A"l(u') with eigenvalue f.l, for every f.l such that o < f.l < lim" Il-l{ -� - Ag' (U,), ) ; O}: A T>-
Hence, there is a phenomenon of continuum spectrum if this limit is positive and, in particular, if jjl { -� - A"l (u" ) ; O} O . To prove Theorem 3.1, we approximate 9 by an increasing sequence of convex functions gn for which the extremal solution u�, is classicaL For this, it is enough to take gn with subcritical growth, by standard regularity theory. We know that f.ll{-� - A�g�(U�)i O} = 0, wherc A;, is the extremal parameter corresponding
>
1 67
EXTREMAL SOLUTIONS AND BLOW-UP
gn' We consider the first eigenfunction 'Pn , which has eigenvalue 0, of -� A�g�(U�), normalized so that II 'Pn ll £l(O) = 1. We show that, up to subsequences, u� and 'Pn converge in LT(O), for every 1 :S r < N/(N - 1), respectively to u* and 'P, with 'P > 0 a weak solution of (3.2) as in the theorem. The key step of passing to the limit as n ---t 00 in the approximate equations is accomplished through an equi integrability result. A similar method was employed by Baras and Cohen [BaCo] to
-
for some nonlinear parabolic problems. Next, we make a detailed study of all weak eigenfunctions of (3.2) whenever o = B1 is the unit ball, g(u) = eU or g(u) = (1 + u)P, with p > 1, and u* is singular. In these cases, we know that u* can be written explicitly, and that the corresponding linearized operator is -� - c lxl 2 , with 0 < c :S (N 2 ) 2 / 4. We express all weak eigenfunctions and eigenvalues of
{
-
- �'P 'P
-
=
=
in terms of harmonic polynomials and Bessel functions. Looking at which of these eigenfunctions belong to HJ (B1 ), we obtain the following improved Hardy inequal ity due to Brezis and Vazquez [BV] : THEOREM 3.2
([BV]). If N 2: 2 then for every 'P E H6 (0),
where H2 > 0 is the first eigenvalue of the Laplacian in the unit ball of �2 , and WN is the measure of the unit ball of �N . 4. Nonlinear elliptic problems: very weak solutions Brezis and the author [Be] have studied in detail how nonexistence develops for A > A * , and also the phenomenon described above in relation with the Implicit Function Theorem, for the following model problem. Let 0 be a smooth bounded domain of �N , N 2: 2, such that 0 E O. For A E �, consider the problem
{
( 4.1)
-�u u
= =
u2 A Ixl2 +
o
in 0 on
aO.
A formal analysis suggests the existence of a small solution of (4.1) for A small, since the linearized operator at the solution (u, A) = (0, 0) is -�, which is bijective. However, we show that for every A > 0 (no matter how small) , (4. 1 ) has no solution in various weak senses. On the other hand, the results of [BS] show that (4. 1 ) has a minimal negative solution for every constant A < O. Hence, for this problem we have u* = 0 and A* = O. 4.1. Non-existence. The nonexistence of generalized solutions of (4. 1 ) for
A > 0 follows from the following result: THEOREM 4.1 ([BC]) . If 0 E 0, u E Lfoc (O \ {O}), u 2: 0 a. e., and -lxl 2 �u 2: u2 in V' (0 \ {O}), then u O. Here V'(O \ {O}) denotes the space of distributions in 0 \ {O} . ==
168
XAVIER CABRE
Theorem 4.1 is proved using appropriate powers of test functions -a method due to Baras and Pierre [BaPJ . An immediate consequence of the theorem is the nonexistence of generalized solutions ( �o called very weak solutions in [BC] ) for the following problem : THEOREM 4.2 ( [BC] ) . If 0 E n , f > 0 a. e., and f =j 0, then - tl.u u
in n on an,
-
has no solution u E Lroc(n \ {O}) in the sense of distributions V'(n \ {O} ) .
Note Lhat, in the previous results, the test functions vanish in a neighborhood of 0, and no assumption is made on the behavior of 'U near O. In particular, these results imply nonexistence of weak solutions in the sense of Definition 2 . 1 in Section 2. A second corollary of Theorem 4.1 is that u = 0 is the only solution of (4. 1) when ).. = 0 (compare this with the uniqueness property of u ' stated in Theo rem 2.3) . A last consequence of Theorem 4.1 is the nonexistence of local solutions (i.e., nonexistence in every neighborhood of D, wiLhout imposing any boundary condition) for a vcry simple nonlinear equation: THEOREM 4.3 ( [BC)) . If 0 E n and )" > 0, then there is no function 'U Lroc (n \ {OJ) such that
E
in V' (n \ {O}).
4.2. Complete blow-up. Next , we study a natural approximation technique for problem (4.1). Consider, for example, the equation -tl. u
with )" > O. For every
n,
min{ u2 n } + ).. in n IxI 2 + ( 1/11,) o on an, ,
Lhere exists a minimal solution
Un.
Vlfe
then have:
THEOREM 4.4 ( [BC] ) . If ).. > 0 then
( )
un x
b(x)
--»
+ 00
uniformly -in x E n, as 11,
-+ 00 .
Hence, approximate solutions blow-up everywhere in n, i.e., there is complete blow-up. This confirms that there is no reasonable notion of weak solution for (4. 1) when ).. > O. The proof of Theorem 4.1 uses the nonexistence result for (4. 1), the mono tonicity property u" < Un+ l , as well as an appropriate lower bound for the Green's function of the Laplacian ill n.
EXTREMAL SOLUTIONS AND BLOW-UP
169
4.3. More general problems. The previous nonexistence and blow-up re sults are extended in [Be] to the problem
{
-�
� : � (x)g(u) + f(x)
in 0 on a�,
g ?: 0 on lR, g is continuous and increasing on [0, 00) , ds < 00, g(s) with a E L fo c (O) , a ?: 0, a(x) dx = 00 x I B,, (O) IN-2 for some 'rJ > 0 small enough, and f E Lfoc (O) , f ?: 0, and f =I'- O. assuming only
1
J
oo
For this problem, we prove nonexistence of solutions in the weak sense of Sec tion 2, using a variant of the method of generalized truncations. Note that here we do not assume g to be convex. 4.4. The existence criterium of Kalton and Verbitsky. We now consider the previous problem in the case of power nonlinearities g, that is
{
= a(x)uP + f(x) in 0 on a�, u = 0 with p > 1, a and f in L foc (O), a ?: 0, a =I'- 0, f ?: 0, and f =I'- 0 in O. (4.2)
-�u
Kalton and Verbitsky [KV] have found an interesting necessary condition for the existence of a weak solution of (4.2) : THEOREM 4.5 ( [KV] ; see also [BC] ) . If (4.2) has a weak solution u ?: 0 (in the sense of Definition 2 . 1 ) , then aG(J)Po E £1 (0) and G(aG(J)P) < _1_ in O . ( 4.3 ) G(J) - p - 1 Here G = (_�) - 1 with zero Dirichlet boundary condition. The explicit constant 1/(p- 1) was found in [Be] . Note that Theorem 4.5 easily
implies some of the previous nonexistence results. For instance, it gives Theorem 4.2 (for weak solutions instead of very weak solutions) whenever f E LOO (O) , since in this case G(J) o. The interest of Theorem 4.5 also lies in the fact that no assumption is made on the set of singularities of the potential a. In particular, a could be singular near some parts of a�. This is in contrast with previous problems considered in this paper, where a had an isolated singularity in an interior point. In [Be] we give a simple proof of condition (4.3 ) using a refinement of the method of generalized truncations described in Section 2. More precisely, we consider the equation satisfied by v1jJ(u/V), where u is a weak solution of (4.2) , v = G(J), and 1jJ is a bounded concave function appropriately chosen. Next, we replace f(x) by Af(x) in (4.2 ) and we study the problem of existence of solution depending on the value of A. We then have: rv
THEOREM 4.6 ( [BC] ) .
Assume that G(aG(J)P) E LOO (O) . G(J)
XAVIER CABRE
170
For A � 0, consider the problem a (x)uP + Af(x)
-�u u
(4.4)
o
in n on an.
Then there exists A' E (0, 00) such that : (i) If 0 < A < A*, then (4.4) has a weak solution U;,. satisfying A
00.
This extends the following result of Peral and Vazquez [PV1 : THEOREM 5 . 1 ( [PV] ) . Let N > 3 . For every Uo > u and for every T > 0, the problem -
(5.2)
Ut - �u u u(O)
2 (N - 2) eU
o
Uo
:=
log (1/IxI2), Uo 'f. u,
in (0, T) x B1 on (0, T) X OBI on B1
has no weak solution u such that u(t, x) > u(x) in (0, T) x BI .
In [BC] we give a simple proof of this theorem by applying the nonexistence results for (5.1) to v : = 'U - U > 0, with u as in Theorem 5. 1 . Indeed, it is easy to check that v satisfies Vt - �v > (N - 2) l x l - 2v2 . We obtain nonexistence of solution of (5 .2) in the distributional sense V' CCO, T) x (B1 \ {O} » . The assumptions on Uo and u in Theorem 5.1 have been improved and studied in detail by Vazquez [Vl .
171
EXTREMAL SOLUTIONS AND BLOW-UP
6. Linear parabolic problems with singular potentials
{
We consider the linear heat equation with potential (6. 1)
Ut - D.. u u u (O)
= a (x) u = 0 = Uo
in (0, T) x 0 on (O, T) x 80 on 0,
0 is a smooth bounded domain of ]RN . We assume that a E Lfoc(O), Uo E Lfoc(O), and that a 2: 0 and Uo 2: 0 a.e. in O. Note that a is time independent.
where
We only consider nonnegative solutions of (6. 1 ) . Note that problem (6. 1 ) with a (x) = 2 ( N - 2) l xl - 2 i s the linearization of (5.2) at the stationary solution u. In 1984, Baras and Goldstein [BaG] considered equation (6. 1 ) with a (x) = )' l x l - 2 , 0 E 0 C ]RN , and N 2: 3. For this problem they prove that if 0 :::; ), :::; (N - 2) 2 /4 =: ),* then there exists a global weak solution of (6.1 ) for every Uo E L 2 (0). Instead, if ), > (N - 2 ) 2 /4 = ),* then, for each T > 0 and each Uo E Lfoc(O) with Uo 2: 0 and Uo :t= 0, (6.1) has no positive weak solution; moreover, in this case there is instantaneous complete blow-up of approximate solutions of (6. 1 ) . Their proof of nonexistence uses Moser's iteration technique, as well as a weighted Sobolev inequality. The critical constant ),* = (N - 2) 2 /4 is related to Hardy's inequality (2.6) . In [CMa2] , the author and Martel consider the case of general potentials a (x ) , and establish relations between the existence of solutions and the validity of a Hardy type inequality with weight a (x ) . In particular, a new and simple proof of the nonexistence result of Baras and Goldstein is given. In order to state the precise results, let 8 (x ) = dist (x, (0) for x E 0, and let LHO) = L 1 (0, 8 ( x ) dx). For 0 < T :::; 00 and Uo E LHO), we say that u 2: 0 is a weak solution of (6. 1 ) if, for each 0 < S < T, we have that u E L 1 ( (0, S) x 0), au8 E L 1 ((0, S) x 0), and
loS in u ( -(t - D..() - in Uo ( (0) loS in a u ( =
for all ( E C2 ([0, 8] x 0) with ( (8) == 0 on 0 and ( = 0 on [0, 8] x 80. If T = 00, we say that u is a global weak solution. Recall the definition of generalized first eigenvalue /1 1 { -D.. - a(x); O} given by (2.4) . The following result states that /1 1 { -D.. - a(x); O} > -00 is a necessary and sufficient condition for the existence of global weak solutions of (6. 1 ) with (at most ) exponential growth. THEOREM 6. 1 ( [CMa2] ) .
(i) Suppose that, for some Uo E LHO) and some constants C and M, there exists a global weak solution u 2: 0 of (6.1) such that Il u(t)81Iu (!1 ) :::; CeMt for all t 2: O. Then /1 1 { -D.. - a (x) ; O} > (ii) Suppose that /1 d -D.. - a(x); O} > -00. Then, for each Uo E L 2 (0) with Uo 2: 0, there exists a global weak solution u E C ( [O (0) ; L 2 (0)) of (6. 1) such that Ilu(t) II £ 2 (!1) :::; Iluo ll £2 (!1) e - I-'l t for all t 2: 0, (6.2) where /1 1 = /1 d -D.. - a(x); O}. The next result states that condition /1 1 { -D.. - a(x); O} > is "almost - 00 .
,
- 00
necessary" for the local existence of weak solutions. More precisely, we have:
172
XAVIER CABRE
6.2 ( [CMa2]) . Snppose that f.L1 { -� (1 - c)a(x); O} = - 00 for some constant c > O. Then, for every T > a and every Uo E L� (O) with Uo > a and Uo =f= 0, there is no weak solution u > 0 of (6.1). Moreover, there is instantaneo1/.S complete blow-up for (6. 1 ) , in the following sense : FaT eveTY 'II > 1 , set an(x) = min(a(x ) , n), uon(X) = min( uo (x) 'II ) , and let Un be the unique global solution of (6.1) with a and uQ replaced, Tcspcctively, hy an and UOn . Then, for all a < 'T < T, T HBOREM
-
,
---> ,
+ 00 Il.1l.iformly ,in ('T, T)
x
0, as n
->
00 ,
The following inequality is the key ingredient in the proof of the necessary conditions in these theorems. Suppose that (6.1) has a positive local solution , and let u be the minimal solution (i.Il., the one obtained as increasing limit of 1tpproximaLe solutions) . Then, for all 0 < t1 < t2 < T, we have
o
a( x)cp2
-
lV'cpl2 ::; r Jo
1 r log t2 - h Jo
�tt2�
1!. u
for every 'f E C� (O).
L cp
l This inequality is provlld IIlultiplying the approximate problems of (6.1) by cp2 /un , then integrating by parts in space, integrating in time, and finally letting 'll 00. We apply the previous theorelllS to some specific equa.tions with "critical" po tentia.ls. First, our method provides a new and quite elementary proof of the result of Baras and Goldstein [BaG] mentioned above. Namely, suppose that 0 E n e ]RN , N > 3, and a(x) = ..\lxl- 2 . Let ..\* = (N - 2)2/4. For this potential, [BaG] establishes the following: (a) If 0 < ..\ < ..\ * , then (6. 1 ) h as a global weak sulut,ion for every Uo E L2 (0), Ito > O. (b) If ). > ).* , then (6. 1 ) has no positive weak solution for every T > 0 and cvery Uo E L H n) with Uo > 0 and Uo =f= O. Moreover, there is insLanLaneous complete blow-np of approximate solutions. We obtain (a) and (b) through direct applications of Theorem 6,1 (ii) and The orem 6.2, respectively. Here the e:;8eIltial role is played by (2.6), i.e., Hardy's inequality with best constant. In case a < ..\ < ..\* = (N - 2? /4, we also obtain that solutions decay exponen tially. More precisely, )'Ie have --->
Ilu (t ) II U (f1) < I l uo Ii U(f1) e-J1.t
for all t >
0,
where f.L = H2 (WN /101)2/N > () This is a consequence of estimate (6.2) and the improved Hardy inequality (Theorem 3.2) . New improved Hardy inequalities and a detailed description of solutions of (6. 1 ) when a(x) = ).1.7: 1 -2 and 0 < ). ::; (N - 2) 2 /4 have been ohtained by Vazquez and Zuazua [VZ] . In [GP] , Garcia Azorero and Peral have extended the results of Baras and Goldstein to the p-Laplace operator. A second application of Theorems 6.1 and 6.2 is the c&;e when 0 c jRN is of elass C2 , N > 2, and a(x) = ..\S ( X)-2 . Here ). 0 is a constant and 5(x) = dist (x, (0) . For th is prohlem, we obtain state ments ( a) a.nd (b) above with ..\* = 1/1. Here we use some Hardy type inequalities proved in [BM] and [D] . ,
>
.
EXTREMAL SOLUTIONS AND BLOW-UP
173
Finally, recall that the results for the potential Alxl- 2 required N 2 3. In dimension N = 2, we prove that the potential a(x) = Alxl- 2 (1 - log Ixl)- 2 is "critical" , in the sense that the dichotomy ( a) - (b) occurs with A * = 1/4 for all domains n with 0 E n e Bl (0) c ]R.2 . References [BaCo]
Baras, P. , Cohen, L . ,
heat equation, [BaG]
Baras, P. , Goldstein, Math. Soc . ,
[BaP]
284,
[BEP]
121-139 (1984).
Baras, P . , Pierre, M., Inst. Fourier,
Complete blow-up after Trnax for the solution of a semilinear 71, 142-174 (1987). J . A . , The heat equation with a singular potential, Trans. Amer.
J. Funct. Anal.,
34,
Singularites eliminables pour des equations semi-lineaires,
185-206 (1984).
Ann.
A Dirichlet problem involving critical exponents, & Appl. , 24, 1639-1648 ( 1995). X., Some simple nonlinear PDE's without solutions, Boll. Unione
Boccardo, L., Escobedo, M., Peral, I., Nonlinear Anal . , Theory, Meth.
[BC]
Brezis, H . , Cabre,
[BCMR]
Brezis, H . , Cazenave,
Mat. Ital . ,
revisited, [BM]
25,
217-237 (1997).
Brezis, H . , Strauss, W.A., Soc. Japan,
[BV]
T . , Martel, Y. , Ramiandrisoa, A . , Blow up for U t - L!.u = g(u) 1 , 73-90 (1996). M . , Hardy 's inequalities revisited, Ann. Scuola Norm. S up . Pisa
Ad. Dilf. Eq.,
Brezis, H . , Marcus Cl. ScL ,
[BS]
I-B, 223-262 (1998).
25,
Semi-linear second-order elliptic equations in L 1 ,
565-590 (1973).
J. Math.
Blow-up solutions of some nonlinear elliptic problems, Rev. 10, 443-469 (1997). Cabre, X. , Capella, A., On the stability of radial solutions of semilinear elliptic equa tions in all of JRn , C. R. Math. Acad. Sci. Paris, 338, 769-774 (2004) . Cabre, X . , Capella, A . , Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, Journal of Functional Analysis, 238, 709-733 (2006). Cabre, X., Capella, A., S anch6n, M . , Regularity of radial minimizers and semi-stable solutions of semilinear problems involving the p-Laplacian, preprint, 2006. Cabre, X., Martel, Y., Weak eigenfunctions for the linearization of extremal elliptic problems, J. Funct. Anal. , 156, 30-56 (1998). Cabre, X., Martel, Y., Existence versus instantaneous blowup for linear heat equations with singular potentials, C. R. Acad. Sci. Paris Ser. I Math. , 329, 973-978 (1999). Cabre, X., Sanch6n, M., Semi-stable and extremal solutions of reaction equations in volving the p-Laplacian, to appear in Comm. Pure Applied Analysis, 2006. Crandall, M .G. , Rabinowitz, P. H . , Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal. , Brezis, H . , Vazquez, J . L . ,
Mat. Univ. Compl. Madrid,
[CCa1] [CCa2] [CCaS] [CMa1] [CMa2] [CS] [CrR]
58, 207-218 (1975).
The Hardy constant,
[D]
Davies, E . B . ,
[GMP]
Gallouet, T., Mignot, F., Puel, J.-P. ,
[G] [JL] [KV] [Mal] [Ma2] [MP]
=
307, Serie I, 289-292 ( 1988). Garcia Azorero, J . , Peral, I. , Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144, 441-476 (1998). Gelfand, I.M., Some problems in the theory of quasilinear equations, Amer. Math. Soc. Trans! . , 29, 295-381 (1963). Joseph, D . D . , Lundgren, T . S . , Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal . , 49, 241-269 (1973) . Kalton, N . J . , Verbitsky, I . E . , Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc . , 351, 3441-3497 (1999) . Martel, Y., Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math. , 23, 161-168 (1997). Martel, Y., Complete blow up and global behavior of solutions of Ut - L!.u = g(u) , Ann. Inst. H. Poincare Anal. Non Lineaire. , 15, 687-723 ( 1998). Mignot, F . , Puel, J.-P. , Sur une classe de problemes non lineaires avec nonlinearite positive, croissante, convexe, Comm. P.D. E . , 5, 791-836 ( 1980). C. R. Acad. Sci. Paris,
[GP]
4 6 , 417-431 ( 1995). Quelques resultats sur Ie probleme -L!.u >.eu ,
Quart. J . Math. Oxford,
174
XAVIER CABRE Nedcv, G . , Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris Ser. I Math . , 330, 997-1002 (2000) . Nedev, G . , Extremal solutions of semilincar elliptic equations, preprint, 2001. Peral, L , Vazquez, J.L., On the stability or instability of the singular solution of the semilinear heat equation with exponential reacti on term, Arch. Rat. Mech. Anal., 129, 201-224 (1995) . Vazquez, J.L., D omain of e.:r:istence and blow-up for the exponential reaction-diffusion equation, Indiana Univ. Math. J., 48, 677-709 (1999). Vazquez, J.L., Zuazua, K, The Hardy inequality and the asymptotic beha1liour of the heat equation with an inverse-square potential., J. Funet. Anal., 173, 103-153 (2000) .
INl] IN2] IPV]
IV] IVZj
POLITECNICA DE CATALUNYA, 1 , Av. DIACONAL 647, 08028 BARCELONA, SPAIN
ICREA AND UNIVERSl'T'A'T'
A PLIC ADA
E-mail address:
,
xavier . cabrelDupc . edu
DEPARTAMENT DE MATEMATICA
Contemporary Mathematics Volume 446, 2007
Capillary Drops on an Inhomogeneous Surface L . A . Caffarelli and A . Mellet
To Haim with admiration and our best wishes. ABSTRACT. This paper is concerned with equilibrium liquid drops lying on a horizontal plane when small periodic perturbations arise in the properties of the support plane (due for example to chemical contamination or roughness). We first establish the existence of a minimizer for the energy and study its regularity. We then show that the free interface stays in a small neighbor hood of a portion of sphere (corresponding to an equilibrium drop lying on a homogeneous plane) , thus showing the existence of sphere-like capillary drops.
1. Introduction The energy of a drop described by the set E c plane (z = 0) is given by (1.1)
a
Jrir
z>o
I D'PE I
-
a
1
z=O
]Rn + 1 and resting on a horizontal
(3 'PE(X, 0) dx +
1
z>O
p f 'PE dx dz
where 'PE is the characteristic function of E, a is the surface tension, (3 is the relative adhesion coefficient between the fluid and the solid, f is the gravitational energy and p is the local density of the fluid. In this paper, we neglect the effect of gravity and assume f = O. The Euler-Lagrange equation associated to the minimization of ( 1 . 1 ) under a vol ume constrain gives rise to a mean-curvature equation, together with a contact angle condition (see [Fl ) . This last condition, known as the Young-Laplace equa tion reads: cos ,",! = (3, where ,",! denotes the angle between the free surface of the drop 8E and the hori zontal plane {z = O} along the contact line 8( E n {z = O}) (measured within the 1991 Mathematics Subject Classification. Primary 49F22; Secondary 74Q05. Key words and phrases. Capillary surfaces, Calculus of variation, Isoperimetric inequality, Homogenization. L. Caffarelli is partially supported by NSF grant DMS-0140338. A. Mellet is partially supported by NSF grant DMS-0456647.
1 75
©2007 American Mathematical Society
1 76
L. A. CAFFARELLI A)lD A. MELLET
fluid) . The coefficient (3 is determined experimentally and depends on the proper ties of the materialt; (solid and liquid) . It is usually assumed to be constant, but it is very sensitive to small perturbations in the properties of solid plane (chemical contamination or roughness) . Thp.',e inhomogeneities are responsible for many in teresting phenomena such as contact angle hysteresis and sticking drop on inclined surfaces (see [JG] , [LJ]). In [HM] , C. Huh and S. G . Mason investigate the effect of roughness of the solid surface on the equilibrium shape of the drop by solving approximately the Young-Laplace equation, for some particular type of periodic roughness (radially symmetric). The purpuse of
this paper is to investigate the properties of equilibrium drops in the case of general periodic inhomogeneities, that. is when the relative adhesion coefficient satisfies (3 (3(x/=:) , with y f-7 /3(11) Zn-periodic. The existence of equilibrium drops (minimizing the energy functional) will be shown, within the class of �d.s of locally finite perimeter. Then, we will investigate the properties of that minimizer, showing in particular that the contact liTle has finite Hausdorff measure. Finally, we will prove that the equilibrium drop converges uniformly to a spherical cap when the size of the inhomogeneities (c) goes to zero (homogeni OJ.
In [eM] , we investigate further consequences of t hose results, justifying in particular the two phcnomena Lhat we mentioned earlier: Contact angle hysteresis and 8ticking drop on an inclined plane (for non-vanishing gravity) .
2. Notations, main results and organization of the paper 2.1. Sets of finite perimeter. We recall here the main facts about sets of
finite perimetcr and BY functions. The standard reference for BY theory is Giusti [Gi] . Let l1 be an open subset of Rn+ l ; B V (l1) denotes the set of all functions in £ 1 (l1) with bounded variation:
BV(l1) = where
ID f l = sup
f E £1 (l1)
:
o
IDf l < + 00
f (x)div g(x )dx : 9 E [C6 (nW + 1 , I gi < 1
•
If E is a Borel set, and l1 is an open set in Rn+1, we recall that the perimetel" of E in l1 is defined bv o
"
o
P(E, l1)
=
!!
I Drp£ i .
1 77
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
A Caccioppoli set is a Borel set E that has locally finite perimeter ( i.e. P(E, B) < 00 for every bounded open subset B of 0). Note that sets of finite perimeter are defined only up to sets of measure O. We shall henceforth normalize E ( as in [Gil) so that 0 < I E n B(x, p) 1 < I B(x, p) 1 for all x E 8E and all p > O. Furthermore, it is well known that if the boundary 80 of 0 is locally Lipschitz, then each function f E BV(O) has a trace f + in £ 1 (80) ( see Giusti [Gil ) . l.From now on, we denote by
0 the upper half space: O = ]Rn x (0, +00), and we denote by (x, z) an arbitrary point in n, with x E ]Rn and z E [0, +00) . We denote by
V > O:
c&'(V) the class of closed Caccioppoli sets in 0 with total volume
{
c&'(V) = E c O :
10 IDipEI < +00, l E I = V } ,
where l E I = In 'PE demotes the Lebesgue measure of E. Since Caccioppoli sets have a trace on 80 = ]Rn x {z = O}, we can define the following functional for every E E c&'(V):
(2 . 1 )
/ (E)
J 1> 0 ID'PEI - 1=0 /3(X) 'PE (X, O)dx P(E, 0) - r }En{z=O} /3(x) dyen(x).
In this framework, equilibrium liquid drops are solutions of the minimization prob lem: (2.2)
inf / (F) / (E) = FEe'(V)
2.2. Constant adhesion coefficient. When /3 = /30 is constant, the ex istence of a minimizer was established by E. Gonzalez [GJ . The corresponding functional reads (2.3)
/o (E) =
10 I D"'E 1 - /30 J 'PE (X, 0) dx.
When /30 = - 1 (hydrophobic surface) , the absolute minimizers are the spheres of volume V in {z > O}. In particular, the equilibrium drop does not touch the support plane. On the contrary, if /30 > - 1 , it is easy to see that, though a sphere of volume V is still a local (degenerate) minimizer of the functional / , the absolute minimizer must touch the solid support. An important tool, when the adhesion coefficient is constant, is the Schwarz symmetrization ( see [Gl ) : For every E E c&'(V), the set (2.4)
E8 = {(x, z) E 0 j I x l < p(z)} ,
is a Caccioppoli set with volume
(
where p(z) = W� 1
V satisfying
/0 (E8 ) ::; /o (E)
J 'PE (X , Z)dX)
1
n-
178
L. A. CAFFARELLI AND A. MELLET
with equality if and only if E was already symmetric. This clearly implies that any minimizer should have axial symmetry. Actually, it can be shown that the minimizers are spherical caps; that is the intersection of a ball Bpo (O, Zo} in ]Rn+1 with the upper-half space O. We denote by Btu (za)
=
Bpa (O, zO } n {z > O}
such a spherical cap. Our main result is a stability/uniqueness result for the minimization problem with constant coefficient: THEOREM 2 . 1 . Let E be such that
(2.5) (2.6)
E E C(V), E lies in a bounded subset BR of 0, 'rIF E C(V) . :38 > 0 s.t. /a (E) < /0 (F) + 8
Then there exists a universal a > 0 and a constant G (depending on R) such that IE6.Bta I :S G8a,
where Bta is such that
lB.;;;' I = V
and the cosine of the contact angle is {3a ·
Note that we recover in particular the fact that the gravity-free equilibrium drop is a constant mean-curvature surface satisfying the Young-Laplace condition. If moreover E satisfies some non-degeneracy conditions, then Theorem 2.1 im plies the uniform stability in the following sense: For any 1) 0, there exists .50 such that if (2.6) holds with .5 < 80 , then
>
B�_1))p C E
c
B�+1))p '
In other words, the free surface aEn {z > O} stays between aB�+1)) p and aB� _1))p ' REMARK 2.2. When {3a = - 1 (so / (E) is the perimeter of E in ]Rn+1 ) , Theorem 2.1 is nothing but a quantitative version of the isoperimetric inequality, also known as Bonnesen inequality. In dimension 2 it is well known (see R.R. Hall [H] ) that we have (with Po such that lEI = I Bpa l ) : I E 6.Bpa l :::; G IEI3/4 (p( E)
-
P(Bpo ) ) �
thus corresponding to a = 1/2 in our theorem. In higher dimension, a similar result was established by Hall with a = 1/4, though it was conjectured that the inequality should hold with a = 1/2 in any dimension. In the last section of this article, we will prove Theorem 2.1 with a = 1/3 in dimension 2. The question of whether the result holds with a = 1 /2 (in dimension 2 at least) is still open.
2.3. Periodic adhesion coefficient. When the relative adhesion coefficient
f3 depends on x , the Schwarz symmetrization (2.4) could increase the wetting energy
J {3(x)'PE(x, O)dx.
However, if {3 = {3(X/E) is periodic with small period E, the wetting energy should not increase by more that GE after symmetrization. If we can prove that fact, Theorem 2.1 will allow us to show that the minimizer associated with a periodic adhpBion coefficient is almost a spherical cap (in L1 and LOO).
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
179
We now make our framework precise: We consider the following energy func tional: (2.7) where and
{3 satisfies
We denote by
/ (E) =
J 1>0 I D0
ID'P E I +
We recall the following estimate (see [Gil):
1=0
I'PE - 'PEj I dx
o 2
-1
If
f3 < 1, then ID T1 , there exists a minimizer E of ,/ in gR,T that satisfies PROPOSITION 4.2.
E E gR,T, (V).
The proof relies on a slight modification of an argument first presented by E. Barozzi in [B] (see also E. Barozzi and E. Gonzalez, [BG] ) . The key lemma is the foHowing: LEMMA 4.3. Let R and To be such that there exists a ball B of volume V lying in r R,To ' Then there exists Tl � To such that for any T > Tl and E minimizer of ,/ in gR,T(V), there exists t, To < t < Tl with -
.7F (E n {z = t}) = 0.
Let us recall here that the coarea formula gives I EI = 1+oc .}t"n (E n {z = t}) dt,
which in particular implies that a.e. t E R
Proof of Proposition 4.2: Let E be a minimizer of ,/ in gR,T(V), and let
t be as in Lemma 4.3. Then, the part of E that lies above z perimeter of E by at least 1 -- I E n {z > t } 1 n+1 , !J,n+ l where !J,n+l is the isoperimetric constant. Thus, if we define
=
t contribute to the
n
Eo =
E
o
in {O < z < t} \ B in {z > t} in B -
E U pB
-
-
with p � 1 such that IEol = l E I , we have and Eo
E gR,T, (V).
,/ (Eo) < ,/ (E) =
FE6'R.T(V)
min
,
,/ (F),
Letting T go to infinity, the compactness property of the minimizing sequence gives the existence of a set E E gR,T, (V) such that ,/ (E) =
FE0"R(V)
min
,/ ( F) .
In other words, E is a vertically-unconstrained minimizer.
Proof of Lemma 4.3: Let t l , t 2 and t 3 be three positive numbers with
0 < h < h < t3 < T
o
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
183
and such that B e {z < td.
VI = IE n {tl < z < t2 } 1 V2 = IE n {t2 < z < t3}1
We denote
and
The isoperimetric inequality yields:
n v t: 1 v2n + 1
J.Ln+1(2m + 81 ) :::; J.Ln + 1 (2m + 82) , :::;
where 8i i s the surface of the lateral boundary:
8i =
1{ti O} ) the area of the free surface in rr and A2 (r) =
'PE(x, O)dx
B;:(xo) the wetted area in rr ' Since E is bounded above, if x E E n {z = O} , going from the slice {z = O} to the slice {z = T}, we must cross 8E. Therefore we have A2 < Al. The isoperimetric inequality then gives: U(rt/ (n+1 ) < ,un +l (2A1 + S(r » . Consider now the set F = E \ fr (xo). It satisfies:
,/ (F) < ,/ (E) - A l + ,SmaxA2 + S, and IFI = IEI - U(r) , and using (4.4), with IFI = Vo and OV = U, we get ,/ (E) = min ,/ = min ,/ < ,/ (F) + C CIEI )U(r). 0"( IEI) S(Vo + U) It follows that
min ( I , 1 - 'smax)A l < A l - 'smaxA2 < S + C(/FI) U(r), with C(F) < �o peE, n) < CCIEj) as long as U < IEI/2. Thus
U (r t/(n+ l )
and if U < c- ( n+1) /2, we deduce
:s;
CU'(r) + CU(r) ,
U(r),,/(n+ l ) < CU' (r),
o
and Gronwall's Lemma gives the result. Using similar arguments, we can also establish the following lemma:
LEMMA 4.5. Let (xo , zo) E BE with Zo > O. There exists c, universal constant, such that for all r < Zo we have IBr(x o, zo) n EI > crn+ 1 I Br (xo, zo) \ EI > crn+ 1
Proof. For
r
< zo, we define
U1 (r) = IBr (xo, zo) n EI U2 (r) = I Br (xo, zo) \ E I
S1 (r) = .non (8Br (Xo , Zo) n E) S2(r) = .non CBBr (xo, zo) \ E)
As in the previous proof, the minimality of E and the fact that Br lies entirely in n for r < Zo give (by estimating ,/ (E \ Br) and ,/ (E U Br) respectively) : pe E, Brexo , zo» < Sl (r) + CUI (r) pe E, Br(xo, ZO» < S2 (r) ,
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
1 87
which together. with the isoperimetric formula yields U1 (r) n+l ::; U2 (r) n+l ::;
Since U: (r)
=
2Iln+ l (Sl (r) + CUI (r)) 2Iln+ S2 (r) , 1
S ( r) , Gronwall's Lemma gives the result.
o
i
4.3. Unconstrained minimizers. We now complete the proof of Proposi tion 4. 1 : Let G denote the projection of E onto {z = O}. As a consequence of Lemma 4.4, there exists Po = OIEI n�l such that if x E G, then
IE n rpo (x) 1 2: Cp�+ l Consider the familly {Bpo (x) I x E G}. We can extract a subfamilly Bpo (xj ) finite overlapping still covering G. In particular, j > C(n) � )E n rpo (xj ) 1 j > C(n) L P�+ l . j This implies that the subfamilly contains at most IEIIpo n +1 odicity of /3, we deduce that G has at most radius
with
balls. Using the peri
1Lpo CIEI n�l . Cp�+ l =
So the proof of Proposition 4.1 is complete.
o
5. Properties of the minimizers In this section, we investigate the regularity of the minimizer E. The regularity of the free surface 8E n n is a consequence of classical regularity results for minimal surface. We then determine the Hausdorff dimension of the contact line 8E n {z = O}. 5.1. Regularity of the free surface. Note that if then E also minimizes the functional
E is a solution of (2.2),
among the Caccioppoli subsets of n satisfying I FI = lEI. We can therefore apply the classical regularity results for minimal surfaces (see E. Gonzalez et al. [GMT] ) : THEOREM 5 . 1 . If E minimizes the functional ,/ in 6"(V), then 8* En n is an analytic {n-l}-manifold and HS [(8E \ 8* E) n n] 0 for all s > n - S. In particular, if n + 1 ::; 7, the singular set is empty. =
HIS
L. A. CAFFARELLI AND A.
MELLET
5.2. Hausdorff measure of the contact line. We now establish the follow ing proposition: 5.2. The contact line aCE n {z Hausdorff me.asure in Rn and PROPOSITION
=
O}) in JRn has finite n
-
1
Yr- l (8( E n {z = O})) < CV ::.:t .
The proof relies on the monotonicity formula and a couple of lemma. We start with the following:
LEMMA 5.3. Let X o be a pO'int in Rn. There exists a critical So > 0 such that if
B;: (xo O ) \ {z < Sor} ,
then
c
E,
B0 2 (XO, 0) c E
Proof, Let us renormalize and take r = 1 . The proof relies on a De Giorgi type argument: Consider the vertical cylinders rk
and
=
B;!k (xo, 0)
x
JR,
and r�
B;'k (xo, 0)
=
X
(0, 05)
with rk
=
consider
� + 2-k ,
Vk = IE" n (r � \ r f+ 1 ) 1 where EC = !l \ E is the complimentary set of the drop. Since
we deduce that there is a cylinder r between r k and r k + 1 such that Ji"n(8rr n EC n {O < z < o}) < 2k+ 1 Vk , r
Next, we consider F
=
E u r� (where r�
r r n {O < z < S}), we have IFI > l E I · =
and
< ",f (E ) - peE, r� ) + ,7t"n(8rr n EC n {o < z < S} ) -fJrninJi"n(Ec n fr n {z = O}I) < ",f (E ) - p( E , r�) + Ji"71 (8rr n EC n {O < z < oJ) + max(O, -fJmin)P(E, r�) < ",f ( E) - min(l, 1 + fJrnin)P(E, r�) + Ji"n (8rr n ec n {O < z < S}) where fJmin = inf fJ (x). In particular, inequality (4.4) and the minimality of E yield k min(l, 1 + fJmin)P(E, f�) < £"' (arr n EC n {O < z < o}) < 2 +l Vk . ",f (F)
-
By the isoperimetric inequality, we get
" < I Ec n rr l J.Ln+ 1 (2P( E, f�) + £"'(8rT n EC n {O < z < S})) �l
n�l k +l V +l < C(2 Vk) . k 0 Therefore, 2k+1 Vk 0 if Vo is small enough (i.e. if 05 is small enough) . This Lemma, together with the monotonicity formula (Lemma 4.5) allows us to control the perimeter of E in the neighborhood on the contact line: and thus
'
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
189
5.4. If (xo, 0) E 8E, then for every r, P(E, B;:(xo, 0)) � Crn where C > 0 is a universal constant. Proof. Let V1 (r) = IE n Br(xo, 0) 1, 81 (r) = £n (E n 8Br(xo, 0)) V2 (r) = IBr(xo, 0) \ EI 82 (r) = £n (8Br(xo, 0) \ E). By the previous lemma, either B: (xo, 0) \ {z < tSor} c Ee, or there exists (Yo, zo) E 8E n Br /2 (xo, 0), with Zo � tSor /2. In the first case, we clearly have V2 (r) � crn+ 1 COROLLARY
and
V1 (r)
rv
IE n fr(xo, O) 1 � crn+ 1
by Lemma 4.4. In the second case Lemma 4.5 gives: and
c(tSor) n+ l
VI
�
IE n Boor(YO , zo) 1
V2
�
IBoor (Yo, zo) \ E I � c(tSor) n + 1 .
�
In either case, we deduce
Vi(r) � crn+ 1
(5.1)
i = 1, 2.
Moreover, the isoperimetric inequality gives n v1n: l � /-Ln +1 (81 + P(E, Br(xo, 0))) v2n +l � /1n+ l (82 + P(E, Br (xo, 0))) and It follows that
�
1
�
v1n+ l + v2n+l
-
T n+ l
n (VI + V2) n+l
which yields the result thanks to (5. 1).
�
2/-Ln+ l P(E, B: (xo, 0)),
D
Proposition 5.2 will now be a consequence of the following Lemma:
There exists a constant C such that P(E, {O < z < t}) � CV nn+ ll t Proof of Proposition 5.2. Let Uj Bo (xj ) be a covering of 8{E n {z = O}} with
LEMMA 5.5.
finite overlapping. Then by Corollary 5.4, we have
P(E, Bo (xj )) � CtSn .
But thanks to the finite overlapping property,
L P(E, Bo (xj )) � CP(E, {O < z < tS}) � CV;:+� tS, -l and therefore the number of balls is less than CV nn+1 tS 1 - n , hence the result. Proof of Lemma 5. 5. Let F the set obtained by cutting E at level t: F = {(x, z) E jRn+ 1 ; (X, Z + t) E E} n { Z > O}.
D
190
L. A. CAFFARELLI AND A. MELLET
Then
IE I - IE n {O < z < t} 1 > lEI wnRnt (where Wn denotes the volwne of the unit ball in JRn) since by Proposition 4.1 we have E c rR with R = cv1/(n+ 1) . Thanks to (4.4) we deduce I FI
=
-
/ (E) < f (F) + CV- n+l Rni n < / (F) + CV- n H V n+' i. 1 1
Moreover
/ (E) - / (F)
=
p eE , {O < z < t })
-
J (3(x/r::) [ipE (X, 0)
-
ipE(X, t) ] dx,
but if x belongs to the symmetric difference of E n {z = O} and E n {z = t}, then, going from the slice {z = O} to the slice {z = t}, we must cross BE, and therefore
lipE ( X, O) - ipe (x, t) ldx < peE, {O < z < i}) . We deduce
( 1 sup( I (3I» P (E, {O < which completes the proof. -
z
n- l < t}) < CV n+l t
5.3. Sphere-like minimizers. We can now prove the following result:
o
PROPOSITION 5.6. There exists a spherical cap Bta such that n -l IEt.Bta l < C( V n+I E)a .
Moreover', I Btu I
=
V and the cosine of the contact angle is «(3) .
Proof. We observe that
(5.2)
« (3) - (3(X/r:: » ipE (X, O) dx ::; L
c,
«(3max
-
(3m in ) dx ,
where we sum on all the cells Ci of r::zn that intersect the contact line. Thanks to the n - ll r:: 1 results of the previous section, the number of such cells cannot exceed V n+ n. Since the area of each cell is En , we deduce: n- 1 (3 (X/E) ipE (X, O)dx + CV ..+ , E. Thus, if we introduce the energy functionnal in which the adhesion coefficient (3(x/E) is replaced by its average «(3) :
/o(F) = (J" we have
z>o
z=o
/o (E) ::; / (E) + CV n + 1 E. n-l
Moreover, if Eo denotes the minimizer of /0 , we also have n -I / (Eo ) < /o(Eo) + CV n +' E, hence (using the fact that / (E) ::; / (Eo» (5 .3 )
n-I
/o(E) ::; /o(Eo) + CV n +1 E,
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
and so
E satisfies
(2.6) with b =
191
-
CV nn+ll 10, and Theorem 2.1 gives the proposition.
o
5.4. Proof of Theorem 2.3- (iii) . We conclude this section by proving that this implies Theorem 2.3-(iii): It is a consequence of the following nondegeneracy result: LEMMA 5.7. Let 0 < ry < 1/2, then (i) If there exists (x, z) E E \ B (H'7) p then
I E \ B: I ::::: C(ryp) n+ l . (ii) If there exists (x, z) E B0 -'7) p \ E then
IB: \ EI ::::: C(ryp) n + l . Proof of Theorem 2.3-(iii). Let 0 < ry < 1/2. = b cv:-::;:i 10 � C(pry) (n+ 1 ) / then Thus, for 10 �
Theorem 2.1 yields that if
C(V)ry (n+ l ) /, Lemma 5.7 implies E \ B0+ '7) p = 0 , and B0 '7) p \ E = 0, -
which gives the last part of Theorem 2.3.
Proof of Lemma
distinguish the case z Lemma 4.5 yields
0
5. 7. We only give the detailed proof of (i). We have to ::::: ryp and z � ryp. When z ::::: ryp, the monotonicity formula
IB'7P (x, z) n EI ::::: C(ryp) n + l , and since B'7P (x, z) n B: = 0 , we have When z
::::: ryp, then the monotonicity formula Lemma 4.4 gives
A simple geometric argument shows that if ry
0 (depending only on the dimension n) such that P ROPOSITION
Moreover, E3 also satisfies (2. 6).
Note that (2.6) and Lemma 3.5 implies that peE) < C for some constant depending only on ,6max and lEI. We will u�c this fact throughout this section. Also, thanks to Giusti [Gil , we know that E can be approximated by a sequence of Coo sets Ej such that --> DO .
as j
In particular, proceeding as in Lemma 3.3, we can show that
Thus Ej satisfies (2.5)-(2.6) with Vj = I Ej I instead of V (where Vj V) and OJ instead of 0 (where OJ --> 0) . It is t.hus enough to establish Theorem 2 . 1 for smooth sets E. For the sake of simplicity, we restrict ourself to the 3-dimensional case ( n = 2) which is the most relevant ease from the point of view of applications. All the arguments are however valid in higher dimension, but with sometime different con stants. In particular, the coefficient a is related to the exponent p in the Bonnesen inequalities (see below) and will thus depend on the dimension. -->
We recall that ES denotes the Schwarz symmetrization of E; E" = { (x, z) ; Ixl < PE (Z)} with (6. 1 )
PE (Z )
(6.2)
AE(Z)
with the notation
(71"-1 AE(zW I
ipe (x, z) dx = .Yt'2 (Ez ) ,
Ft = F n {z = t}.
We recall that ES is a Cacciopolli set satisfying I E s l sation preserves the volume) and so (2.6) yields
= lEI
(the Schwarz symmetri
fo (E S ) < /o (E ) ::; /o (ES) + o. Since we deduce (6.3)
130
J ips (x , 0) dx = f30
ipE" (x, 0) dx,
P ( E", O) < P (E O ) < P(ES , O) + 0, ,
To deduce something regarding the symmetry of the drop, we establish the following proposition;
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
1 93
P ROPOSITION 6 . 2 . Let F be a subset in n, such that F c rR, T, and let F* be the set obtained by replacing each horizontal slices of F by a disk with same area and same center of grovity:
with
\ J
ap(z) = Ap z) x'Pp(x, z) dx Then there exists a constant C(R, T, P(F)) such that I F�F* I :::; C(P(F) p ( Fs ) ) 1 /3 . -
Note that by an approximation argument, it is once again enough to establish this proposition for smooth set F. The proof relies on the following Bonnesen type inequality: The Fraenkel asym metry of a set G is defined by
IG n B(a, p) 1 = inf IG�B(a, p) 1 2IB(a, p) 1 IB(a, p) 1 where p is such that l E I = I B(a, p) l. Then R. R. Hall [H] (see also Osserman dimension 2) showed that P(G) ::::: P(B(a, p)) ( 1 + /1 1 A* (G) P ) with p = 2 in dimension 2 and p = 4 in higher dimension. In dimension 2, it follows that there exists a constant /1 1 such that A* (G) = I
_
sup
a
a
[?] in
(6.4) with p = pp, ,\* = ,\} and p(z) = P(Fz, IRn). We also define the following quantity for each horizontal slices Fz of F:
(6.5) where
a(z) is the center of gravity of Fz .
It is readily seen that
Ap(z) :::; A* (FJ +
:(�) A*(Fz).
In order t o estimate F�F* = J Ap(z)H2(Fz) dz, we thus need t o control the isoperimetric default p(z) - 21l"p(z), which will be done using the following lemma (the proof of which is postponed to the appendix) : LEMMA 6.3. Let F be a smooth set in n, let p(z) denotes the (n - I)-perimeter of the slice Fz . Let p(z) be defined by {6. 1} with E = F, and let F8 be the Schwarz symmetrization of F. Then
P(F) ::::: and
J Vp2
+
(21l"pp' )2 dz
1 94
L. A. CAFFARELLI AND A. MELLET
Proof of Proposition 6.2. Lemma 6.3, implies
P(F) - PcPS) >
(p - 2rrp)
VpZ
2rrp dz. + (2rrpp') 2
For any borel set Ao in �+ , we have Ao
1 /2 < So if Ao is the set Ao =
CKTl /2
Ao
(p - 2rrp ) p dz
•
{ z ; Vp2 + (2rrpp')2 > (P(F) - P(F8» - 1 /3 } ,
I Aol < P(F) (P(F) - P(F8» 1 /3, and 1-{2 ( Fz�D (a( z) , p(z » ) dz = r >'F (Z)1-{2 (Fz) dz Iii!.\ Ao R\Ao l Z S / { P F ) P F) ( 1/ < - CKT 2 (P(F) - P(F·» 1/3 :5 CRT 1 /2 (p {F) - p(F8» 1 /3 .
we have
It follows that
l /2 )(P (F) p( F8 » 1 /3 < ) RT R2 F + F (F C( P I I � * _
which gives Proposition 6.2.
o
Proof of Proposition 6. 1. If we tried to apply Proposition 6.2 directly to E, we would still have to determine how ES differs from E*. This means that we need to control the variations of the center of gravity a{z), which appears to be a delicate task. Instead, we make use of a different approach: Let H be an hyperplane in jRn+ 1 , perpendicular to {z = a}, and let H+ and H- be the two half space defined by H. By sliding H in the normal direction, we can ensure that H cuts the set E into two sets E+ = E n H+ and E- = E n H with same volume V/2. If we denote E1 the set formed by adjoining to E+ its reflexion with respect to H, and Ez the set form by adjoining to E- its reflexion with respect to H, we obtain two set El and Ez such that Repeating the same operation with El and Ez, with respect to an hyperplane H' perpendicular to H and {z = a}, we obtained four sets satisfying:
I El l = I Ez l = I E3 1 = I E41 = V , p eEl , 0) + P(E2 ' 0) + P(E3 , 0) + P{E4 , 0) = 4P{E, 0).
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
1 95
Moreover, each of those set is symmetric with respect to the axis H n H' , which we assume to be given by x = 0, and therefore
(6.6)
Now, if we denote
S( E) = we have
r
J{z=O}
IPE (X, 0) dx,
S(E1 ) + S (E2 ) + S(E3 ) + S (E4 ) = 4S (E) ,
(2.6) , we deduce: /o (Ed + /0 (E2 ) + /0 (E3 ) + /0( E4 ) = 4 /0 (E) � 4 /0 (Eo) + 8, for any Eo E 6"(V). If we choose Eo to be a minimizer for /0 , we also /o (Eo) � /O (Ei ) for each i = 1 · · · 4, and thus i = 1 · . · 4. /O (Ei ) � /o(Eo) + 8, But P(Ef ) � P(Ei ) and S(Ef ) = S(Ei); therefore i = 1 · . · 4. P(Ei ) - P(Ef ) � 8, Proposition 6.2, together with (6.6) implies (6. 7) i = 1 · · · 4. I Ei�Ei l � C81/3 and using
have
In order to conclude, we now reconstruct the set
ES = (Ef n H+ n H'+ ) U (E� n H+ n H'- ) u (E3 n H- n H'+ ) U (EX n H- n H'- ) , We clearly have IE�Es l � C81 /3, so we only have t o check that IEs �Es I � C81/3 . To that purpose, we need to show that for a given z, the slices Ei z have almost the same radii for i = 1 · . . 4. This is a consequence of the strict convexity of the square function: We note that the sum of the area of the slices Ei z is equal to four times the area of Ez, and the same is true for the perimeter. In other words: 2:Pi(Z) 4p(z ) 2: Pi(Z) 2 = 4p(z) 2 , where Pi and Pi denote respectively the perimeter of Eiz and the radius of Eiz . =
Thus, if we denote by
J-Li = Pi - 27rPi
Eiz, we have: 4 4 1 1 27rp � P = 4 L Pi � 4 L (27rPi + J-Li ) . i=l i= l
the defect in the isoperimetric inequality for the slice
It follows that
196
L . A. CAFFARELLT AND A . MELLET
and a direct computation shows that:
L I Pi - Pi 1 2 i<j
4
0 .J (/30) (/30 1)4/3 (/30 + 2)5/3 we have that for every /30 E (-1, 1 ) , there exists a constant C such that /o (B:) - /o (B:O) > CI cos , - /301 2 and so (6,10) gives that the contact angle i of B: satisfies 1 cos , - /30 1 < C6 1 /2 , I
=
_
=
/I
=
_
199
CAPILLARY DROPS ON AN INHOMOGENEOUS SURFACE
any
Finally, (6. 11) implies that P is a Lipschitz function of cos l' on (-1, 1 - 'T/) for 'T/ > O . Since /30 < 1 we deduce that
I p - Po l � C (/3o) I cos l' - /30 1 , if cos l' � (1 + (30 ) /2. When cos l' � (1 + (30 ) /2, then I cos l' - /301
yields
o�
� (1 - (30 ) /2 which
C(/3o ) .
In either case, using (6. 10), we deduce the existence of constants
IB+ !:l.B+ I P Po
O . When v = 0 we recover formally the Euler cquationH (2.1), and Dv1v=0 = Dt. The vorticity w = V x u obeys an equation similar to (2.4): (3 . 3) Dvw = w . Vu. The Eulerian-Lagrangian equations (2.6) and (2. 10) have viscous counterparts ([6]). The equation corresponding to (2.6) is Dv A
=
0, D"v = 2vCVv, u = W[A, v]
(3.4)
The u = W[A, v] is the Weber formula (2.8), the same as in the ca.e of v = O. The equation D" (u, V)A = 0 describes advection and diffusion of labels. The right hand side of (3. 4) is given terms of the connection coefficients C;:;�i =
( VA ) - I )
ji
(Oj Ok Am ) .
The detailed form of the virtual velocity equation in (3 . 4) is Duv;
=
2vCriokVm' ,
The connection coefficients are related to the Christoffel coefficients of the flat Riemannian connection in R3 computed using the change of variables a = A (x, t}:
;
( A t) a x , CJ:\(x, t) = -rj7(A( x , t)) X k
The Eulerian-Lagrangian label gradient VA is the pull back of the Eulerian gradient under A.
vt
=
« VA )-l){aj.
The connection coefficients are commutator coefficients,
[V t , OkJ
=
Cri V! ,
and ( note case v = 0)
[Du(u, V), VfJ
=
2VCr;ak V!.
As in the inviscid ease, we associate to the virtual velocity v the Eulerian-Lagrangian curl of v (3.5)
( = VA
X v.
The viscous analogue of the Eulerian-Lagrangian Cauchy invariant active scalar system (2. 10) is
(3.6)
DvA = 0,
Dv(q = 2vG�k {)k (P + vT:(P, u = V X (_L\)-l (e [VA, (])
209
DIFFUSIVE LAGRANGIAN TRANSFORMATIONS The Cauchy formula
l C[V'A, (] = (det(V'A » (V' A) ( . is the same as the one used in the Euler equations, in the form (2.9). The specific form of the two terms on the right hand side of the Cauchy invariant's evolution are ·gckim Gpqk - up '" CkqiP ' (3 .7 ) and -
Tq - EqJ"t'rmp ern p kji er ./kjj " The system (3.4) is equivalent to the Navier-Stokes system. When v = 0 the system reduces to (2.6). The system (3.6) is equivalent to the Navier-Stokes system, and reduces to (2.10) when v = O.
(3.8)
-
. .
THEOREM 1 . The system
Dv (u, V')A = 0,
A(x , O) = x,
Dv (u, V')v = 2vCV'v, v(x, O) = uo (x) , u = P ( V'A?'v) has local smooth solutions. The function u given by the Weber formula above obeys the incompressible Navier-Stokes equations OtU + u . V'u - vflu + V'p = 0, V' . u = O. 4. Diffusive Lagrangian Invariants
One cannot expect conservation of all invariants in the presence of viscosity. In order to describe the behavior of the invariants, let us recall that, under the change of variables A = x + P., the Euclidean Riemannian metric becomes g mn (a ) = (ok A m ) (Ok An)
A(x, t). We say that a function F is diffusively Lagrangian under the Navier-S tokes flow if F = rjJ A and rjJ obeys a a second order linear parabolic PDE with coefficients determined locally by the Euclidean Riemannian metric induced by the change of variables A, and which vanish when v = 0: orjJ(a, t) = v£[g, oal rjJ(a, t) ot where erg , o"jrjJ = g ij 0i2j
a = A{x, t); da = (Det(\7A))dx,
it is natural to define a density pdx to be diffusively Lagrangian if p(x, t) = f(x, t) (Dct(\7 A))
with f a diffusively Lagrangian function. In order to verify that the helicity has a diffusively Lagrangian density it is convenient to use the helicity density w·w
where w = {\7A)Tv. The fact that this is a helicity density follows from the Weber formula (2.8) . From the Cauchy formula (2.9) we have that w
. w = (Det(\7A)) (v . ().
Now, from the definition it is easy to verify that products of diffusively Lagrangian functions are diffusively Lagrangian, and therefore v . ( is diffusively Lagrangian. The definition is reasonable, because in view of vex) = veal, ((x ) = zeal
it follows that T
(v.! . w)dx =
A(T)
V · zda z
expresses the helicity in terms of the solution v . of a parabolic equation which starts as a perturbation of the heat equation with diffusivity 1/.
211
DIFFUSIVE LAGRA NGIAN TRANSFORMATIONS
5. Resettings and vortex reconnect ion The determinant of
A
obeys
D.,(u, \7 ) (log
(5.1 )
When v =
\7
Det(\7A»
=v
{ CL CZ;i } .
0 this equation states the conservation of incompressibility,
as the initial
data is zero, and the evolution does not change this. In the presence of viscosity
numerical calculations
( [1 7)), ([18])
show that viscosity has a dramatic effect, and
the non-dimensional integrals V
tt
{ Ck;s CZ;J
to
dt
can be bOWlded away from zero far vaflishingly small viscosity. This is of course a statement about the development of large second order derivatives of the diffusive Lagraflgian flow map, but it is also related to the physical phenomenon of vortex reconnect ion . One can consider resetting times
o
0 and a control function u E H l (0, T) such that the solution y in Loo « O, T) x (O, L)) of
Yt Yxx = f e y) , y(t , O) = 0, y et , L) = u(t) , y eO , x) = Yo (x), -
(10) satisfies y e T, ) = Y1 ' REMARK
.
3. This is a (partial) global exact controllability result. The time needed in the proof of Theorem 2 given in [20) is large, but on the other hand there are indeed cases where the time T of controllability cannot be taken arbitrarily small. For instance in the case where f ey) = _ y3 , any solution of (1 0) in £"" « O, T) x (0, L) ) starting from 0 satisfies the inequality
10 (L - x)4y(T, x)2dx � 8LT, L
and hence, if YO = 0, a minimal time is needed to Teach a. gi'IJen Yl i= O . This result is due to Bamberger [47] , see also [36, Lemma 2.1] . Note that Diaz has found in [23] an obstruction to global controllability even in large time. In [20) it is proved that PROPOSITION 4. If Yo and Yl belong to di.�tl:nct connected components of S, then it is actually impossible to move either from Yo to Yl or from Yl to Yo, whatever the tim.e and the control are. The result of Theorem 2 may be achieved directly by using repeatedly a lo cal exact controllability theorem, see Fursikov-Imanuvilov [36, Theorem 4.4] or Imanuvilov [50, Theorem 3.3] . But in [20) we pre�ent a new controllability strat egy, based on a feedback stabilization procedure, which is therefore more robust to perturbations. It is clear also that this approach may be applied to other problems, without requiring controllability of the linearized system around an equilibrium; see [14] for an example. Let us now turn to our open problems related to the previous results. We take
( 11) As mentioned above, it is one of the cases where, for every T > 0, there are initial data Yo E LOO(O, L) such that Cauchy problem (9) ha.s no solution y E L oo ( (O, T) x (O, L)) . One may ask, if for this special nonlinearity, the blow-up phenomenon can be avoided with a control at x = L, that is
21 8
JEAN-MICHEL CORON
OPEN PROBLEM 1 . Let us assume that (11) holds. Let Yo E LOO (O, L) and let T > O . Does there exists U E LOO(O, T) .mch that the Cauchy problem (2)- (3)-(4) has a solution
Note that it seems to be a case where one cannot use the proof of [32, Theorem 1 . 1] to get a negative answer to Open Problem 1 . One could be even more "am bitious" and ask if our control system is exactly controllable in an arbitrary time, i.e. ask the following question OPEN PROBLEM 2. Let us assume that (11) holds. Let T > O. Let Yo E Loo(O, L) and let y E LOO « O, T) x (0, L)) be a solution of Cauchy problem (2)-(3) (4) for some initial data Yo E L oo (O, L) and some control u E Loo(O, T) . Does there exists u E Loo(O, T) such that the solution y E L= « O, T) x (0, L)) of the solution of Cauchy problem (2)-(3)-(4) exists and satisfies such that y(T, x) = y(T, x), x E (O, L)?
Let us recall ( see Remark 3) that, if one replaces (11) by f ey) :- _y3, the answer to this question is negative. One could also ask questions about the connectedness of S in higher dimen sion. Indeed the proof of Theorem 2 given in [20] can be easily adapted to higher dimension (see also [21]). Let us recall ( see the proof of Proposition 3.1 in [20]) that, for every smooth bounded open subset n JR, c
is a connected subset of 02 (0).
In particular:
One may ask if the same holds in larger dimension.
OPEN PROBLEM 3. Let 0 be a smooth bounded open set of JR2 . Let
Is 6(0) a connected subset of C� (O) ? 2. Viscous Burgers equation
In this section we consider the following control Burgers ( 12)
Yt - Yxx + YYx = 0, (t, x)
E (O, T) x (0, 1). :
For this control system, the state at time t is y (0, 1) JR. We consider two case::; (i) The ca::;e where one requires also that yet, 0) = 0 (Subsection 2.1), ( ii) The case where no more condition is required ( Subsection 2.2). We could also consider the case when one requires (besides (12)) y(t, 1) O. But this case can be reduced to the case where one requires y(t, 0) = 0 just by making the following change of variables x := 1 x and fj := -yo ->
=
-
2 19
OPEN PROBLEMS IN CONTROL THEORY
2.1. Control at x = 1 . In this subsection the control systeIJl considered is Yt - Yxx + YYx y(t, O)
( 13)
= =
0, 0,
(t , x ) E (0, T) x (0, 1), t E (0, T),
where, at time t E (0, t), the t;tate is y(t , ·) : (0, 1) --+ R For the control at time t , we can take, for example yet, 1 ) . But other choices are possible, as, for example, Yx (t , I ) and we prefer to not specify the choice: the control system (40) is just considered as an underdetermined equation . The problem of controllability is the following one. Given T > 0, g E L2« 0, T) x (0, 1 » satisfying ( 13) and yO E L2(0, 1), does there exists y E L2 « (0, T) x (0, 1 » satisfying ( 13) together with (14)
y (O , x )
=
yo ( x) , y(T, x )
=
y (T, x ) , x E (0 , 1 ) ,
REMARK 5. Throughout Section 2, the equation ( 15)
Yt - Yxx
+ YYx
=
0 , (t, x) E (0, T) x (0, 1),
stands, as usual, for ( 16)
Yt - Yxx +
=
° in V' «(O , T)
x
(0 , 1 » ,
where D' « (O, T) x (0, 1» denotes the set of distributions in (0, T) L2 « 0, T) x (0, 1 » satisfies (16), then ( 17)
x
(0, 1 ) . If y E
Yt E L2« 0, T), H-2 (0, 1 » ,
Yxx E L2 ( (0, 1), H-1(0, T» .
(18) From (1 7), one gets
Y E C([O , T] , H-2 (0, 1 » .
(19) From (18), one gets
Y E C([O, 1] , H- 1 (0, T» .
(20)
Hence, by (20), the boundary condition y(t , O)
has to be understood, as usual,
as
Y (-, O)
=
=
0, t E (0, T)
° in H-1 (0, T) .
Similarly, using now (19), (14) has to be understood, as usual, as yeO, . )
=
Yo in H-2(0, 1), y (T, ·)
=
Y(T, ·) in H-2(0 , 1 ) .
We use this convention, and similar usual conventions, throughout the whole paper. (We have already used such conventions in Section 1.) Let, for T > 0, W�,2 « 0, T) that
x
(0, 1 ) be the set of y E L2 ( 0, T) x (0, 1 » such
Yt and Yxx al·e in L2 ((0, T) x (0, 1)).
One has the following result of local controllability, due to Fursikov and Imanuvilov [37, Theorem 6.1].
JEAN-MICHEL CORON
220
THEOREM 6. Let T > 0, let i)
E
Wg,2 « 0 , T)
(0, 1» be such that (1 3) holds (for y : = y). Then there exists c > ° such that, for every Yo E HI (O, 1) such that Yo(O) = ° and lIyo yeO, ') I IHl(o,l) < c, there exists y E Wi ,2 « 0. T) (0, 1» x
x
-
satisfying (13) together with
yeO , x) = yo (x ), yeT x ) = yeT, x), ,
x E (0, 1),
(In fact [31, Theorem 6.1] deals with interior control but the proof can be easily adapted to treat the case uf boundary control. See also [36, Section 5 . 1].) Fernandez-Cara and Guerrero have proved in [30] the following property: what ever is £ > 0, there exists T > ° and Yo E L2(0, 1) with I IYollL2(O,I) � c Sllch that there is no y E L2« 0, T) x (0, 1» satisfying ( 13) together with
y(o, i ) = yo (x) , y (T, x )
(2 1 )
=
0, x E (0, 1 ).
In fact, if '1'(c) is defined as the infimum of the time T such that, for every yo E L2(0 , 1) such that II Yoll p(o,l) � c, there exists y E L2« 0, '1') x (0 , 1» satisfying (13) together with (21), they have got the behavior of T(c) as c 0+: --)
THEOREM 7. (130] .) There exists 8 > ° such that, for every c < {), 8
T (c) � � In(I/c)
1 8 1n(l/c) '
Let us make some comments on this theorem. Nole that Theorem 6 is a local controllability result for a fixed time T > 0. Theorem 7 shows that, as '1' 0+ , the local controllability becomes quite bad: One has, if we denote by c(T) the (best) c; in Theorem 6 for fj := 0, --)
lim £('1') = O. T-+OT We expect that this type of behavior holds for many nonlinear partial differential equation.". In fact, it already holds for many control systems in finite dimension. For example, let us consider the following very simple control system
(22)
x=
(23)
sin(u),
where the control is u E JR and the state is x E JR. The linearized control system at the equilibrium (x , u ) : = (0, 0) is the linear control system
x = u,
(24)
•
where the control is u E R and the state is x E JR. The linear control system (24) is controllable. Hence, by a classical theorem (see, e.g., [16, Theorem 3.8]) the nonlinear control system (23) is locally null-controllable in arbitrary t ime T 0 , 0, there exists c;(T) > 0 such that, for every a E R such that Le., for every T lal � c(T), there exists u E LOO (O, T) such that the solution x E C([O, '1'] , IR) of the Cauchy problem
>
>
(25) (26 ) satisfie!:l
(27)
x
=
sin(u{t» ,
x(O) = a, x ( T) = O.
OPEN PROBLEMS IN CONTROL THEORY
221
(In fact, for this very simple nonlinear control system, this null�controllability result is obvious.) From (25) and (27) , one has T I sin (u( t» ldt � T. Ix(O) 1 � o
Hence, taking a = c:(T), one gets
c:(T) � T. In particular (22) holds. The reason for this phenomenon is not mysterious and is the following one. Let us start with a :;tate which is not O. In order to steer the linearized control system from this point to 0 in small time, we need to usc large controls. However, if one uses large controls, the linearized control has nothing to do with the nonlinear control system and the nonlincarity is going to play a crucial role. In this example, the nonlinearity is in the control. Nevertheless, the same phenomenon holds for the control affinc system Xl = sin(x2 ) ' X2 = u,
where the control is u E lR and the :;tate is ( Xl > X 2 ) E IR? (For this control system, the nonlinearity is in the state.) In conclusion, the nonlinear terms play a crucial role in general for t.he behavior of e(T) as T 0+. Concerning the global null controllability, it has been proved by Fursikov and Imanuvilov in [36J that, for every Yo E L2(0, 1), there exists T > 0 depending on Yo such that there exists y E L2« 0, T) x (0, 1» satisfying (13) together with (21) . The next theorem, which, as far as we know, seems to be new, tells us that one can choose T independent of Yo (and so there exists C > ° such that T(r) � C for every r > 0). ---+
THEOREM 8. Ther'e e.rists T > 0 such that, for every Yo E L2(0, 1), there exists y E L2« 0, T) x (0, 1 » satisfying (13) together with (21). The key lemma to prove Theorem 8 is the following one. LEMMA 9. Let T' > 0 and let y E L2« 0, T') (28)
+ YYx = 0, (t, x)
x
(0, 1» be such that
(0, T') X (0, 1), y (t, 0) = y et, 1) = 0, t E (0 , T').
Yt - Yxx
(29)
Then
-I
(30)
t
-x
�
y(t, x)
E
x
�
t ' (t , x) E (O, T' )
x
(0, 1 ) .
Let us prove this lemma. Let T' > 0 and let. Y E L2 « 0, T') x (0, 1» be such that (28) and (29) hold. By density of Loo (O, 1) in L2 (0, 1) we may assume that y(O, ·) E LOO(O, L). Let B > 0 he such that
(31) Let
c:
> O. Let y
E
yeO, x) COO ([O, T'J
y(t, x) _
:=
x
� B, x E (0, 1).
[0, 1]) be defined by
Be + x
c+l
,
Vet, x) E [0, T ] I
x
[0, 1].
222
JEAN-MICHEL CaRON
Then (see (29) for (33) and (3 1 ) for (34)) Yt - Yxx
(32) (33) (34)
+ YYx = 0, in [0, T']
x
[0 1 ] ,
0 = yet, 0) � yet , 0), 0 = yet, 1 ) � y et , 1), t E (0, T') , y ( O, x ) � y (O, x), x E (0, 1).
From (28) , (32), (33) , (34) and the maximum principle for parabolic equations, one gets y(t, x) � y(t, x) �
(35)
Bf: + X
, (t, x)
E (O, T ) I
x
(0, 1).
t Letting f: 0+ in (35) one gets the second inequality of (30) . Finally the first inequality of (30) follows from the second one applied to the function (t , x) E (0, T') X (0, 1) t--> - yet , 1 - x). This concludes the proof of Lemma 9. .....,
REMARK 10. Inequalities (30) are independent of the viscosity: They hold with (28) replaced by Yt - VYxx + yYx = 0, whatever is v > 0. When v = 0, they are the Oleinik inequality (see, e.g. [71, (2.32)] or [55, (3.31)]), which hold for enf1'opic solutions of Yt + YYx = 0.
These inequalities hold for much more general nonlinearity, even with nonlinear viscosity adapted to the flux (see, e.g. , [26]). Let us go back to the proof of Theorem 8. Note that (30) implies that
(36)
1 lIy (T , X) IIL2(O, 1 ) � " T I
By Theorem 7, there exists v > ° such that, for every Yo E L2 (0, 1) such that II Yo llu(O , l) � v, there exists y E L2 « 0, 1) x (0, 1)) satisfying (13),
(37) (38)
yeO , x ) = yo(x ) , y ( l , x) = 0, x E (0, 1), y (t, O) = 0, t E (0, 1).
Then, taking T' = l/v and using also (36), on gets that Theorem 8 holds with T : = T' + 1 = ( l/v) + 1 . In contrast with Theorem 8 one has also the following results, due to Diaz [24, Theorem 1], and to F'ursikov and Imanuvilov [37, Lemma 6.2 page 59], which give obstructions to global (approximate) controllability in a uniform time for the (viscous) Burgers control system (13).
THEOREM 11. [24 , Theorem 1] Let Yo E L OO ( O, 1 ) . Then there exists C > ° such that, for every T > ° and for every solution y E L OO « O, T) x (0, T)) of (1S) satisfying y (O x) = Yo (x), x E (0, 1), one has
,
ye t, x) �
C I-x
, (t, x) E (0, T)
x
(0, 1).
OPEN PROBLEMS IN CONTROL THEORY
,
223
THEOREM 1 2 . [37, 6. 2 59J There exists I< > ° such that, for every T > ° and for every solution of the control system (13),
Lemma page
d 1 (39) ( 1 - x)6yt (t: x)dx < I 0. Then there exists I< := I< (7) such that, for every '{J LOO (O, 1), there exists z = Z'T ('{J) E LOO« 0, 7) x (0, 1)) such that Zx E
L2« 0, 7)
E
x
(0, 1 )) ,
(42) (43) (44)
Zt - Zxx = 0, (t, x) E (0, T)
(0, 1), z (O , x) = '{J (x), Z (7, x) = 0, x E (0 1 ) X
,
,
Let T > 0. Let us choose 71 > ° and T2 > 0 in such a way that T = 71 + 72 . Let us2 also point out that, if y E L2((0, T) x (0 , 1 )) satisfies ( 40) and (41), then y E L « 0, T) x (0 1 ) ) defined by yet, x) := -yet, 1 - x ) satisfies also (40) as well as ,
y(O, ')
=
0, y(T, · ) = -C.
JEAN-MICHEL CaRON
224
Hence, we may assume that C < O. Let h E Loo « O, T) requIrIng •
x
•
(0, 1 ) ) be defined by
ht - hxx = 0, (t, x) E (0, Td x (0, 1 ) , h(t, 0) = 0 , h(t, 1) = 0 , t E (0, TI ) , h (O , x )
= e 1c 1 / 2 - e I C l x /2
,
x E (0, 1).
By the maximum principle for paraholic equations, (45)
Note also that hx
(46) Let w
E
0 ::;; h(t , x ) ::;; e1c1 /2 , (t, x) E ( O, TI ) x (0, 1 ) . L 2 «0, T) x (0, 1 ) ) . Let us extend h to h , T] x [0, 1] by
h(t, x) : = ZT2 (hh , · ) )(t - TI , X) , (t , x) E ( TI T) x (0 , 1) . L OO « O T) x (0, 1)) be defined by 4 / w et , x ) : = e IC l2 t/ e lc lx 2 + h (t, x) , (t, x ) E (0, T) x (0, 1). ,
E
(47) One has
,
Wx E L 2« 0, T) x (0, 1)), Wt - Wxx = 0, (t, x) E (0, T) x (0, 1 ) ) .
(48) (49)
Let us cheek that, if ICI is large enough,
wet, x ) �
(50) By (4 5) and (47) ,
wet , x)
1,
(t , x) E
(a, T)
� e ICI ' t / 4elcl x/2 � 1 ,
(t, x)
x
(0, 1 ) .
E (0, TI )
By ( 44) , ( 15) , (46) and (47) , we have, on h , T) x (0, 1 ) ,
x
(0 1 ) . ,
2 C 1 1 117 (t , x) � e t/ 4 - K (T2) llh(Tl , ·) IILOO(O.1) � e l C l Td4 K ( T2 ) e 1 c 1/2 2
_
.
Henee (50) indeed holds if 101 is large enough. Therefore, if 101 is large enough,
y := -
2117x
w
E
L2 ( ( O, T)
x
(0, 1 ) ) .
Moreover, by the property of the Hopf-Cole transformation recalled above, (49) and (50), (40) holds. Finally it follows easily from our construction that, for every x in [0, 1 J , x/ 2 ) C I C l x/ 2 _ i 2 elcl 1 / x + e (e yeO, x) = -2 = 0, x E ( 0 1 )
y eT, x)
=
e1c1/2 (e IC I2T/4 e 1 c 1 x/ 2 ) " -2 2eICI2T/4 eIClx/2
,
,
= C, x E (0, 1).
Hence (41) also holds. This concludes the proof of Theorem 13. 15. One can provide a different proof of Theor·em 1 3 by proceeding in the following way: Take, for t E [0, T/ 2] y(t, 0) = y ( t, 1 ) = C. Then straightfor ward estimates show that Ily(T/2, ·) - CIILOO (O,I ) < ", -1 exp( -",C2 T) , for a suitable 1) independent of C > 1 and 0 < T < 1 with C2 T > 1 . Then one concludes by using an analogous of Theorem 6 for the control system (40) with fi = C, but with a careful estimate of the dependance of f. on C and T similar to the ones obtained in [19] . This new proof is more complicated, but is also much more flexible. In REMARK
,
225
OPEN PROBLEMS IN CONTROL THEORY
particular it works for many other equations where the Hopf- Cole transformation cannot be used. It also shows that, with the notations of Theorem 1 3, there p..'Eists A > ° such that one can take M = AIT (this can also been obtained with our Hopf- Cole 's approach).
Looking at Theorem 6 and at Theorem 13 leads naturally to the following open problem. 4. Let T > ° and C E IR. Does there exists y E L2 « 0, T) (0, 1 » satisfying (40) such that y(O, ') = 0, y(T, · ) = C? OPEN P ROBLEM
x
Recently Guerrero and Imanuvilov have got in [46] the two following theorems, which provide uncontrollability results for the control system (40).
16. There exist T > 0, yO E HI (0, 1) and c > ° such that, for every y E L2« 0, T) x (0, 1» satisfying (40) and THEOREM
y (O, ') one has
= yO,
lI y (t , ' ) II H1 (U,1) �
C.
17 . For evenJ T > 0, there exist yO E H 1 (0, 1), yl E H1 (0, 1 ) and c > ° such that, for every y E L2« 0, T) x (0, 1» satisfying (4 0) and THEOREM
y(O, .)
one has
= yO ,
I l y(t, ' ) - y I IlHl( ,l) � C. O
18. For the nonviscous Burgers equation (i. e. without the term -Yxx), results have been obtained by Ancona and Marson in [2] and by Horsin in [48] . Note REMARK
that it is surely important to have a good understanding of the contmllability of the nonviscous Burgers equation if one wants to study the global controllability of the viscous Burgers equation in small time. Indeed for large states and fixed time or fixed states and small time the nonlinear term y Yx is surely a key term compaTl�d to Yxx ' Similarly, it is by using results on the controllability of the Euler equations [11, 13, 42] that one gets global results for the contmllability of Navier-Stokes equations (see [12, 18] and section 6).
3. Singular optimal control: A linear I-D parabolic-hyperbolic example
Let ( T, L, M) control system f: ,
(51)
E (0, + 00)3
X
R.
We consider the following parabolic linear
(t, x) E (0 , T) Yt - f: Yxx + M Yx = ° y(t , O) = u(t) , yet, L) = ° t E (0, T), y (O , x) = yO (x) x E (O, L),
x
(0, L) ,
where the state is y(t, ·) E L2(0, L) and the control is u(t) E R We are interested in the dependence of the cost of the null controllability of system (51) with respect to the four parameter::; T, L, M. It is already known that, for every T > 0, the control system (51) is controllable to the null final state at time t = T. This means that, for every yO E L2 (0, L) and E,
226
JEAN-MICHEL CORON
for every (e, T, M) E (0, +00)2 X JR, there exists u E L2(0, T) such that the (weak) solution of (51 ) satisfies yeT, . ) == O. This result is due to Fattorini and Russell [28, Theorem 3.3] . See also Imanuvilov [49, 50], Fursikov-Imanuvilov [37] , and Lebeau Robbiano [56] for parabolic control systems in dimension larger than 1. ( The last reference does not explicitly deal with transport terms; but the proof given in [56] can perhaps be adapted to treat transport terms. ) For yO E L2(0, L), we denote by U (c, T, L, M, yO) the set of controls u E L2 (0, T) such that the corresponding solution of (51 ) satisfies y(T, · ) = O. Next, we define the quantity which measures the cost of the null controllability for system (51) : K(e, T, L, M)
(52)
e
-->
:=
lIyO I I L' (O , L ) � l
sup
{ min{ll u ll £2 (O,T) : u E Ute, T, L, M, yO) } }.
In this section we are looking for estimates on K(e, T, L, M), in particular as 0+ . Let us point out that simple scaling arguments lead to the relations K(e, T, L, M)
(53 )
=
1
M / 1 2 c, a T, a L, 1 / K 1 /4 a 2 a
and (54)
for every (a, e, T, L, M) E (0, +00)4
X
JR.
In order to understand the behavior of K(c, T, L, M) as e -> 0+ , it is natural 0+. This is to look at the limits of trajectories of the control system (51) as c done in the following proposition, proved in [19] . --t
PROPOSITION 19. Let (T, L, M) be given in (0, +00)2 xJR' and let yO E L2 (0, L) . Let (Cn)nEN be a sequence of positive real numbers which tends to 0 as n -> +00. Let (U,,)nEN be a sequence of junctions in L2(0, T) such that, for some u E L2(0, T), (55)
Un
converges weakly to U in L 2 (0, T)
as
n
--t
+00.
For n E N, let us denote by Yn E CO([O, T) ; H- 1 (0, L» the weak solution of (56) (57) (58)
Ynt - Cn Ynxx + M Ynx = 0, (t, x ) E (0, T) x ( O, L) , Yn (t, O) = un (t) , Yn (t, L) = 0, t E (0, T),
O Yn (O, X) = y (x ), x E ( O, L ) .
For M > 0, let Y E CU([O, T] ; L2(0, L)) Yt + M yx = 0 y (t, O) = u(t) (59) yeO, x ) = yO (x ) and, for M
0 such that, for every (e , T, L, M) E (0, +00)3 X
(62) K(e, T, L, M) � C1
l -3/2 -1/ 2 2 e T L M /2
(63)
1
L3 M3 + -".e3
exp
M (L 2c:
_
TM)
_
7r2eT L2
if M > 0,
C3 /2 T- l/2 L2 1 M1 1/2 T MI ( I 2L T I M I) 7r2 c: K(e, T, L, M) � C1 exp if M < o. 2c: L2 1MI'-3 L 3 '-0;-1 + -' 3 e Concerning upper bounds of K(c:, T, L, M), let us point out that • If M > 0 and T > LIM, the control u = 0 steers any state to 0 in time T for the control system (59), where the state is y(t, ·) E L 2 (0, L) and the control is u(t) E R ( This means that, if M > 0, T > LIM and u = 0, then, for the function y defined in Proposition 19, y(T, ·) == 0 whatever is yO E L2 (0, L).) • If M < 0 and T > LIIMI then, for the function y defined in Proposition 19, y(T, ·) = 0 whatever is yO E L2 (0, L ) . This could have given the hope that, for every CT, L, M ) E (0, +00) 2 X JR" with T > L/IMI, _
_
K(c:, T, L, M)
(64)
-->
0
as
e
-->
0+ .
As shown by Theorem 20, this turns out to be false for M < 0 and T E (LI I M I , 2L/IMI). Our next theorem, proved in [19] , shows that (64) holds if TIMI/L is large enough.
THEOREM 2 1 . There exists C2 > 0 such that, for every (e, T, L, M) E (0, +00) 3 x R* with
(65) •
If M > 0 and
(66) (67)
then
L T � (4 . 3) M'
3
-
4
2 2TM 2.61 1 3L -
•
228
JEAN-MICHEL CORON •
If M
° such that, for every nO, nf E N", for every ('l/Jo, So, Do), (1Pf , Sf, Dj ) E [§ n H(O) (I, C)) x lR x R with (76) (77) there exists
( 78 ) (79)
(80)
a
I I'l/Jo - r.pno ll H7 + I So l + I Do l < TJno ' II 'l/Jj r.p", llw + I Sf l + I Dj l < TJnf ' time T > ° and ('l/J , S, D, u ) such that 'l/J E CO([O, T] , H2 n HJ (1, C) n C1 ( [0, TJ , L2 (1, C) , u E HJ (O, T), S E Cl ([O, TD , D E C2 ( [O T]) , -
(70) , (71) and
(81)
(82)
,
(72) hold,
('l/J (O) , S(O ) , D(O» = ('l/Jo, So, Do), ('l/J (T) , S(T) , D(T) ) ( 'l/Jj, Sf , Dj). =
(83)
Thus, we also have the following corollary.
CO ROLLARY 23. For every no, nf E W" , there exists a time T > ° and ('l/J, S, D, u ) satisJ'l/ing (78) to (81) such that ('l/J (O), S(O), D(O» = ( r.pn , O , O) ('l/J(T), S(T), D(T» = a
(r.pn" O, O) .
,
Note that, if one does not care of S and D and if (no , nf ) = (1, 1), Theorem 22 is due to Beauchard [4] . There are lea.st three points that one might want to improve in Theorem 22: (i) Remove the a.ssumptions (76) and (77) on the initial data and the final data, (il) Weaken the regularity a.ssumptions on the the initial data and the final data, (iii) Estimate the time T of controllability. Concerning (i), on can propose, for example, the following open problem. OPEN PROBLEM 6 . Let ('l/Jo, So, Do ) , ('l/Jf, Sj, Df) E [§ n H(o) (I, C l] Does there e.'lits T > 0 and ( 'l/J , S, D, u) such that (78) to (83) hold?
x
lR x
R
Concerning (il), one can propose, for example, the following open problem. OPEN PROBLF,M 7. Let
Hro) ( I, C) ; = {r.p E H7 (1, C); r.p(2k) (0)
=
r.p(2kl (l)
=
0 for k
=
0, 1}.
Let ('l/Jo, So, Do), ('l/Jf , Sf, Df) E [§ n HroP' C)] x lR x R Does there exits and ('l/J , S, D, u) with u E L2(0, T) such that (78) to (83) hold?
T>°
230
JEAN-MICHEL CaRON
REMARK The regularity conjectured in Open Problem 7 comes from the regularity for the controllability of linearized control systems (see, in particular, [4, 336]). 5, 862] and [5,
24. Theorem page Proposition 2, page Concerning (iii) , since the speed of propagation for the Schrodinger equation is infinite, one could expect that the following small-time local controllability property. For every c > 0, there exists T} > ° such that, for every (,00, So , Do ) , (,0f, Sf, Df ) E [§ n H[O) (I, C)] x R x R with 11'1/10 - 11'1 (0, ' ) IIH7 + I So l + I Do l < T},
(84) (85)
II'I/If - 'I/Il ( c, ' ) llw + I Sf l + I Df l < T} ,
there exists ('1/1, S, D, u) such that (78) to (83) hold with T = c and Il u IIHl (U,e)
:(
e.
However this is false. Indeed, one has the following theorem ([16, Theorem 9.8]; see also [15, Theorem 1 .1] for a weaker result) . THEOREM 25. Let T > ° be such that
T
0 and k E N \ to} . Let us assume that (95) (i + 12 + jl = 3k2 and (j, l) E (N \ { 0}) 2 ) =} (j = I k) . =
2k1l". There exists r > 0 such that, for every Yo , YT E L2 (0, L) with Let L II Yo l i ,,2 (0,L) < r and Ii YT IIL2(0 ,L) < r, there exists y E G([0 , TJ , L2 (0, 2k11"» n L2 ( (0, T), Hi (O, L» satisfying (KdV) such that y(O, ·) Yo and y(T, ' ) = YT . =
REMARK 28. Assumption (95) holds for k � 6. There are an infinite number of positive integers k such that (95) does not hold: see [16, Remark 8.2) . How ever, there are also infinitely many positive integers k such that (95) holds: see [16, Proposition 8.3)}. Unfortunately, we have forgotten assumption (95) in [1 7, Theorem 2] . Thi,s is a mistake: the proof of [17, Theorem 2) requires (95). =
When
L = 2k1l"
0
the linearhled control system of (KdV) around
Yt + Yx + Yxxx = 0 , y(t, O) = y(t, 2k11") = o.
(KdVL) It has been shown by Rosier in
is
[66] that this linear control system is not controllable.
To prove that the nonlinear term
yYx
gives the local controllability, a first idea
could be to use the exact controllability of the nonlinear equation around nontrivial stationary solutions proved in
[22)
and to apply the method introduced in
[14]
(that
is, use the return method together with quasi-static deformations) . But, with this method, we could only obtain the local exact controllability in proving Theorem 2 7 one uses in
[17]
large
time.
For
a different strategy that we briefly describe
now. One first points out that in this theorem we may assume that
Yo = 0:
this
follows easily from the invariance of the control system (KdV) by the change of variables
r
= T - t , � = 2br - x.
Then one uses the following result for the linear
control system.
THEOREM 29 . (166,
Remark
Let T > 0 and let H :=
3.6
ii)l ')
y E L2 (0, 2k11"),
Let k E N \ to} be such that (95) holds.
2k1r
o
y ( l - cos(x»
dx
=0 .
For' every (Yo, YT ) E H x H, there exists y E G([O, T] , L2(0, 2k11"» nL2((0, T), HI (O, 2k1!"» satisfying (KdVL) such that y(O, ·) = Yo and y (T, . ) = YT . Next one can see that the nonlinear term
yYx
allows us to "go" in the two
directions ± ( l - cos(x» which are missed by the linearized control system Finally one derives Theorem
27 by means
of a fixed point theorem.
(KdV L) .
OPEN PROBLEMS IN CONTROL THEORY
233
For the other critical lengths, the situation is more compLicated: there are now four noncontrollable (oriented) directions of the linearized control system around ° and there are Less explicit than ±(1 cos(x)). But our guess is that Theorem 27 also holds for all the other critical Lengths, that is the answer to the following open problem should be positive. -
10. Let L > ° and T > 0. Does there exist r > ° such that, for every YO, YT E L2(0, L) with IIYo l l u (O,L) < r and IlyT il L2 (0,L) < 1', ther'e exists Y E G(IO, TJ, L2(0, L)) n L2« 0, T), HI (0, L)) satisfying (Kd V) such that y(O, · ) = 110 and y (T, · ) = YT . OPEN PROBLEM
All the previous results are local controllability results. Concerning global controllability results one has the following result due to R.osier [65J .
T
THEOREM 30. For every Yo E L2(0, L), for every YI E £2(0, L), there exist > ° and 11 E G([O, T], L2(0, L)) n L2« 0, T), HI(O, L)) satisfying
such that y(0,
.
Yt + Yx + YXXI + YYx ) = Yo and y eT, . ) = YI ·
=
° in D'«
O, T) x (0, L))
Note that in this theorem (1) One does not require y(t, 0) = yet, L) = 0, (2) A priori the time T depends on Yo and YI . lt is natural to see if one can remDve these restrictions. For example, one may ask
T > 0 , Yo E L2(0, L) and YI E L2(0, L) . Docs there exists Y E G(IO, T], L2 (0, L)) n L2« 0, T), HI (O, L)) satisfying (KdV) such that OPEN PROBLEM 1 1 . Let L > 0,
y(O, · )
= Yo and y (T, ') = Yl ?
If one does not care of T (Le., if we allow T to be as large as wc want, depending on Yo and Yl), a classical way to attack this open problem is the following one (this is the way which is already used by R.osier to prove Theorem 30). Step 1 Use the reversibility with respect to time of the equation to show that one may assume that YI = 0. (In fact this part holds even if one deals with the case where one wants T > ° to be small.) Step 2 Use a suitable stabilizing feedback to go from yO into a givcn ncighborhood of O. Step 3 Conclude using a suitable local controllability around y : = O. Step 1 indeed holds (perform the change of variables (i, x) : = (T - t , L - x)) . Let us assume that L +oo
There are available results showing that (100) holds if there is some interior damping, more precisely with (96) replaced by
Yt + Yx + Yxxx + YYx + a(x)y
where a E LOO (O, L) is such that
=
0,
t E (0, +(0), x E (0, L),
a(x) � 0, x E (O L )
the support of a has a nonempty interior. ,
,
Tbese results are in [64] by Perla Menzala, Vasconcellos and Zuazua, in [63] by Pazoto, and in [68] by Rosier and Zhang. 6. Navier-Stokes equations In this section we present some results and open problems on the Navier-Stokes equations of incompressible viscous fluids. Let us introduce some notations. Let I E {2, 3} and let f! be a bounded nonempty connected open subset of ]Rl of class ceo. Let fo be an open subset of f af! and let f!o be an open subset of 12. We assume that
:=
(101)
The set f 0 is the part of the boundary f and 120 is the part of the domain 12 on which the control acts. The fluid that we consider is incompressible so that the velocity field satisfies div On the part of the boundary f\fo where there is no control, the fluid does not cross the boundary: it satisfies
y
(102)
y = O.
n
where denotes the outward unit normal vector field on f. Besides (102), the fluid satisfies on f\fo, , some extra conditions which will bc specified later on. For the moment, let us just call by Be all the boundary conditions satisfied by the fluid on r\fo. For simplicity, often we omit to specify the regularities of the functions considered (one can find these regularities iII the papers mentioned) . Let us introduce the following definition.
(102))
(including
OPEN PROBLEMS IN CONTROL THEORY
235
DEFINITION 31 . A trajectory of the Navier-Stokes control system on the time interval [0, T] is a map y : [0, T] x f/ Rl such that, for some function p : [0, T] x f/ lit, ---+
---+
(103)
Yt
-
t1y + (y 'V) y + 'Vp = 0 in [0, T] .
div Y = 0 in [0, T]
(104)
x
TI,
x
(f/\f/o),
(105 ) y(t, . ) satisfies the boundary conditions Be on 1'\1'0, "It E [0, TJ . J.-L. Lions's problem of controllability is the following one: let T > 0, let Yo : f/ lItl and Yl : f/ lItl be :mch that div Yo = 0 in 0, (106) ---+
--t
(107) ( 108)
div Yl
=
° in 0,
Yo sat.isfies the boundary conditions BC on n1'o ,
(109)
Yl satisfies the boundary conditions Be on n1'o , does there exist a trajectory y of the Navier-Stokes or the Euler control system such that
y(O, · )
(1 10)
=
Yo in f/,
and, for an appropriate topology -see [59] , [60]-,
(111)
y(T, ) is "close" to Yl ? .
That is to say, starting with the initial data Yo for the velocity field, we ask whether there are trajectories of the Navier-Stokes control system considered which, at a fixed time T, are arbitrarily close to the given velocity field Yl . If this problem has always a solution one says that the control system considered is approximately controllable. Note that (103), ( 104) , (105) and (1 10) have many solutions. In order to have uniqueness one needs to add extra conditions. These extra conditions are the controls. We will see below a way to replace (111) in order to recover a natural definition of (exact) controllability of the Navier-Stokes equations. Let us now specify the boundary conditions BC. Three types of conditions are considered: • Stokes' boundary condition, • Navier's boundary condition, • Curl condition. The Stokes boundary condition is the well-known no-slip boundary condition
(112) which of course implies (102). The Navier boundary condition [62] imposes, condition (102) , which is always assumed, and
( 11 3)
l7y.T + (1
-
(7)ni
TJ
=
0 on 1'\1'0,
236
JEAN-MICHEL CORON
where (1 is a constant in [0, 1), n = (n1 , , nl ) and T = (7 1 , . , Tl ) is any tangent vector field on the boundary r. In (113) we also have used the usual summation convention. Note that the Stokes boundary condition (112) corresponds to the case (1 = 1 , which we will not include in the Navier boundary condition considered here. The boundary condition (113) with a ° corresponds to the case where the fluid slips on the wall without friction. It is the appropriate physical model for some flow problems; see [41] for example. The case (J E (0, 1) corresponds to a case where there the fluid slips on the wall with friction; it is also used in models of turbulence with rough walls; see, e.g. , [54]. Note that in [9] F. Coron has derived rigorously the Navier boundary condition (113) from the boundary condition at the kinetic level (Boltzmann equation) for compressible fluids. Let us also recall that Bardos, Golse, and Levermore have derived in [3] the incompressible Navier-Stokes equations from a Boltzmann equation. Let us point out that , using (102), one sees that, if I = 2 and if T is the unit tangent vector field on 80. such that ( T, n) is a direct basis of JR2, (113) is equivalent to (1y.T + curl Y = 0 on r\ro, •
.
.
.
.
=
with 0'
E COO(r; JR) defined by
2(1 - a) lI: (x) - a \..I r ) . _ _ vX E , 0' ( x .-
( 1 14)
1 - 0'
,
where II: is the curvature of r defined through the relation 8n/8T = 11:7. Finally the curl condition is considered in dimension 2 (l = 2). This condition is condition (102), which is always assumed, and curl y = ° on
( 115)
r\ro.
It corresponds to the case (J = ° in (114). Due to smoothing properties of the Navier-Stokes equations, one cannot expect to get y(T, ') = Yl , at least for general Yl . For these equations, as for the heat equation and the vi::;cous Burgers equation considered above, the good notion for exact controllability is not passing from a given state Yo to another given state Yt : the good definition for exact controllability is passing from a given state Yo to a given trajecto'T7J y. This leads to the following, still open, problem of exact controllability of the Navier-Stokes equation with the Stokes, or Navier, or curl condition.
O PEN
13. Let T > 0. Let y be
a
tmjecto'T7J of the Navier-Stokes control .5y8tem on [0, T] . Let Yo : 0. -> R!) satisfying (106) and (108). Does there exist a trajecto'T7J y of the Navier-Stokes control system on [0, T] such. that
( 1 16) (117)
PROBLEM
E 0., = y(T, x) , \Ix E m
y(O, x) = Yo (x) , 'Ix y(T, x)
Let us point out that the (global) approximate controllability of the Navier Stokes control system is also an open problem. Related to the Open Problem 13 one knows two types of results • local results, • global results, which we briefly describe in the next ::;ubsections
OPEN PROBLEMS IN CONTROL THEORY
237
6.1. Local results. Let us introduce the following (informal) definition. The Navicr-Stokes control system is locally exactly controllable if, for every (smooth enough) trajectory y on [0, TJ of the Navier-Stokes control system, there exists (' > ° such that, for every ( Hmooth enough) Yo : D Il�J satisfying (106) , (108) and -t
I Yo - yeO , ) 1 < € , for a suitable norm I . I , there existH a trajectory y of the Navier-Stokes control sYRt.em un [0, TJ satisfying (1 16) and ( 1 17) . Then one has the following results (with precise choices of I . ! and of smoothness requirements) . .
THEOREM
32. The Navier-Stokes control .�ystem is locally exactly controllable.
The case of the curl condition has been obtained by Fursikov and Imanuvilov in [39J for l = 2. The case of the Navier boundary condition in every dimension has been obtained by Guerrero in [45J . The case of the Stokes condition has been obtained by Imanuvilov in [51 J and [52J (see also the prior work by F'tU'sikov in [34] as well as the paper [31] by Fernandez-Cara, Guerrero, Imanuvilov and Puel for less regular spaces) . 6.2. Global results. Let d E CO(D; JR) be defined by
d(x) = dist (x, r)
=
Min { I x - xi i ; x' E r}.
In [12] tbe following theorem is proved. T HEOREM 33. Let (10"/) hold and
T > 0, ld Yo and YI in HI (D, JR2) be such that (1 06) and
curl Yo E LOO(D) and cud YI E L= (D), Yo ' n = 0, YI n = 0 on l' \ ro o .
Then, thc7'c e.xists a sequence « yk , Pk ) ; k E N) of solutions of the Navier-Stokes control system on [0, TJ with the Navier boundary condition (1 13) such that (1 18) ( 1 19)
yk and Pk arc of class COO in (0, TJ yk E L2« 0, T) , H2 (D?) ,
(120)
y�
(121)
Pk E L2« 0, T) , HI(D» ,
and such that, as k (122)
-t
+00,
E
x
D,
L2 « 0, T), L2(D?),
l dl1 l yk (T, ·) - Yl ! -t 0, 'riM > 0,
0, lI yk (T, . ) - YI l l w-l . � (rl) and, for all compact K included in D u ro , (124) Il yk (T, · ) - YI IIL� (K) + II curl yk (T, ·) - curl YI II L� (K)
(123)
-t
�
O.
In this theorem, W -I,OO (D) denotes the usual Sobolev space of first derivatives of functions in LOO (D) and II Ilw- l'�(rl) one of its usual norms, for example the norm given in [1, Section 3 . 12, page 64] . As in the proof of the controllability of the 2-D Euler equations of incompress ible inviscid fluids [11, 13J , one uses the return method in order to prove Theorem
JEAN- MICHEL CORON
238
33. Let us recall that it consists in looking for a trajectory of the Navier-Stokes control system y such that y (O, ·)
(125 )
=
y (T, ·)
=
0 in 0,
and such that the linearized control system around the trajectory y has a control lability in a "good" sense. With such a y one may hope that there exist.s y close to y satisfying the required conditions, at. least if Yo and Yl are "small" . Note that the linearized control system around y is -
-
(126)
at
oz -
6.z + (y . V) z + (z . V) y + V7r = 0 in [0, T] x (0\0 0 ) ,
(12 7)
divz = 0 in [0, T]
z.n = 0 on [0, T]
(128) ( 129)
�Z.T
+ curl z
=
x
x
0,
(r\ro ) ,
0 on [0, T]
x
(T'\ro).
In [38, 35] Fursikov and Imanuvilov have proved that this linear control system is controllable (see also [58] for the approximate controllability). Of course it is t.empting to consider the case y = O. Unfortunately, it is not clear how to deduce from the controllability of the linear system ( 126) with y = 0, the existence of a trajectory y of t.he Navier-Stokes control system (with the Navier boundary condition) satisfying ( 1 10) and ( 1 1 1) if Yo and Yl are not small. For this reason, one does not use y = 0 , but y similar to the one constructed in [13] to prove the controllability of the 2-D Euler equations of incompressible inviscid fluids; these y are chosen to be "large" so that, in some sense, "6." is small compared to "(y . V) + ( . 'V')y" . REMARK 34. In fact, with the y we use, one does not have {125}: we have only the weaker property
(130)
y(O, ')
=
0, y (T, ·) is "close " to 0 in 0 .
However the controllability of the linearized control system around y is strong enough to take care of the fact that y(-, T) is not equal to 0 but only close to O. A related observation can be found, in a different setting, in the paper [72] by Sussmann. With the notation of [72], y plays the role of �. in the proof of [72 Theorem 12] and the Euler control system plays for' the Navier-Stokes control system the role played by 9 for F. ,
Note that (122) , ( 123) , and (124) are not strong enough to imply
( 131)
ii yk (T . ) - Yl iiv(!!) ,
-t
0,
i.e. to get the approximate controllability in L2 of the Navier-Stokes control system. But, in the special case where ro = r, (122), (123), and (124) are strong enough to imply (131). Moreover, gluing together the proofs of Theorem 32 and of Theorem 33, one gets
THEOREM 35. ([IB, 40] ') The Open Problem 13 has a positive answer when ro = r .
239
OPEN PROBLEMS IN CONTROL THEORY
6.3. Control on the tangential component of the velocity on the boundary. It would be interesting to know if the Navier-Stokes equations are controllable even if the control is only on the tangent component of the velocity on roo For example, in the case of the Stokes boundary condition, this leads to the following open problem, where ro is assumed to be a non empty open subset of r. OPEN PROBLEM 14. Let T > 0. Let fi
and P E L2 « 0, T), Hl (n» be such that
E £",° « 0, T) , Hl (n» n L2 « 0, T), H2(0»
iit E L2«
Let Yo
E
0, T) , L2(0» , div y = ° in (0, T) x 0, fi(t, x) = 0, (t , x ) E (0, T) x (r \ r a), fi(t, x) · n (x) = 0 , (t, x) E (O , T) x ro, Yt - �Y + ( Y 'V)ii + 'Vp = 0 in (0, T) x n. .
Hl (Oi Rl ) be such that
div Yo = ° in 0 , Yo(x) = 0, Vx E r \ ro, yo(x) · n(x) = 0, Vet, x) E [0, TJ
x
roo
Does there exist y E Loo « O, T), Hl (0» nL2« 0, T) , H2(0» andp E L2 « 0, T), Hl (n» such that Yt E L2« 0, T) , L2 (n» , div y = ° in (0, T) x 0, y(t, x) = 0, (t, x) E (O, T) x (r \ ro ) , yet, x) . n(x) = 0, (t , x) E (0, T) x ro, Yt - b.y + (y . 'Il)y + 'Ilp = 0 in (O, T) x 0, yeO, x) = Yo (x) , x E 0, y(T, x) = fi(T, x), x E m
Acknowledgement. We thank Eduardo Cerpa and an anonymous referee for useful comments.
References [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jo vanovich, Publishers] ' Elsevier, Oxford, 2005, Pure and Applied Mathematics, VoL 1 40, Second edition. MR MR0450957 (56 #9247) [2] Fabio Ancona and Andrea Marson, On the attainable set for scalar nonlinear conserva tion laws with boundary control, SIAM J. Control Optim. 36 ( 1998 ) no. 1 , 290-3 12. MR MR1616586 (99h:93008) [3] Claude Bardos, F\"an 3. The Harnack ineqnality states that for each V CC U there exists C C(V) such that if u > 0, then max u < Cmin1L. I proved in [E3] that if u is smooth, then I Dul also satisfies a Harnack inequality: m�x I Dul < CmJn IDul; and this has the remarkable consequence that a smooth, nonconstant solution of (3.3) can have no c,..itical point within U. Aronsson had earlier proved this for n 2 dimensions. However, this conclusion is dead false for general weak ( that is, viscosity) solutions, a fundamental example of which for n = 2 is =
v
v
-
=
1.1
(3.5)
-
XS1 4
x23 , 4
-
which Aronsson found by separating variables. Blow-up limits. Crandall, Gariepy and I have shown in [C-E-G] and [C-E] that for each point E U that lim u(xo+r;x)-u(xo) if the limit v(x) := rj-O exists, rj (3.6) then vex) (a, x) is linear. This is an interesting assertion in view of the basic open question of regularity for weak solutions. Note carefully that (3.6) does not assert to be differentiable at the point since different rescaled, blow-up sequences could possibly converge to different linear functions. Xo
=
Xo
u
THE
THE
I-LAPLACIAN,
oo-LAPLACIAN
249
AND DIFFERENTIAL GAMES
4. Game theory interpretations 4.1 Minimax formulas_ I again indulge myself in this section with a bit of
philosophy concerning nonlinear problems. me
review convexity. A key point is that convexity is a kind of "one-sided" linearity. More precisely, if : �n -+ � is convex, we can write
Convexity. First, let
(x )
(4.1 )
=
max {(a"', x) + ba }
for appropriate a'" E �n , b" E R This representation formula suggests why con vexity is a fundamental hypothesis for many nonlinear problems: linear and affine functions are simple, and "max" is a simple operation. Conseqllent.iy, convex func tions are (relatively) simple to study. General nonlinearities. Suppose now IJI : ]Rn � is an arbitrary, say Lipschitz continuous, function. Then we can write C
u.
Our dividing
1
IDul2 Since we are in n (4. 9)
(4.8) by E2 and letting
.1 2 .1 Du , D uDu ) (
=
t ->
0 yields
1.
= 2 dimensions, we can rewrite, t,o obtain - IDul�lU = 1 u=O
in U on au.
Our formal derivation is again very suspicious; but the conchlsion is valid, as proved by Kohn and Serfaty, provided we interpret u as a viscosity solution. J.
Spruck and I in [E-SpJ have previously in vestigated the boundary-value problem (4. 9). The geometric interpretation is that the level curves of u evolve by curvature motion, starting at au = {xlu(x) = O}. Consult Kohn and Serfaty's paper [K-SJ for some pictures and more explanation ali to why curvature motion is in fact relevant for their pusher-choose game.
Geometric interpretation.
Questions. In summary, boundary-value problem (4. 9) for the pusher-chooser
(4.5)
for the tug game entails the I-Laplacian, whereas the corresponding problem ot�war game involves the oo-Laplacian. Is there a philosophical explanation for this? Are these problems somehow "dual" ? In view of my previous comments about minimax representations, H is perhaps not so surprising thaL our PDE have game theoretic interpretations, but in what sense are these particular games "natural" ?
252
LAWRENCE C. '"VANS
5. Generalizations
Generalization: other dynamics, running costs. Barron, Jensen and I in [B-E-J] have generalized the Lug-of-war problem by assuming now the dynamics
to be
d:�) d��)
(5. 1 )
= f(X(T) , 1J (T» =
for player I for player II,
-f(X(T) , « T»
for times 0 < T < t" . We also introduce both a terminal payoff and a running cost:
P(1J(') ' « '» := g(x(T:,,» +
(5.2)
o
t*
±h(X(T), O (T» dT.
Here 0 ( ' ) denotes 1J (-) when I iii playing and « . ) when II is playing. The "±" term means that the running payoff is -h(X(T) , 'T}(T» when Player I controls the dynamics, and is h(X(T), ( T» when Player II controls the dynamics. The dynamic programming formula analogous to (4 . 4) for the value function now reads
(,').:�)
uf (x)
1 =
max [u (x + d(x, y» - f.h(x, y)]
2 Ivl< l
'
+ min [uE(x - Ef(x, z» + Eh (x , z)] .
Izl;eR the expression (f(x, w) , p) - h(x, w). u uniformly, as E -+ O. It turns out that then the Now suppose that u' function u is a viscosity solution of the PDE . . (5.4) - !' (x, w ) P (x, w) UX i Xj = O in U, ,
�
for each W E argmax { (f(:c, w), Du) - h(x, w) : Iwl < I } .
We can interpret (5.4) as a generalization of the oo-Laplacian.
Aronsson's equation, forward and hackwards Hamilton·Jacobi flows. Another generalization of the oo-Laplacian is Aronsson's operator'
(5 .5)
defined for H : IRn
�
R, H = H(p) .
We sketch a situation where the Aronsson operator arises. Given H consider the pair of Hamilton-Jacobi equations
(5.6)
Vi
+ H (Dv) = 0 v=g
(t > 0) (t = 0)
0
( t > 0) (t = 0)
and
(5 .7 )
Wt - H(Dw)
=
w=g
as
above,
THE I-LAPLACIAN, THE oo-LAPLACIAN AND DIFFERENTIAL GAMES
253
A degenerate nonlinear wave equation. We formally differentiate (5.6) with respect to t and Xk , to find
for k =
1 , . . . , n.
Vll + Hpk VXk t = 0, V"'kt + Hpj VXkXj = 0
Substitute the second equation into the first, to discover that 1!tt -
AH [v]
=
0,
for the Aronsson operator (5.5) . Similarly, we deduce thaL So Lhe solution� v, tv of the "forward" and "backwards" Hamilton-Jacobi equa tions (5 .6 ) and (5.7) formally solve the same nonlinear wave equation; a deduction that is certainly in general false, since v and tv are not usually smooth. But we do have the rigorous conclusions that v = 9 - tH(Dg) + w as
"2 AH[g] + o(e) t2 = 9 + tH(Dg) + "2AH[g] + 0(t2 )
0, valid for smooth initial data equations to knock out the OCt) terms: t
--7
t2
g.
And we can average these expansions
A nonlinear parabolic PDE. To be more precise, we now introduce nonlinear semigroup notation, writing v : = R(t)g to denote the unique viscosity solution of (5.6), and tv : = 8(t)g to denote the unique viscosity solution Qf (5.7) . Next, set
F(t)
R e .;t) + S e .;t) 2
:=
(t > 0) ,
to record an average of the two dynamics, over the long time scale .;t.
\Vhat happens if we repeatedly apply t.he nonlinear operator F over shorter and shorter time intervals?? In [B-E-J] we demonstrate that the limit 71. : =
lim F(t/n)ng
n � DC
(t > 0)
exists uniformly for eaeh time t > 0; and u is the unique viscosity solution of the parabolic equation Ut -
�AH[U]
=
0
u=g
(t > 0 ) (t = 0) .
The proof uses one of my favorite tools of nonlinear analysis, the "Chernoff formula" for nonlinear semigroups, due to Brezis and Pa:q [B-P .
]
254
LAWRENCE C. EVANS
References G. A ronsson , Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6 (1967) 551-56l. (A2] O. C. Aronsson, O n the partial differential equation ux�xx + 2uxuy uxy + Uy�yy Ark. Mat. 7 ( 1968) , 395-425. G. Aronsson, M. Crandall , and P. Juutinen A tour of the theory of absol utely mini [A- C-J ] mizing functions , Bnll. Amer. Math. Soc. 41 (2004), 439-505. [B-E-J] E. N. Barron , L. C. Evans and R. Jensen, The infinity Laplacian, Aronsson s equation and their generalizations, paper to appear. H. Brezis and A. Pazy, Convergence and approximation of semigronps of nonlinear [B-P] operators in Banach spaces, J. Funct ional Analysis 9 (1972), 63-74. M .Crandall, L. C. Evans and R. Gariepy, Opt imal Lipschitz extensions and the infinity [C-E-G] Laplacian, Calculus of Variations and Partial Differential Equat ions 13 (2001), 123139. M .Crandall and L. C. Evans, A remark on infinity harmonic functions, Electronic [C-E] Journal of Differential Equations, Conf. 06, (2001), 123-129. L. C. Evans , A new proof of local c1,a regularity of solutions of certain degenerate [E 1] partial differential equations, Journal of Differential Equations 45 (1982), 356--373. [E2] L. C. Evans, Some min-max methods for the Hamilton-Jacobi equation, Indiana U Math J 33 (1984), 31-SO. L. C. Evans, Est imates for smooth absolut ely m inimiz ing Lips ch itz extensions, Elec [E3] tronic Journal of D ifferent ial Equations 1 (1993) . L. C. Evans and P. E. Sougarridis, Differential games and representation formulas for [E-Sol solutions of Hamilton-Jacohi equations Indiana University Mathematics Journal 33 (1984), 773-797. [E-Sp] L. C. Evans and J . Spruck , Motion of level sets by mean curvature I, J ournal of Differential Geometry 33 (1991), 635-681 . R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradi [J] ent , Arch. Rational Mech. Analysis 123 (1 993) , 51 74 . R. Kohn and S. Serfaty, A deterministic control-based approach to motion by mean [K-S] curvature, to appear in Commun ications Pure and Applied Math [LG] E. Le Gruyer, On absolutely minimizing Lipschitz extensions and the PDE L:.oo (u) = 0, preprint, 2004 A. Oberman, Convergent difference schemes for the infinity Laplacian: construction of [0] absolutely minimizing Lipschitz extensions, preprint Y. Peres and S. Sheffield, Tug of war with noise: a game theoretic view of the � [P-S] Laplacian preprint, 2006. [P-S-S-W] Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-lVar and the infinity Lapl",. cian, preprint , 2005. O. Sav in , C1 regularity for infinity harmonic functions in two dimensions, Arch. Ra [S] tional Mech . Analysis 176 (2005), 3,; 1 - 361.
(AI]
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OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, B ERKELEY , CA 94720 E-mail address: evanslDmath . berkeley . edu
DEPARTMENT
Contemporary Mathematics
Volume 446,
2007
Probabilistic Approach to a Class of Semilinear Partial Differential Equations Jean-Franc;ois Le Gall Dedicated to Profe.... or
Hairn Bn!zis.
We discuss the recent progress about positive solutions uf the semi linear equation .6.'U = 'UP in a domain, which has involved a combination of probabilistic and analytic methods. We emphasize the main ideas that have been used in the probabilistic approach. Special attention is given to the boundary trace problem, which consists in obtaining a one-to-one correspon dence between the set of all solutions and a suitable set of admissible traces on the boundary. A few important open questions are also listed.
ABSTRACT.
1.
Introduction
It has been known for a long time that properties of random systems of branch ing particles are related to solutions of certain semilinear partial differential equa tions. In the last 15 years, these connections have given rise to fruitful developments in the setting of the theory of measure-valued branching processes, also called su perprocesses. A major step was accomplished by Dynkin [7J, who provided a simple probabilistic representation of the solution of the Dirichlet problem for the equation �u = uP, 1 < p < 2 in a domain of jR d , in terms of the so-called exit measure of the associated superprocess (see Theorem 2.1 below) . A very interesting feat.ure of this representation, in contrast to other probabilistic approaches, is its robustness: A formula that is a priori only valid for solutions with a given continuous boundary value can be generalized, by means of various limiting procedures, to yield similar representations for many other solutions whose behavior at the boundary can be very singular. In fact, as will be explained in Section 6 below, a generalized ver sion of the probabilistic formula applies to any nonnegative solution of �u = uP, 1 < p :::; 2. Our goal in this work is to give an account of these developments, including the important recent contributions of Dynkin and Kuznetsov from the probabilistic side, and Marcus and Veron from the analytic side. We made no attempt at exhaustivity, 1991 Mathematics Su bject Classification. Primary
35J60, 35J65; Secondary 60J45, 60J80. Key words and phrases. SemiJinear partial differential equation, removable singularity, boundary blow-up, boundary trace, superProcess , Brownian snake, exit measure, polar set. ©2007
250
A meric an Mathemat ical Soci�ty
256
JEAN-FRANQOIS LE GALL
ami for instance we do not discuss parabolic equations which can also be handled by the same probabilistic tools. Rather, we try to explain as simply as possible the basic probabilistic ideas and the way these ideas can lead to analytic results. For this reason, we often concentrate on the particular ca.�e p = 2, where the random process called the Brownian snake can be used in the probabilistic representation of solutions. The Brownian snake was introduced in [22] , and its connections with equation �u = u2 were first discussed in [24] (see also the monograph [27]) . The Brownian snake is a simpler object than superprocesses and is sometimes more tractable for analytic applications, even though most of the analytic results that have been obtained for p = 2 via the Brownian tinake could then be extended to the case 1 < p < 2 using superprocesses. We do not discuss analytic methods herc. The reader who is interested in the analytic approach to the problems discussed below should look at Laurent Veron's recent paper [40]. Above all, we tried to emphasize the nice interplay between analytic and prob abilistic concepts. Already in [7] , Dynkin used the characteri�ation of removable singularities from Baras and Pierre [2] to solve the important problem of the de scription of polar sets fur super-Brownian motion (see Theorem 4 . 1 below) . In the reverse direction, we give examples of theorems that were finit proved for p = 2 via the probabilistic approach, and then ext.ended to arbitrary p > 1 by analytic methods. See Thcorem 3.3 and its generalization Theorem 3.4 by Labutin, or The orem 6.1 and its generalization Theorem 6.2 by Marcus and Veron. Obviously, the probabilistic approach, which docs not apply to the case p > 2, does not replace analytic methods. Still we believe that in some particular cases the probabilistic intuition can help guessing or even proving new analytic results, which can then be generalized. Section 2 below gives a brief presentation of the Brownian snake and i:itates the key Theorem 2.1, from which the different probabilistic representation formulas can be deduced. This section should provide sufficient background to understand the probabilistic ideas that are explained in the remainder of the paper. Analytic ques tions, namely solutions with boundary blow-up, removable singularities, solutions with measure boundary data, and the trace problem are discussed in Sections 3 to 6. We have emphasized the boundary trace problem, which has given rise to recent major advances by Dynkin, Kuznetsov, Mselati, Marcus and Veron.
2. A probabilistic tool: The Brownian snake
In this section, wc give a brief presentation of the Brownian snake, which will be our main tuol in the probabilistic analysis of semilinear partial differential equations. We refer to the monograph [27] for a more detailed presentation. At an informal level, our aim is to construct a "tree of Browrnan paths" originating from a given point x E JRd . More precisely, we will construct a collection (W8) of random paths, indexed by a real parameter s varying iII some interval. For each fixed value of the parameter s, Ws = (W.(t) ° < t < (s ) is thus a finite path in ]Rd starting from 1.: : , with lifetime denoted by (s . If s i= s', the paths W8 and Ws' coincide over an interval of the form [O, m(s, s')], where m(s, s') < (8 /\ (." In this sense, the collection (�V., ) forms a "tree" of paths. Assuming that x belongs to a domain D, a key role in our applications is played by the set E D of all exit points from D of the paths Ws (more precisely of thoi:ie paths W8 that do exit D), and by the exit measure from D, whkh is a finite measure supported on ED ,
PROBABILISTIC APPROACH TO SEMILINEAH
257
PDE
Let US turn to more rigorous definit.ions. The Brownian snake is a Markov process taking values in the set of finite paths in ]Rd . By definition, a finite path in lR.d is a continuous mapping w : [0, (] -7 lR.d . The number ( ((w ) > 0 is called the lifetime of the path. We denote by W the set of all finite paths in Rd. ThiH set is equipped with the distance d (w, w' ) = I ((w) - ((w') I + sup Iw(t II ((w» - w' (t II ((w,» I · -c
t> o
Let us fix x E ]Rd and denote by Wx the set of all finite paths with initial point w(O) = x. The Brownian snake with initial point x is the continuous strong Markov process W = (Ws , s > 0) in Wx whose law is characterized as follows. 1 . If ( = ((w.) denotes the lifetime of W., the process (( s > 0) is a reflecting Brownian motion in ]R + . 2. Conditionally on ((. , s > 0), the process W is a (time-inhomogeneous) Markov process. Its conditional transition kernels are described by the following properties: For s < s' , • W., (t) = W.(t) for every t < m(s, s' ) : = infIs , s'l (,. ; • (Ws. (m(s, s') + t) - Ws, (m(s, s'» , O < t « s, - m(s, s'» is a standard Brownian motion in ]Rd independent of W Informally, one should think of W.. as a Brownian path in lR.d with a random lifetime (. evolving like (reflecting) linear Brownian motion. When (s decreases, the path Ws is "erased" from its tip. When (. increases, the path W. is extended (independently of the past) by adding "small pieces" of Brownian motion at its tip. From this informal explanation, it should be clear that the evolution of the Brown ian snake generates a "tree of Brownian paths" in the sense that was explained at the beginning of this section. Denote by x the trivial path in Wx with lifetime O. It. is immediate that x is a regular recurrent point for the Markov process W. We denote by Nx the associat.ed excursion measure. Under Nx the law of W is deHcribed by properties analogous to 1. and 2., wit.h the only difference that the law of reflecting Brownian motion in 1. is replaced by the (infinite) Ito measure of positive excursions of linear Brownian motion (see [27] ) . In other words, the "lifetime process" ((s, s ). 0) is under Nx a positive Brownian excursion: It starts from 0, comes back to 0 at a finite time TJ > 0 called the duration of the excursion (and then stays at 0 over the time interval [TJ, +00» , whereas between times 0 and TJ it takes positive values and behaves like linear Brownian motion. Knowing the lifetime process ((., s > 0) , the behavior of the Brownian snake under 1"1", is given by property 2., as was informally described above. From our definitions, it is clear that W. = x for all s > TJ' Nx a.e. Thus under 1"1"" we will only be interested in the paths Ws for 0 < s S 7]. We can normalize Nx so that, for every 10 > 0, Nx (sups>o (. > c) = (210) - 1 . Although Nx is an infinite measure, we have for every 6 > 0 .
"
•.
( 2. 1 )
Nx (
s>O, O 6) = Cd 6- 2 < 00,
where Cd is a positive constant (see [27] , Proposition V.g) . For every fixed s > 0, conditionally on (.. Ws is distributed under Nx as a d dimensional Brownian path started at x and stopped at time ( If 0 < s < s ' < Ti, the paths W. and W.' coincide up to time m(s, s ' ) > 0, by Property 2., and this •.
JEAN-FRAN O, T(W.) < oo}. Notice that [ D is a random closed subset of aD. The exit measure ZD from D is a random finite measure supported on the set [D. This measure can be defined by the following approximation ([27], Chapter V) : =
:
=
(ZD , 2. Remark.
=
=
=
PROBABILISTIC APPROACH TO SEMILINEAR POE
259
In order to prove Theorem 2 .1, one establishes the equivalent integral equation
dy GD(X, Y) U2 (y) = J( KD ( x, dz) g (z) , x E D aD D where GD is the usual Green funct.ion (for � �) in D and Kv(x, dz) is the harmonic measure on aD relative to the point x. In probabilistic terms, this equation can be (2 .3)
u(x) + 2
rewritten in the form (2.4)
where (Bt, t � 0) is a d-dimensional Brownian motion that starts from x under the probability measure Px , and T = inf{ t � 0 : Bt ¢: D}. A computational way of proving ( 2 .4) is to expand the exponential in the formula u(x) = Nx(l exp - (ZD , g) , and then to use recursion formulas for the moments of (ZD , g), which follow from the tree structure of the Brownian snake paths. Let us summarize the contents of this section. Under the measure Nx , the paths (W., 0 � s < 'I) form a tree of Brownian paths started from x, each individual path Ws having a finite lifetime (s ' The set E D consists of all exit points from D of the paths Ws (for those that do e.xit D), and the exit measure ZD is in a sense uniformly spread over ED. We will also use the range R, which is defined by -
R
:= {y = Wa (t); 0 < s :::; '1), 0 < t < (a}.
This is simply the union of the Brownian snake paths.
3. Solutions with boundary blow-up According to Keller [18] and Osserman [38] , if D is a bounded smooth domain and 1jJ is a function that satisfies an appropriate integral condition, there exists a nonnegative solution of equation �u = 1jJ (u) in D that blows up everywhere at the boundary. This holds in particular if 1jJ(u) = uP for some p > 1 . This raises the following two questions: (a) For which non-smooth domains does there exist a solution that blows up everywhere at the boundary ? (b) Assuming that there exists a solution \vith boundary blow-up, is it unique ? The probabilistic approach turns out to be rather efficient in providing answers to t.hese questions. Let us start by reformulating in terms of the Brownian snake two key theorems again due to Dynkin [1]. THEOREM 3 . 1 . Let D be a bounded domain. Assume that D is Dirichlet regular.
Then Ul (X) = Nx(ZD i= 0), x E D is the minimal nonnegative solution of the problem in D �u = 4u2 (3. 1 ) UraD = +00 . The proof of this theorem is ea.')y from Theorem 2.1. Simply consider for every n > 1 the function vn(x) = Nx( 1 - exp -n(ZD , 1) that solves (2.2) with 9 = n. Clearly, Vn r Ul as n r 00 , and it follows that Ul also solves .6.u = 4u2 in D. Since Ul > Vn for every n, we have ullaD = + 00. Finally, if U is any (nonnegative) solution of (3. 1 ) , the maximum principle implies that u > Vn for every n and so
u > Ul.
260
JEAN-FRANCOIS LE GALL
To state the second theorem, recall our notation R for the range of the Brownian snake.
3.2. Let D be any open set in ]Rd and U2 (X) = Nx(R n DC t= 0) for x E D . Then Uz is the maximal nonnegative solution of !.::m = 1u2 in D (in the sense that u < 1L2 for any other nonnegative funci't on u of class C2 in D such that l!.1/. = 4'u2 in D). THEOREM
Remark. By combining Theorem 3.2 and (2.1), we recover the classical a priori bound u(x) < U2 (X) < cd dist(x, oD)-2 , x E D which holds for any nonnegative solution of l!.U = 4 u2 in D. Again the proof of Theorem 3 . 2 is relatively eai:iY from the preceding theorem. One can argue separately on each connected component of D, and thus a.."lm me that D is connected (notice that by construction the range R is also connected) . It is then easy to construct an increasing sequence (Dn )n> l of bounded Dirichlet regular subdomains of D such that Dn C Dn+ 1 for every n, and D = UDn. By Theorem 3.1, ui' (x) = Nx(ZDn t= 0) is a solution in Dn with infinite boundary conditions, On the other hand, from the probabilistic formulas of Theorems 3.1 and 3.2, one can check that. ul(x) 1 U2 (X) as n i 00, for every x E D. It follows that Uz is also a solution of l!.U = 4u2 in D. Moreover any other nonnegative solution 'U is bounded above by uf in Dn (by the maximum principle in Dn) anu therefore is bounded above by U2 . The previous theorems already shed some light. on questions (a) and (b). From Theorem 3. 1 , a solution with boundary blow-up exists as soon as D is Dirichlet regular. Unuer this assumption, question (b) reduces to giving conditions ensuring that Ul = U2 ' In the case when D is not Dirichlet regular, it. is easy to construct examples where Ul t= uz . Let B(x, r) denote the open ball with radius r eentered at x. Then if d = 2 or 3 and if D = B(O, 1 ) \{0} is the punctured unit ball (which is not Dirichlet regular) , the fUIIcLion U2 blows up near the origin, as a consequence of Theorem 3.3 below, whereas the function Ul stays bounded near the origin, because the exit measure "does not see" the origin, To state conditions ensuring that Ul = U2 , assume that d > 2 (the case d = 1 is trivial) and denote by Cd-Z (K) the Newtonian capacity (or the logarithmic capacity if d = 2) of a compact subset K of Rd. According to Theorem IV.9 of [27], the answer to question (b) is positive under the following assumption: For every y E aD, there exists a positive constant c(y) such that the inequality (3 . 2)
holds for all n belonging to a sequence of positive density in N (here B (x, 1') is the closed ball of radius r centered at x ) . See also Marcu� and Veron [29] for related results obtained by analytic methods for the more general equation l!.U = vY . It is interesting to compare (3.2) with the classical Wiener test, which gives a necessary and sufficient condition for the Dirichlet regularity: The bounded domain D is Dirichlet regular if and only if for every y E aD, (3.3)
261
PROBABILISTIC APPROACH TO SEMILINEAR PDE
Clearly, assumption C3.2) is stronger than C3.3). However, it is very plausible that (3.2) is not the best possible assumption, and this leads to the following question.
Open problem. Is the solution with boundary blow-up unique in the case of a
general Dirichlet regular domain ? ( ) JRd, the that t the
Let us discuss question a . Here Theorem 3.2 immediately tells us that for a general open set D in existence of a nonnegative solution of t;.u = u2 in D blows up everywhere a boundary of D is equivalent to I.he property DC =J. 0) = +00 (3.4) n lim Nx(n x-Y,xE D for every y E aD. It is not hard to see that this condition holds if and only if for every y E aD, Ny(Ws (t) r;. D for some s > 0 and t E ( 0, (8]) = +00. In this form, (3.2) is quite similar to the probabilistic version of the characterization of the Dirichlet regularity, with the difference that a single Brownian path started from y is replaced by a tree of Brownian paths started from the same point.. In order to sLaLe the next result, we need to introduce some notation. If > 0 and K is a compact subset of JRd, we define the capacity Ca (K) by setting l inf 1/ v (dY) !I(dz)fa C l y - zl) ) Cu(K) = ( vEM,(K) wehere M l (K) is the set of all probability measures on K, and if 0 , = = faCr) if > 0 . THEOREM 3.3. [4] Let D be a domain in IRd . Then the following two properties a
a
a
are equivalent. (i) The problem
(ii
in t;.u = u2 u l8D = + 00 has a nonnegative solution. ) Either d < 3, or d > 4 and for every y E aD,
D
L 2n(d-2) Cd_ 4 (DC n B (y, Tn » 00
= +00.
n=l
This theorem, which was proved in [4] by probabilistic methods involving the Brownian snake, thus gives a complete answer to question (a) ahove. Theorem 8.:3 wa.'> generalized a few years later by Labutin [21] using purely analytic methods. To state Labutin's result, and in view of further statements, we introduce Bessel capacities in JRd. For every I > 0, consider the classical Bessel kernel dt 2 -t 1[" 1 1 G�d) (x) = ) (3 .5) t 2 exp ( 2 o 41[" t where a-y = (41[") --y /2r ('y/ 2) -1 . For any compact subset K of IRd, and every p > 1 , we then set, P/P' (3.6) C-y,p(K) = sup ( Jr dx ( J /1(dy) G�d) ( y _ x ) r') }.tEM ,(K) llf.d a
-y
00
7- d
x
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JEAN-FRANQOIS LE GALL
where ! + ; = 1 as usual. The capacity Cr,p can also be viewed associated with the Sobolov space W')" P: See Theorem 2.2.7 in [1) . ,
as
the capacity
1.
THEOREM 3.4. [21J Let D be a bounded domain in ]Rd d > 3 and p > Let p' be defined by ! + ; = 1 . Then the following two pmperties are equivalent. ,
,
(i) The problem
�u = uP in D U l 8D = +00 has a nonnegative solution. (ii) Either p < �2 ' or p > d�2 and for every y E aD, d
.L 2n(d-2 ) C2 ,p' (D C n B( y, 2- n) ) = +00 . DC
'0.=1
In the case p = 2, we recover the preceding result. Indeed, a few lines of calculations show that, if d > 4, there exist two positive constants al and a2 such that, for every compact subset K of the unit ball,
(3 7) .
4. Removable singularities
Let K be a compact subset of ]Rd. We say that K is an interior removable singularity for �u = u2 if the only nonnegative solution of �u = u2 in JRd \K is the function identically equal to O. This turns out to be equivalent to saying that for any open set 0 containing K , any nonnegative solution on O\K can be extended to a solution on O. From the probabilistic point of view, interior removable singularities correspond to (interior) polar sets. The compact set K is said to be polar if for every x E ]Rd\K,
NxCR. n K # 0) = o.
In other words, the cornpaet set K will never be hit by the tree of Brownian paths which is the range of the Brownian snake.
THEOREM 4.1. Let d > 4 and let K be a compact subset o/ Rd. The following
are equivalent. (i) K is an interior removable singularity for �u = u2 . (ii) K is polar. (iii) Cd - 4 (K ) O. =
In dimension d < 3, the equivalence (i){c}(ii) also holds trivially, since (i) or (ii) can only be true if K is empty. The equivalence (i){c}(ii) is an immediate consequence of Theorem 3.2 above applied with D = KC• The equivalence (i){c}(iii) was obtained by Baras and Pierre [2): More generally, Baras and Pierre have shown that K is a removable :;ingularity for �u = uP if and only if C2 ,p' (K) = 0 (see also [3J for an earlier discussion of removable singularities for sernilinear equations). From the probabilistic viewpoint, it is worthwile to look for a direct proof of the equivalence (ii){c}(iii). A simple argument gives the implication (ii)=> (iii) (this implication was first obtained, independently of [2] , by Perkins [39) , and later the
263
PROBABILISTIC APPROACH TO SEMILINEAR PDE connection with [2] was made by Dynkin [7] ) . Indeed, suppose that Cd- 4 (K ) and so that there is a probability measure v supported on K such that
> 0,
JJ v(dy)v(dz)fd_4(ly - zl) < 00
(4 . 1 )
where fd- 4 (r) is as above in the definition of Ca (K) . Let h be a radial nonnegative continuous function on ]ftd with compact support contained in the unit ball, and for every c E (0, 1]' set he (x) = c- d h(x/c). Finally let I be the "total occupation" measure of the Brownian snake defined by
" t (I, g ) = 10 ds g(Ws ) -
-
where Ws = Ws «(s) is the terminal point of the finite path W. , and we recall that TJ is the duration of the excursion under Nx • Notice that by construction I is supported on R. If x E ]ftd\K is fixed, explicit moment calculations using (4. 1) give the bounds
Nx «I, he * v» > C1 > ° Nx ( (I, he * v) 2 ) < C2 < CXJ , where the constants Cl and C2 do not depend on c E (0, 1] (see Chapter VI in [27] for details). Let Ke denote the closed tubular neighborhood of radius c of the set K. From all application of the Cauchy-Schwarz inequality, it follows that ,
* c� . h I « v) N ? e , x ( 0) * > N (R n K -'v) > > T 0) > N « I h Nx( (I, hE * v) 2 ) C2 By letting c go tu 0, it follows that Nx (R n K i= 0) > 0, and thus K is not polar. In view of the simplicity of the preceding argument, une would expect that similar probabilistic proof should also give the converse implication (iii)=>(ii). Sur prisingly t his is not the case, and the only known way to obtain this implication is ' via Baras and Pierre's result (i),*(iii). Open problem. Give a direct probabilistic proof of the implication (iii)=>(ii) in Theorem 4-1. Finding such a proof would be of interest for other related problems where the analogues of the results of [2] are not always available. An example of such problems is provided by the notion of boundary removable singularity. From now on, consider a bounded domain D in ]Rd , with a smooth (COO) boundary aV. A compaet subset K of av is called boundary removable for Llu = uP (in V) if the only nonnegative function u of class C2 in V such that. Llu = uP and u tends to ° pointwise at every point of aD\K is the function identically equal to 0. Boundary singularities were studied first by Gmira and Veron [17] , who proved in particular that singletons are removable if p 2 �±t. To introduce the corresponding probabilistic notion, recall that £ D is the set of all exit points from V of the Brownian snake paths. The compact set K c av is said to be boundary polar if x
e
-
x
,
E
-
-
N", (£D n K i= 0) = 0
for every x E V. The following analogue of Theorem 4 . 1 was obtained in [25] , confirming a eonjecture of Dynkin [8] . THEOREM 4.2. Suppose that d > 3 and let K be a compact subset of ]Rd. Then the following are equivalent.
JEAN-FRANQOIS LE GALL
204
(i) K is a boundary removable singularity for .6.u = 1.12 in D . (ii) K is boundary polar. (iii) Cd-3 (K) = o. If d < 3, (i) and (ii) only hold if K = 0. The equivalence (i){o}(ii) is an immediate consequence of the following lemma (Proposition VII.I in [27] ) , which is analogous to Theorem 3.2. LEMMA 4.3. If K is a compact subset of D, the function
UK(X) = N:x: ([D n K # 0),
xED
is the maximal nonnegative solul'ion of the problem .6.1.1 = 41.12
(4.2)
in D
UI8 D\ K = O.
Lemma 4.3 is essentially a consequence of Theorem 2.1 above. Roughly speak ing, one can find a sequence a sequence (gn) of continuous functions on aD, such that (ZD, gn) converges to +00 on the event {C D n K # 0}, and to 0 on the com plementary event. It follows that UK solves .6.1.1 = 41.12 and it is also not hard to l>p.e that UK vanishes on aD\K. A suitable application of the maximum principle gives the maximality property stated in the lemma. Coming back to Theorcm 4.2, the implication (ii)=>(iii) can be p.stablished in a way very similar to the probabilistic proof of (ii)=>(iii) in Theorem 4. 1 that was described above (compute the first and second moments of (ZD , g) for suitable func tions 9 that vanish outside a small neighborhood of K). The implication (iii)=>(i) was obtained in [25] by Fourier analytic methods, using some ideas from [2J . The analytic part of Theorem 4.2, that is the equivalence (i){o}(iii) , can in fact be extended to equation .6.1.1 = up. This extension again involves the Bessel capac ities that were introduced above, but now considered for subsets of the boundary aD. If K is a compact subset of aD, we set ' / P P C�� (K) = sup ( 4. 3) a(dx) f.L(dY) G�d - l) (y _ X) (JD /LEM , ( K ) where a(dx) stands for Lebesgue measure on aD, and the Bessel kernels G(d) were defined in (3.5). This is of course analogous to (3.6), but lRd is replaced by the (d - I )-dimensional manifold aD, and consequently G(d) is replaced by G(d- l) .
(
(J
r' )
THEOREM 4.4. Let K be a compact subset of aD . Then the following are equivalent. (i) K is a bounda.ry removable singularity for .6.u = uP in D. (ii) cggp, (K) = O. In the case p = 2, we recover the preceding theorem, since an easy calculation shows that Cff (K) = 0 if and only if Cd- 3 (K) = O. Theorem 4.4 was proved in the case 1 < p < 2 by Dynkin and Kuznetsov [13J using a combination of probabilistic and analytic techniques (in the ca.. 2 of Theore m 4.4 was obtained by Marcus and Veron [31) . Rather surprisingly, the analytic techniques of [31] did not apply to the case p < 2 treated in [13J . In a subsequent paper [32J , Marcus and Veron •
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PROBAI3ILISTIC APPROACH TO SEMILINEAR PDE
developed a different approach that allowed them to give a unified treatment of all cases of Theorem 4.4.
5. Solutions with measure boundary data In this section, as well as in the next one, we keep assuming that D is a bounded domain in lRd with a smooth boundary aD. Many of the subsequent results hold under weaker regularity assumptions on D, but for the sake of ::;implicity we will omit the precise minimal assumptions. We are now interested in the problem in D , D.u = uP (5.1) Ul 8D = V , where v is a finite (positive ) measure on aD. Similarly as for (2.2) , the boundary condition Ul 8D = v may be interpreted via t,he int.egral equation (5.2)
u(x) +
1
dy GD (x , lI) UP (y) =
v(dz) PD (x , z) ,
xED,
2 D 8D where CD is as in (2.3) the Green function of D, and PD is the Poisson kernel of D (in the notation of (2.3) , KD(X, dz) = PD (x, z)a(dz)). (5.2 ) makes it obvious that u is bounded above by the harmonic function PDv. Conversely, any nonnegative solution of D.u = uP in D which is bounded above by a harmonic function solves a problem of the type (5.2), for some finite measure v on the boundary: See e.g. Proposition 4.1 in [25], for an argument in the case p = 2 which is easily extended. A nonnegative solution that is bounded above by a harmonic function will be called moderate. Gmira and Veron [17] considered the problem (5.1) ( in fact. for more general nonlinearities ) . They proved in partieular that (5.1) has a unique solution for any 'o finite measure f.1. on the boundary if p < ��i. Notice r,hat this conditi n corresponds to the case when singletons are not boundary polar. We fix p > 1 and to simplify terminology, we call boundary polar any compact subset K of aD that satisfies the equivalent conditions of Theorem 4.4. This is of course consistent with our preceding terminology for p 2.
5. 1 . Suppose that p > �+� and let f.1. be a finite measure on aD. The following two conditions are equivalent: (i) The problem (5. 1), or equivalently the integral equation (5.2), has a unique nonnegative solution. (ii) The measure f.1. does not charge boundary polar sets. Consequently, there is a one-to-one correspondence between the set of all moderate solutions of D.u = uP in D and the class of all finite measures on aD that do not charge boundary polar sets. =
THEOREM
In the case p = 2, this theorem was proved in [25] (again confirming a conjecture of Dynkin [8]) using both analytic and probabilistic arguments. In thaI. case, there is a probabilistic representation of the solution in terms of the Brownian snake: This is analogous to Theorem 2.1 with the difference thaL the quantity (ZD , g) should be replaced by a suitable additive functional of the Brownian snake. Similarly as for Theorem 4.4, the general form of Theorem 5.1 was obtained by Dynkin and Kuznetsov (see [13] and [14]) when 1 < p < 2 and by Marcus and Veron [31] when p > 2. A unified treatment was provided in [32] .
266
J EAN-FRAN!;:OIS LE GALL
6. The boundary trace probleID The classical Poisson representation states that nonnegative harmonic fWlctions h in D are in one-to-one correspondence with finite measures v on aD, and this correspondence is made explicit by the formula h = PD v, where PD is as above the Poisson kernel of v. We may say that the measure v is the trace of the harmonic fundion h on the boundary. Our goal in this section is to discuss a similar trace representation for nonneg ative solutions of �u = uP in D. ¥le will deal separately with the sub critical case p < ��i (where there are no nonempty boundary polar sets ) and the supercritical d±l . case p > - d-l
6.1. The subcritical case. We first consider p
2, so that the Brownian snake approach is available. Then the subcritical case holds if and only if d < 2 . Since the case d = 1 is trivial, we concentrate on d = 2 , where we have the folluwing theorem ([23], [26]). Recall that a(dz) denotes Lebellgue measure on aD. =
THEOREM 6.1 . Ass'llme that d = 2 , There is a one-to-one correspondence between nonnegative solutions of �u = 4 u2 in D and pairs (K, v), where K is a (possibly empty) compact subset of aD, and v is a Radon measure on aD\K. If a solution u i,� given, the associated pair (K, v) is determined as follows. For eveTY z E aD, denote by Nz the inward-pointing normal unit 1Jector to aD at z, then: (i) A point y E aD belongs to K if and only if, for every neighborhood U of y in aD,
lim a(dz) u (z + rNz) = +00. dO u (ii) For every contimtous function g with compact support on aD\K,
r
a(dz) u(z + rNz ) g (z) lim rJO JaD \ K
=
j V(dZ) 9(Z) '
Conversely, if the pair (K, v) is given, the solution u can be obtained by the fonnula (6.1 )
whel'e (ZD(z), z E aD) is the continuous density of the exit meaSU1'e ZlJ with respect to Lebesgue measure a(dz) on aD.
The pair (K, v) will be called the trace of u on the boundary. Informally, K is a set of singular points on the boundary (this is the set of points where u blows up as the squarc of the inverse of the distance from the boundary) and 11 is a mea.'mre corresponding to the boundary value of u on 8D\K. The formula (6. 1) contains as special cases the other probabilistic representations that have appeared previously. The formula of Theorem 2.1 corresponds to K = 0, v(dz) = g(z)a(dz). The function U2 of Theorem 3.2 (which here coincides with Ul of Theorem 3 . 1) is obtained by taking K = aD, More generally the functions UK in Lemma 4.3 correspond to the case v = O. Finally, the moderate solutions of Theorem 5.1 are obtained when K = 0. Let us outline the proof of the probabililltic representation formula (6, 1 ) , as suming for simplicity that D is the unit disk of the plane. Fix a sequence rn of
.
.
PROBABILISTIC APPR.OACH TO SEMILINEAR PDE
2 67
real numbers in (0, 1) �Ilch that rn i 1 as n i 00. For every n > 1 and x E D, set Un (X ) = r�u(rn x), so that we have .6.un = 4n� in D. Since Un obviously has a continuous boundary value on aD, we may use Theorem 2.1 to write, for every x E D, (6 . 2)
( )
Un X
=
Nx ( 1 - exp _ (ZD, un )
=
Nx (1 - exp
-
J
u
(dz) ZD (z)un (z»
using the fact that the exit measure ZD has a continuous density ZD with respect to u (this property only holds when d = 2). Note that we can identify aD with lR/Z. Using a compactness argument and replacing (rn) by a subsequence if necessary, we may assume that for every open subinterval ] of aD with rational ends, we have
nlim -co
I
u
(dz) un ( z)
=
a (J)
where a(I) E [0, +00] . We then set
+00 if y E I}. Replacing again (rn ) by a subsequence, we may also assume that the sequence of measures la \K (z)un(z)O"(dz) converges to a limiting measure v(dz), in the sense D of vague convergence of Radon measures on aD\K. From the definition of K and v, it can then be proved that, for every x E D, K
(6 .3 ) and (6.4)
j. n--.oo a lim
r
--+oo }a
lim n
D
D
=
{y E aD : a (I)
O"(dz) un (z ) ZD(z)
u(dz) un (z) ZD (z)
=
=
=
+00
(v, ZD)
,
Nx a.e. on
{En n K # 0}
, Nx a.e. on
{ED n K = 0}.
Indeed, (6.4) is easy if we observe that the support of ZD is contained in aD\K, on the event {E D n K = 0}. The proof of (6.3) reduces to checking that on the event {ED n K # 0} there is a (random) point z E aD such that ZD(z) > o. Using (6.3) and (6.4) , we can pass to the limit n ----> 00 in the right-hand side of (6.2), and we arrive at the representation formula (6. 1 ) . The other assertions of Theorem 6.1 then follow rather easily. Let us come back to the general case of equation .6.u = up. Marcus and Veron ([30] , Theorem 1) proved that, for any p > 1 and in any dimension d � 2, the trace (K, v) of a nonnegative solution U of .6.u = uP in D can be defined by properties (i) and (ii) of Theorem 6.1. Independently, Dynkin and Kuznetsov [15] gave a slightly different but equivalent definition of the trace. The one-ta-one correspondence between solutions and their traces can in fact be extended to the general subcritical case. The following theorem was proved by Marcus and Veron [30] . THEOREM 6.2. Assume that d < ��i . Then the mapping u • (K, v) associating with u its trace (K, v) (definp4 by (i) and (ii) of Theorem 6. 1) gives a one-to-one correspondence between the set of all nonnegative solut'ions of .6.u = uP in D and the set of all pairs (K, v), where K is a (possibly empty) compact subset of aD, and v is a Radon measure on aD\K.
When 1 < p < 2, the probabilistic representation formula (6.1) can be extended to this more general setting: See Theorem 1 .3 in [28] (which elaborates on preceding results of Dynkin and Kuznetsov [15], [16] ) .
JEAN-FRANC;OIS LE GALL
�68
6.2. The supercritical case. The supercritical case p > �+� is more compli cated and in a sense more interesting. As Weu> mentioned above, properties ( i) and (ii) of Theorem 6. 1 r:an still be used to define the trace of any nonnegative solution of �u = uP in D. However, the fact that there are nontrivial boundary polar sets now suggests that all pai rs (K, v) cannot occur as possible traces. More precisely, Theorem 4.4 indicates that the pair (K, O) cannot be a possible trace if K is boundary polar, and similarly, Theorem 5.1 suggests that, v should not charge boundary polar sets in order for (0, v) to be a possible trace. The characterization of possible traces was obtained independently by Marcus and Veron [31) and Dynkin and Kuznetsov [15) (the latter in the case p < 2) .
THEOREM 6.3. Let K be a compact subset of aD, and let v be a Radun measure on aD \ K. Then the pair' (K, I) ) is lhe trace of a nonnegative solution of �u = uP in D if and only if: ( i) The measure 1/ does not charge boundary polar sets. ( ii) The set K is the union of the two sets
K; = {y E K K n U is not boundary polar for every neighborhood U of y} and
:
avK = {y E K : v ( K n U)
=
00 fur evel'g neighborhood U of y}.
Another problem in the supercritical case is the lack of uniqueness of the so lution corresponding to a given (admissible) trace. To give an example of this phenomenon, consider the case p = 2, d 2: 3. Let (y,, ) be a dense sequence in aD and, for every n, let (r�, k = 1 , 2, . . . ) be a decreasing sequence of positive numbers. For every k > 1 , set
Hk
=
00
U { y E aD
n� l
:
iy
-
Yn i < r� } ,
and x E D.
Then it is easy to see that, for every k > 1 , 'Uk is a solution with trace (aD, O). On the ot.her hand, the fact that singletons are boundary polar implies that Uk 1 0 as k i 00 , provided that the sequences (r�, k = 1 , 2 , . . . ) decrease sufficiently feu>t. Therefore infinitely many of the fUIlctions 'Uk must be different. Tn view of this nonuniqueness problem, Dynkin and Kuznetsov [19J , [16] have proposed to use a finer definition of the trace, where the set K is no IOIlger closed with respect to the Euclidean topology. We will explain this definition in the general case of equation �u = up. 'vVe first need to introduce the analogue of the singular part for the fine trace of a solution u. Let b be a nonnegative continuously differentiable function on D. We can then eonsider the Poisson kernel (Pb(x , y) , x E D y E aD) associated with the operator �u bu in D (see Section 11.1.2 in Dynkin [9] for a detailed construction of pi»). A point y of the boundary aD is called singular for b if Pb (x, y) = 0 for some, or equivalently for every, x E D. Informally, this corresponds to points of rapid growth of b. A simple equivalent probabilistic definition can be given a::; follows. If ,
-
PROBABILISTIC APPROACH
TO
269
SEMILINEAR POE
( Bt , O < t < T ) is under Px-y a Brownian motion started from x and conditioned to exit D at y (in the sense of [5]), the point y is singular for b if and only if
dt b(Bt} = +00 )
Px-y a.s.
Consider now a nonnegative solution u of �u = up . The singular set of 'U, which is denoted by SC (1t ) is the set of all boundary points that are singular for up- I . Note that SG(u) is a Borel subset of aD, but needs not be closed in generaL We denote by N the set of all finite meaSUrf'B on the bowldary that do not charge boundary polar sets. For every v E N, we denote by u" the unique solu tion of the problem (5 . 1 ) , or equivalently the solution associated with v via I.he correspondence of Theorem 5.l. o
DEFINITION 6.4. Let u be a nonnegative solution of �n = 'uP in D . The fine trace of u is the pair (r, 'l) that is defined as follows: (i) f = SG(u). (ii) /1 is the O'-finite measure on DD\f sllch that, for every Borel subset A of 8D\f,
(6.5)
i1C4)
=
sup{v(A) : v E N, UV < u} .
Remark. It is not obvious that formula (6.5) defines a measure. Sec Theorem in [16]. It is clear from (ii) that v docs not charge bowldary polar sets. It can be
1 .3
checked that in the subcritical case this definition is equivalent to the one given by (i) and (ii) of Theorem 6.1. The interest of this definition comes from the following theorem (Theorem 1 .4 in [16] ) .
THEOREM 6.5. [16] Let us call O'-moderatc any nonnego,tive sulution of �u = uP that is the inc'{'f>.asing limit of a sequence of moderate solutions. Then O'-moderate solution are characterized by their fine traces.
This theorem shows that the lack of uniqueness mentioned above disappears if one considers the fine trace instead of the (rough) trace discussed in the previous subsection. Dynkin and Kuznetsov [16] also give a description of those pairs (r, v) that can occur as fine traces of solutions. Provided one considers only iT-moderate solutions, the fine trace thus yields a one-to-one correspondence between solutions and admissible pairs (f, v) . The obvious question, which was stated in the epilogue of [9] is thus:
Are all nonnegative solutions O'-moderate
?
This question was answered positively first in the case p = 2 in Msclati's thesi� [36], [37] . In addition, Mselati's work gives 1-1. probabilistic representation of solu tions, which is analogous to Theorem 6.1. To state this representation, we need to introduce some additional notation. Let v E JV, and let ltv be the harmonic function in J) associated with 1/ (h" = FD V in our previous notation) . Then, if (Dn ) is an increasing sequence of smooth subdomains of D such that Dn C Dn+l and D = UDn, we can define
Z" := lim (ZD" , h,,) ,
Nx a.e.
n T oo and the resulting variable Zv does not depend on the choice of the sequence (Dn ) . Note that the existence of the limit defining Zv is easy because (ZDn ' hv) is a nonnegative martingale. Then, if v is a O'-finite measure on aD that does not
270
JEAN-FRANyOJS LE GALL
charge boundary polar sets, we can find an increasing sequence (Vk ) in N such that v = lim 1 Vk, and we set ZI/ = lim i ZVk ( again this does not depend on the choice of the sequence (Vk)).
4
THEOREM 6.6. [37] All nonnegative solutions of .6.1.1 = 1.12 are a-modemte. Moreover, if 1.1 is a solution and (r, v) is its fine tmce, we have for every x E D, u(x ) = Nx ( 1 - 1{£Dn r=0} exp( - Zv ) .
)
(6.6)
A major step in the proof of Theorem 6.6 was to prove that the solution UK defined in Lemma 4.3 is O"-moderate, for any compact subset K of aD. The proof depends on delicate upper bounds on UK near the boundary, and analogous lower bounds for certain a-moderate solutions, which are obtained via probabilistic meth ods. Motivated by Mselati's work, Marcus and Veron [33] , [34] were able to obtain very precise capacitary estimates in the general case p > 1 . Part of their results is summarized in the following theorem.
THEOREM 6.7. [34J Consider the general case p > 1 . Let K be a compact
subset of aD and assume that K is not boundary polar. Let UK be the maximal nonnegative solutwn of .6.1.1 = uP in D that vanishes on aD\K . Then UK is 0" moderate. Moreover, for every x E D, UK (X)
2. The very recent paper [35] by Marcus and Veron collt.ains important progress towards the solution of this open problem.
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DMA-ENS,
L.
45 RUE
D'ULM
75005 PARIS, FRANCE
E-mail address : legalHldma . ens . f r
Contemporary Mathem1l.tic.s Volume 446, 2007
Variational methods in image processing Ali Haddad and Yves Meyer This paper is dedicated to Haim Brezis, with our deepest admiration and respect.
Several algorithms have been proposed to unveil the geometrical structure of a given image. We will mostly focus on the ROF algorithm designed by Leonid Rudin, Stanley Osher, and Emad Fatemi. The ROF model can be used to detect the objects which are contained in an image and also to analyze its textured components. We then comment on some improved versions of the ROF algorithm. ABSTRACT.
1. Introduction Conventional wisdom says that natural images contain many objects with .simpIe geometrical forms. Detecting these objects is a main issue in image processing. When they addressed this issue in 1985 David Mumford and Jayant Shah ([29), [30]) proposed to model objects by functions of bounded variation. In the Mum ford & Shah model any imagc f is decomposed into a sum u + v between a sketch u and a second term v which takes care of the textured components and of some addit.ive noise. The working hypothesis of the Mumford & Shah model says that the objects we are looking for belong to the sketch u. These objects are assumed to be delimited by contours with finite lengths and u is a geometric-type image. It is then natural to assume that u is a function of bounded variation. This assumption will however be questioned in this paper. Jean-Michel Morel [27) told us that Ennio de Giorgi elucidated the role of the space B V (consisting of functions of bounded variation) which was implicit in the Mumford & Shah model. '
In 1992, Stanley Osher, Leonid Rudin, and Emad Fatemi simplified the Mum ford & Shah model. Their model is named the ROF model and is also aimed at splitting a given image f into a sum f = u + v . In the Mumford & Shah model as in the ORF model the decomposition f = u + v minimizes a certain energy. 1991 Mathematics Subject Classification. Primary 68U10, 94A 08 ; Secondary 42C4U, 65T60. Key words and phrases. Functions of bounded variation, Image processing, RO F algorithm. AH acknowledges support from the University of California. YM acknowledges support from CNRS, France. 273
274
ALI HADDAD AND YVES MEYER
In these notes we will mostly focus on the ROF model. Here is our first result: Let us assume that we a priori know the objects which are contained in an image f and that the ROF algorithm is applied to f. It will be proved that some pieces belonging to the (physical) objects are always incorporated into the v component by the algorithm (see Theorem 3 below) . That is why Stanley Osher, Luminita Vese, and the second author have been looking for a third model. This model is analyzed in these notes and, in some cases, performs much better than the ROF model, as proved in Theorems 10 and 1 1 . These notes are organized as follows. Sections 2 contains introductory remarks on image processing, neurophysiology, and modeling. Sections 3, 4, and 5 are devoted to the standard ROF model. A first alternative to the ROF model is studied in Section 6. The BV norm is replaced by a Besov norm in this new model. This does not change the norm of indicator functions of rectifiable domains. This alternative model leads to an algorithm which is similar to Donoho's wavelet shrinkage. Therefore atomic decompositions and variational approaches have been reconciled. In Section 7 to 9 the ROF model is applied to a class of textured images and the qualities and drawbacks of the ROF model are carefully analyzed. Section 10 is devoted to the new Osher-Vese model. This material is a continuation of a program initiated in [26].
2. The primary visual cortex
A black and white image E is defined as a function f(x) = f (XI , X2 ) which is named the grey-level of x E E. We set f(x) = 0 if the point x is black in thc given image and f (x ) = 1 if x is bright white. Then f(x) E [0, 1] and the value of f(x) is the grey-level at the point x. The pair (x, f (x» is named a pixel. The bounded function f(x) is not continuous in general. Jump discontinuities of f(x) are playing a key role in image processing. Such discontinuities are generated by the edges of the objects which are present in the given image. At this level of the discussion the grey-level f is a bounded measurable function. This immediately raises a fundamental issue, since it is clear that most bounded measurable functions do not correspond to natural images. Similarly a random se quence of letters is not a poem or a novel. A natural image has a meaning and meaningful images should be adequately modeled. A first approach to modeling natural images consists in studying the laws which govern their production. The geometrical organization of the surrounding world and the laws of optics are playing a seminal role. But one should not forget. to take in account the processing achieved by the human vision system. This processing can be used as a clue for modeling natural images. David Rubel writes in [22] : In collaboration with Torsten Wiesel and Margaret Livingstone, I have attempted to build upon the work of Ramon y Cajal in an attempt to obtain a detailed understanding of the physiology of one small part of the cerebral cortex-the striate cortex, o r the primary visual cortex. . . Our main effort has been to determine how the visual information coming from the eye is handled and transformed by the brain. . . Cells in the primary visual cortex, to
VARIATIONAl, METHODS IN IMAGE PROCESSING
, • •
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which the optic nerve projects (with one intermediate nucleus in terposed) are far more exacting in their stimulus requirements. The commonest type of cells fires most vigorously not to a cir cular spot, but to a short line segment-to a dark line, a bright line, or to an edge boundary between dark and light. Further more each cell is influenced in its firing by a restricted range of line orientations: a line more than about 15 to 30 degrees form the optimum generally evokes no response. Different cells prefer d-iffc7'ent orientations, and no one orientation, vertical, horizontal or oblique is represented more than any other. These observations, made in 1 958, had not been predicted and came as a complete surprise . . . A cell in the left hemisphere might respond to a bright red line oriented at 45 degrees to the horizontal, in a small region of the right visual field, but fail to respond to vertical lines, horizontal lines, or white and black lines. Are these discoveries bridging the gap between neurophysiology and image pro cessing ? D. Hubel is providing us with two mesHages. Hubel emphasizes the role of edge detection in perception. This will agree with the ROF model or in the Mumford & Shah model. We will return to this point in Section 3. The second message is even subtler. D . Hubel says that edges are not captured as a whole by the primary visual cortex. They are split into tiny components or "atoms" . The criterion is given by the orientation. D. Hubel tells us that some cells of the primary visual cortex are detecting the simplest geometric entities which still have a mean ing in an image. Can this processing be viewed as an atomic decomposition ? Is it ' possible to model the cells in the primary visual cortex by some wavelets 1/J). which would combine the frequency localization of the Gabor wavelets [19] together with the localization of the Grossmann-Morlel wavelets in the space domain ? In a striking paper [:31] B. A. Olshausen and D. J. Field proposed the following paradigm: the highly specialized tasks of the cells st.udied by David Hubel could be the result of an evolutionary process which selected this solution. The selection criterion is concision and robustness. That is why Olshausen and Fields aimed at finding the most concise representation of the class of all natural images. Instead of processing all nat.ural images, they used some pictures of landscapes available in a data basis. For discovering the most concise representation, they used an algorithm named lCA, or Independent Component Analysis. References to ICA are given in [1] , [13], and [23] . It happened that the building blocks found by Olshausen and Fields are close to some of the wavelets belonging to Coifman's libraries. These wavelets are described in [24] . In [5] H. B . Barlow also suggests that mammals use concise and efficient representations to process natural images. Before closing this section , let us mention a sriking discovery by Emmanuel Candes and David Donoho (1 1], [12] where atomic decompositions and geometry are truly reconciled. In full contrast with Coifman's best basis search (see [24] ) , Candes and Donoho where looking for a single hasis ej , .i E N, which would optimally compress all cartoon images. A cartoon image is defined as followH. We are given finitely many regions Om, 1 < m < N, delimited by smooth boundaries rm ' The cartoon image is required to be smooth inside each Om with jump discontinuities
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ALI HADDAD AND YVES MEYER
across the boundary I'm of Om . If ej , j E N, is an orthonormal basis, we denote by Pj (f) the error norm Ilf - fj ll2 where fj denotes the best approximation to f by a linear combination fj = (Yl en1 + . . . + lYjCnj of j vectors picked in this basis. Using a standard wavelet expansion, we obtain the est.imat.e Pi (I) = j-l/� . Using a Fourier series expansion, t.he error would have been j-1/4 and wit.h Donoho-Candes curvelets, the error (still computed in L2) is reduced to j-1 (log j)3/2 . Up to the logarithmic term, this convergence rate is optimal. I cannot resist quoting Ca.ndes and Donoho. In [12] they write In fact neuroscientists have identified edge-processing neurons in the ea.rliest and most fundamental stages of the processing pipeline upon which mammalian visual proce.�8ing is built. . . This article is motivated by fundamental questions concerning the math ematical representation of objects containg edges: what is the sparsest representation offunctions f(x1, X2 ) that contain smooth regions but also edges ?
It is time to conclude. D . Hubel is telling us that edge detcetion is playing a key role in perception. Edge detection is also pivotal in the seminal paper [29] by Mumford & Shah, which is fully consistent with D. Hubel's discoveries. Moreover D. Marr [25] and S . Mallat [24] successfully bridged the gap between edge detection and atomic decompositions. This issue will addreslled again in Section 6. Cross fertilization between neurophysiulogy and image processing is not a dream. 3. The Rudin-Osher-FateIlli Illodel
The ROF modcl is a simplified version of the Mumford & Shah model. In both models, a given image is optimally split into two parts u and v . The first component u takes care of the objects which arc included in the given image f. These objects can be human beings, animals, some furniture or other items. The working hypoth esill says j,hat objects are delimited by boundaries whith finite lengths. Therefore they can be drawn by a painter. Neurophysiology is replaced by art which also relies on some remarkable properties of the human brain. Textures and noise are included in the second component. The space BV of funct.ions of bounded varia tion will play a key role in the the Mumford & Shah and in the ROF models. The co-area theorem by De Giorgi will explain this role. In the Mumford & Shah model, the given image f is defined over a domain n, the u component helongs to the subspace SBV of BV, which consists of functions in BV whose distributional gradient does not contain a singular diffuse measure. In other words, this distributional gradient V'u is the sum between an L l function and a measure carried by a one dimensional singular set K. Then t.he Mumford & Shah penalty on the u(x) component is a sum J(u) between two terms. The first term is the one-dimcnsional Hausdorff measure of K. The second one is the square of the L2 norm of the gradient of u (x) calculated on the complement of this singular set K. The third term of the J (u ) functional is the square of the L2 norm of vex). Then the functional to be minimized is (1 )
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277
where H l (K) denotes the I-dimensional Hausdorff measure of K. The two positive parameters a and f3 need to be tuned effectively. Indeed too many objects are detected the Mumford & Shah model if 0: is small. The same will happen with the Osher-Rudin-Fatemi model if the parameter A is small. The the Mumford & Shah model is raising beautiful mathematical problems which have much to do with the theory of minimal surfaces [16] . In the Mumford and Shah model, textures are treated as noise. The ROF model is similar. An image f(x) is again a sum between two compo nents 7t and v . In the seminal paper [32] the ROF model is motivated by ill-posed inverse problems and the BV norm wa.'> viewed as a regularization which preserves edges. Denoising is a particular example of such ill-posed problems. We now list some properties of the space BV(R2 ) . Our definition of the space R V slightly differs from the usual ones since do not demand that a function in BV should belong to L1 . This only concerns the behavior at infinity. One is tempted to say that the indicator function XE (X) of a domain E belongs to the space BV if and only if its boundary oE has a finite length and that the BV norm of XE(X) equals the length of the boundary. This statement is clearly valid in the (;1 case but is not true in general. In order to treat the general case, De Giorgi defined the reduced boundary 0* E of a measurable set E and proved that the BV norm of XE is the I-dimensional Hausdorff measure of its reduced boundary. References can be fonnd in [2] , or [10] . For defining this reduced boundary, let us denote by B(x, 1' ) the ball centered at x with radim; r. We then follow De Giorgi:
DEFINITION 1 . The reduced boundary 0* E of E is the set of points x belonging
the closed support of i-' = VXE snch that the following limit exists i-'{ B (x r ) } = vex) 1i-'I{B(x, r) } ,
(2)
An indicator function XE belongs to BV if and only if 8* E has a finite 1dimensional Hausdorff measure. More generally a function f(x) defined on R2 belongs to BV if (a) f (x) vanishes at infinity in a weak sense and (b) the dis tributional gradient V f of f (:1:) is a (vector valued) bounded Borel measure. An apparently weaker definition reads as follows: f belongs to BV if its distributional gradient is a (vector valued) bounded Borel measure. Then is is easily proved that f = g + c where c is a constant and g tends to 0 at infinity in the weak sense. The condition at infinity says that f * rp tends to 0 at infinity whenever rp is a function in the Schwartz class. When the Rudin-Osher-Faterni model is being used, a specific definition of the BV norm is crucially needed. Indeed the ROF model amounts to minimizing a functional which contains this BV norm. We will impose that this norm be isotropic. Let Wl begin by the simple case where Vf belongs to L 1 (R2 ) . Then the BV-norm of f will be defined as IIfll Bv = flV f(x) 1 dx. We then write f E W l,l and this function space will be useful in what follows. If Vf is a general Borel
ALl HADDAD AND YVES MEYER
278
measure, our simply minded approach does not work but paves the way to the following definition. We write p,j = oj f and we define the Borel measure (j by (j = 1p,1 1 + 1p,2 1 · By the Radon-Nikodym theorem we have p,j = 6j (x)rJ, j = 1 , 2 , where 6j (x) are Borel functions with values in [-1 , 1] . Finally the Borel measure IV(f)1 is defined by (3) We can conclude:
DEFINITION 2. The BV norm of f it! the total mass of the Borel measure IV(f) I · With an obvious abuse of language, we write Ilfll BV = J vl p,1 1 2 + 1 p,2 12 dx. We now return to the RO F algorithm. It splits an image f into a sum u + v between two components. As it was already said the first component u takes care of what might be drawn by a painter using a pencil. Therefore this component is adequately modeled by a function in BV. The texture and the noise belong to the component v . Then u is a sketch of the given image and our working hypothesis says that this sketch u captures the main geometric fcatures of the given image f. The v componcnt is more complex and is not described by a functional Banach space in the ROF model. In the ROF model, v E £2(R2) since both f and �l are square-integrable. The RO F algorithm depcnds on a tuning parameter >. > O. Objects with size less than -ix will be treated as some texture and wiped out from 'U (see Corollary 2 of Theorem 1 ) . We now arrive to the definition of the algorithm.
DEFINITION 3. Let f E £2 (R2 ) . Then the ROF decomposition f = 'tt + V of f minimizes the functional J(u) = lI'IlII Bv + >' l l v ll� among all decompositions of f as a sum between a function u E flV and v E £2(R2 ).
Later on the ROF model will be generalized and the space BV will be replaced by other functional spaces. This will pave the way between the ROF algorithm and the famous wavelet. shrinkage.
4. Properties of the ROF algorithm As will be proved in these notes, The Rudin- Osher-Fatemi algorithm is indeed performing a thresholding which is similar in spirit to the wavelet .shrinkage defined by David Donoho and lain Johnstone. More precisely the ROF algorithm has the following property : there exists a norm I I . II . and a threshold 2\ such that an image f with a norm IIfli. < A is put to 0 and an image with a norm larger than the threshold is reduced by a fixed amount (see [26] or Theorem 2 and its two corollaries). This is similar to Donoho's wavelet shrinkage. However the ROF al gorithm and Donoho's wavelet shrinkage present the same drawback. Pieces which belong to the true objects contained in an image are wiped ont by the algoritlun and viewed as belonging to the textured component (see Corollary 3 of Theorem 1 ) . The Osher-Vese algorithm performs better (Theorem 1 1) . We now define the dual norm II . II Let us denote by W I , I the closure in BV of the linear space of smooth functions with compact support. In other words f E W1,l means Vf E £1 . Here W I . I denotes the homogeneous version of the •.
VARIATIONAL METHODS IN IMAGE PROCESSING
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standard Sobolev space. The dual space of W 1,1 is the Banach space G consisting of all generalized functions 9 which can be written as 9 =
divH
(4)
where H = (hI, h 2 ) E Loo x LOO. The norm of g in G is denoted by I l g ll . and is defined as being the infimum of I I H lloo where this infimum is computed over all decompositions (4) of g. Here and in what follows IIHlloo = sUPXER2 I H(x) l , I H I = v'l h112 + I h2 1 2 . Then L2 C G and the space Go is defined as the closure of L2 in G. We have [26] LEMMA l . The Banach space BV is the dual space of Go and the norm in BV is the dual norm. This implies the following property: if a sequence Uj of functions belonging to BV converges to U in the distributional sense and if I l uj IIBv < 1, then u belongs to BV and lI u l I BV < 1 . This weak compactness property implies the existence of the optimal ROF decomposition and uniqueness is standard. The following lemma will be needed. LEMMA 2. If both u and v belong to L2, then we have u(x)v(x) dx l < Il u l l Bv l l v ll *
(5 )
For proving it, it suffices to approach u in L2 by a sequence of functions in virl ,l. The details can be found in [26] . An image f satisfies 0 < f(x) < 1 and the u component inherits this prop erty. The proof of this remark is easy. If f = u + v i� the ROF decompo sition, we have f = B(f) = B(u) + w. We then have II B(u) IIBv < I l u l l Bv and I w(x) 1 = I B(f) B(u) 1 < I I(x) u(x) l · Therefore II w l l 2 < Il v l l z and the uniqueness of the ROF decomposition implies u = B(u) as announced. �
�
Antonin Chambolle made the following crucial remark. Let FA be the closed convex subset of L2 (R2) defined by I I , 11. < A. Then we have THEOREM 1 . The ROF decomposition I = u + v of a function I E L2(R2 ) is given by (6) v = Arg inf { 1 1 1 v11 2 ; v E FA } �
The proof of this theorem will be given in the following section. For the time being, let us comment on Theorem l.
COROLLARY 1 . Let 1 E L2 (R2). If 11/11* < A , then the ROF decomposition of f is given by u = 0, v = I. If 11111. > A, it is given by f = u + v where Il v l l . = \ 2 and Il f - v l1 is minimal under that constraint. 2 Corollary 1 is now rephrased in a more intuitive way :
COROLLARY 2. If 11/11* < A, then f can be interpreted as a texture corrupted by an additive noise and does not contain any object.
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Returning to Corollary 1 and assuming I l f l l . > -ft' we have 1 = u+v , IIvll. = A. and lIull 2 is minimal under these requirements. Another characterization of the optimal pair (u, v) is given by I = u + v, lIvll. = A, and J uv = IluIIBv llvll This leads to the following definition: • .
DEFINITION 4 . A pair (u, v) of two junctions in L2 (R2 ) is named an extremal
pair if u E BV and J uv dx = lIu Il Bv l l vll. ·
Theorem 1 can be rephrased into the following assertion
COROLLARY 3. If 1 1 1 11 . >
-ft'
the ROF decomposition f = u + v 01 f is chamcterized by the following two properties: (a) IIvll . = -A and (b) (u, v) is an extremal pair.
The reader is referred to (26] where Corollary 3 is given a direct proof. Corol laries 1, 2 and 3 are telling us that the ROF algorithm is a shrinkage where the threshold is }" .
5. The abstract formulation of the ROF algorithm A proof of Theorem 1 is given now. This proof is valid in a more general con text which reads as follows. Let H be a real Hilbert space. The norm in H is denoted by I . I and the corre sponding inner product is x y. Let F be a non-empty closed convex subset of H. Let us define p : H I-t R U { +oo} by .
p(x) = sup {x . y ; Y E F }
( 7) This functional p is convex, lower semi-continuous and satisfies p(>.x) = >.p(x) for ). > 0, x E H. Let I E H be given. Among all decompositions I = x + y of I we want to find the one for which the energy J(x) = p(x) + ).lyl 2 is minimal. We denote by I = x + y this optimal decomposition. Without loosing generality we can assume ). = 1 /2. It suffices to replace F by A. F to obtain the general case. Following Antonin Chambolle we have
THEOREM 2. With the preceding notations the optimal decomposition I = x + Y is given by (8) y = Arg inf{ 1 f - YI; y E F}
The proof of Theorem 2 is not difficult . It consists in applying von Neumann's minimax theorem to the functional 1 V(x, y) = x ' Y + I I - x I 2 , x E H, y E F ) (9 '2 The minimax theorem says the following. Let E be a compact and convex set. Let F be a convex set which does not need to be given a topological structure. We consider a functional V : E x F I-t R. We define P : E I-t R U { oo } and Q : F I--t { - oo} U R by (10) P(x) = sup{V(x, V) ; Y E F} and similarly
Q(y) = inf{V(x, V) ; x E E} (11) We obviously have Q(y) ::; V(x, y) < P(x) which implies sup{Q(y) , y E F} = (3 ::; Q: = inf{P(x), x E E}. The minimax theorem (Theorem 3 below) implies Q: = {3.
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VARlATIONAL METHODS IN IMAGE PROCESSING
THEOREM :3. Let u.s assume that x 1-+ VeX, y) is convex and lower semi continuous on E for every y E F. Let us also assume thai, y t-t V(:c, y) is concave on F for every x E E. Then there exists an element x E E such that P(x) = {3.
Let us fix a large R > 1 and let E c H be the closed ball 1 :1: 1 < R equipped with the weak topology. The functional V is defined by (9) . Theorem 3 says that P(x) = p(x) + � If - xl2 reaches its minimum a at x . Then Q( y) reaches its maximum f3 at y and y is defined by (8). We then have f = x + y and Theorem 2 is proved. The details of t.his proof arc left to the reader. For proving Theorem 1 it suffices to observe that p(u) = II u II Bv by Lemma 2 .
6. Replacing the BV norm by a Besov norm in the ROF algorithm. We denote by S(R2) the Schwart,;o; c\a.;;ll and by S'(R2 ) the dual space of tempered distributions. A functional Banach space E is defined by the property S(R2) C E C S' (R2) where the two embeddings are continuous ones. In general S(R2) is not dense in E and we let Eo denoLe the closure in FJ of the space of testing functions. The space E* is the dual space of Eo and not of E in general. The norm in E* is denoted by II . II If E' = BV, then Eo is defined by f E L2 and V' f E L l . The generalized RO F models we have in mind are based all the energy •.
K( u) = Il u ll E + A l l f - u ll�
,
( 1 2)
The optimal decomposition f = u + v is the one which minimizes K(1.) . This op timal decomposition exists and is unique whenever the Banach space E has the following property: for every sequence Uj E E, j E N, such that l I uj li E < 1 and lIuj - ulb ---> 0, j ---> 00 , we have llullE < 1 . With these notations Theorem 3 is still valid and v is the argument of inf{ II f - v 1 l 2 ; IIvll. < 2\ }.
The Besov space Z = iJi'oo is close to nv as will be proved below. Moreover this Besov space is isomorphic to a trivial sequence space and this isomorphism is given by the wavelet expansion. Let us consider now the variant of the ROF algorithm where the BV norm is replaced by the norm in Z. It will then be proved that the u component is obtained by a variant on the standard wavelet shrinkage.
The spaee BV can be defined by the existence of a constant C such that for every y E R2 we have ( 13) II!(- - Y) - f O ll l < Gly l The lower bound of these constants G is a norm which is equivalent to the usual BV norm of f. Similarly the homogeneous Besov space Z = st, oo i� dcfincd by the existence of a constant G such that for every y R2 we have
E
11f( - - y ) + f( · + y ) - 2f O l h < Glyl
(1 4 )
The embedding BV c Z is obvious and should be compared to the well known fact that a Lipschitz function always belongs to the Zygmund class, the converse not being true. It is therefore surprising that for a large eolleetion of functions the two norms are equivalent. Here is the full story. Let SN denote the collection of all step functions with N levels. In other terms .f = CI X E1 + . . . + CNXEN where Eb · · · , EN are Borel sets and ct , · . . , CN are N real coefficients. We then have
ALI HADDAD AND
282
YVES MEYER
THEOREM 4. There exist two positive constants Co and C1 such that for every f E SN we have Co llfll z < I l f l l BV < C1 N I l f li z ( 15) The proof of Theorem 4 relies on a deep theorem by Gerard Bourdaud [8] , [9] , [35] . If A E R, we define O;,(t) = (t - A)+ . We then have
THEOREM 5. There exists a constant Co such that, for every f E Z and A E R, we have (16) I I B;, U) ll z < Co llfll z Before returning to Theorem 4, let us restate Theorem 5 under the following form: COROLLARY 4. If b > a > 0, we write I = [a, b] and define a function TI (t) of the real variable t by TI (t) = ° if t < a, TI (t) = t - a if a < t < b, and finally TI (t ) = b - a if t > b. Then we have Ih (f) lI z < 2 Co llfll z
Indeed TI (t) case:
=
(1 7)
Ba (t) - (h (t). Then the proof of Theorem 4 begins with a simpler
LEMMA 3. Let E Then we have
C
R2 be a Borel set and let X E the indicator function of E. 1
ll z < I l xE ll BV < I I XE ll z xe 1 1 2
(18)
The proof of ( 18) relies on an obvious remark: If a, b, and c all belong to {O, I } , then we have la - bl < l a - 2b + ci ( 19) The proof is trivial and left to the reader. Returning to ( 18), we apply (19) to a = X E (X + y), b = Xe(x ) , and C = X E (X - y). This yields the right-hand estimate in (18) . The left-hand estimate is true for any function f E BV. We now treat the general case in Theorem 4. Without loosing generality, we can assume Cl < C2 < . . . < CN . The corollary of Theorem 5 is applied with IN = [CN-l, CN] . We then obtain I I ( CN - CN-l) XEN li z < Co llfll z Since Ilxe IIBV < Il xE ll z , it implies II (CN - CN - l )XEN I I BV < Co llfll z
We then consider Ilk = Ek U · · · U EN 5 is applied to h = [Ck-l > Ck] (with Co
= =
=
L:� (Ck - Ck-l) xO k
(21)
{x; f(x) > cd. The corollary of Theorem ° by convention) and we obtain
I I ( Ck - Ck - l ) XO JBV < Co llfll z
But f
(20)
(22)
and thc triangle inequality ends the proof.
Since the BV norm and the Besov norm are equivalent ones for simple functions, it is tempting to replace the standard ROF energy by K(u)
=
lIuli z + A l l v ll �
(23)
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VAlUATIONAL METHODS IN IMAGE PROCESSING
where v = f - u as before. A main difference with the standard ROF algorithm is coming from the fact that the Banach space Z is not contained in L2 (R2). Instead Z is contained in the Lorentz space L2,00. A typical example of a function in Z is Ix l -l which is not square integrable. Good news are coming. The space Z admits a simple wavelet characterization. Let (1/;1 , 1/; , '1j;3 ) be three functions in the Schwartz class such 2 that the eolledion 'lj!j,1c = 2j'lj!(2jx - k ) , j E Z, k E Z2, 1/; E {1j;! , 1/;2 , 1/;3 } , is an orthonormal basis of L2 . We then have
LEMMA 4. A function f E L2,00 belongs to Z coefficients c(j, k) = J f1/;j ,1c dx satisfy sup
(L
j EZ IcEZ2
Ic(j, k) l )
=
=
.si'oo
if and only if its wavelet
C < 00
Similarly a tempered distribution f belongs to the dual space Z· only if its wavelet coefficients c(j, k) satisfy
(24) =
B- l , 1 if and 00
L sup Ic(j, k) 1 = 0' < 00 .J k
(25)
This is proved in [26] and an excellent introduction to Besov spaces can be found in [36]. We remind the reader that Z· is not the dual space of Z but rather the dual of the closure in Z of the spaee of testing functions. The rtorm of f in Bi ' oo and the infimum of C in (24) are equivalent norms and a similar remark applies to 13001,1 and C'. In other words the two Besov spaces Bi'oo and Bool,l are identified to simple sequence spaces. This is not the ease for the space BV of functions of bounded variation. We have by [15] and [26] ;
THEOREM 6. The wavelet coefficients c(j, k) of a function f E BV belong to
weak-II . More precisely there exists a constant 0 which only depends on the wavelet basis such that for every positive A, we have o 3 # { (j, k) E Z ; Ic(j, k) 1 > A } < >: lIflIBV
(26)
However ( 26) does not characterize RV and (26) does not even imply f We now turn to the variant of the ROF algorithm. We write Z the new energy is defined by K (u) = lI u liz + A ll f - u ll�
=
E
Bi ' oo .
B�'oo and (2 7)
The optimal decomposition f = u + v is the one which minimizes K( u). It exists and is unique. For simplifying the following discussion the norm in Z will not be defined by (14) but instead by
lI u li z
=
sup
L
jEZ kEZ'
ICu (j, k ) 1
(28)
which is an equivalent norm. Here Cu (j, k) are the wavelet coefficients of u . With these notations K ( u) lIuli z + Ao-(u) where O'(u) 2::j 2:: 1: icf (j, k) - cu(j, k) 1 2 . This variant on the ROF algorithm leads to the following algorithm. =
=
ALI HADDAD AND YVES MEYER
284
y (j k) which mimimi7.es (29) sup ( L ly (j, k) l ) + L L i cf (j , k) - y (j, k W k JEZ kEZ2 .i Then the wavelet coeffi.ci!mt� of are cu (j, k) = y(j, k), (j, k) E Z3. This optimiza We want to find the sequence
,
u
tion problem can be solved by the following procedure. Let a positive number 7) be
j,
given and let us assume that for each by a limited budget defined by
the sequence
y(j, k), k E Z2, is constricted
L k Ez2 I y(j, k) 1 < 7). We are asked to minimize CT(U)
within this budget limitation. This problem is named P(ry) . The minimum of can be decoupled into
is denoted by w(7) . Problem P(7)
a
CT(U)
:;equence of problems.
We are given a sequence Then we want to minimize L k IX k - Yk 1 2 under
Each one of them is standard and reads as follows :
Xk , k E Z, and a positive number 7). the constraint L k I Ykl < 7) . This is a variant on the standard wavelet shrinkage. Indeed Yk and ;C k should have the same sign and satisfy 0 $ I Ykl < I X kl. These two over T/ E
requirements say that the wavelet coefficients are shrunk towards step consists in minimizing 7)
+ AW(7)
[0, ) (0
O.
The second
and is left to the reader.
depends on the wavelet basi:;. It is not t r ans l ation and rotation invariant.
This approach can be questioned. When defined by (24), the norm in
.si'oo
.
7. Analysis of textured images As it was already said, the f into a sum
ROF
algorithm is aimed at decomposing an image
u + v where u represents the objects contained in f while v models the
textured components. A collection of images will be tested to confirm t hi s working hypothesis. These images are explicitely given as a
9
and a texture
h. More
precisely we consider a sum
simplest sketch we can figure while component . For instance
swn
h
between a cartoon image
f = 9+h
where 9 is the
is an oscillating function modeling a textured
hex) = m(Nx)x(x)
which is 27f-periodic in one variable and has
where a
m(x)
is a continuous function
vanishing mean while
X
is the in
9. We write iN (x) = g(x) + x(x)m(Nx) and N will b e arbitrarily large. Can the ROF algorithm be trusted ? Does the ROF algorithm yield a decomposition iN = UN + VN where the sketch UN is close to the original sketch 9 ? More precisely we expect UN = 9 + EN , VN = h - EN where the L2 norm of the error term EN tends to O. dicator function of a rectifiable domain. This example of textures will be given a
systematic treatment in Section
This
fairy tale
is untrue, since the
ROF
algorithm applied to f = 9 does not
g back but instead a new function g. A more precise statement will be given in Lemma 9. The best to be expected in the general setting is UN = 9 + EN, VN = II, + 9 9 - EN with EN -> O. This is true, as Theorem 7 will tell us. get
-
Theorem
7 will apply to the RO F algorithm, but also to more general contexts.
We begin with a Hilbert space
II . II.- If
are given a dense subspace V denoted by
x
H and II · 11 will denote the corresponding norm. We c H together with a norm which is finite on V and
tt. V, then
lower semi-continuous on
H:
I l xli . lim
J - OO
=
00 . Let us assume that the norm
I l x - xj ll = 0
II . II . is (30)
285
VARIATIONAL METHODS IN IMAGE PROCESSING
Let us denote by F c V the closed convex set defined by II . I I . < 1 and let Pp : H >-> F be the orthogonal projection; PF (:);) = z i� thc point in F which minimizes liz - I II. We know from Theorem 1 that the decomposition given by the ROF algorithm is x = y + z, z = PF(X) . Writing RF = I - Pp, we have Rp(x) = y. The main theorem of this section is the following: TlIEOREM 7. For every x, x' E H we have
II Rdx') - RF(x) 1 I
< 13(11 1: 11 + II x' ll h/l l x' - x II .
(31)
Before proving Theorem 7, let us return to the ROF algorithm. Then F will bc dcfined by I I · II . < A where II . II . is the dual norm as in Section 6. Theorem 7 reads COROLLARY 5. We consider the ROF model 'with a given vahw of the param eter A. Let it and h be two functions in L2(R2). Let h = U, i + Vj be the ROF decomposition of Ij , j = 1 , 2 . Then we have (32)
\Ve now assume that h is a sum h = it + It hetween a cartoon image II and a textured component h satisfying II h ll . < c . We then have ,
(33)
In other words the ROF algorithm does not perform what could have been dreamed, since the cartoon component II is not preserved. Instead it is modified into U I . This being said, the RO F algorithm acts in a consistent way: when it is applicd to h it yields Jl = U\ + VI and when it is applied to 12 = it + h it yields 12 = U2 + V where 'U = 1l.} + O( y'C) , V2 = V I + h - O( y'€) , the errors being 2 2 measured in L2 . The relevance of (31) comes from the fact that in many applications the norm which controls x' - x is much weaker than the norm which is used in the left-hand side of (31). It is indeed the ca. 1112 1 while bump fUIlctiuIl and w is arbitrarily large. Then l l h ll < � C > O. 2 l I I .
.
Finally the weight given by II x ll + I lx' ll in the right-hand side of (31) cannot be erased. The simplest counter-example does not concern the standard ROF al gorithm but the wavelet shrinkage. It is given by H = 12(N) and IIxll* = 11 1: 11 00 = sup I x nl . Here PF (x ) = i is defined by in = Xn if I xn l < 1 while in = signxn if not. In other words RF(X) = x is defllled by xn = 0 if I xn l < 1 and xn = Xn - signxn if not. This operator RF is a shrinkage which pulls the coefficients back to O. We now check (31) on the twu sequences (xn ) and (x�.) defined by x� = 1 + € if 1 < n < N and x� = 0 if n > N + 1 . \\'hen Xn = 1 if 1 < n < N and In = 0 if n > N + 1, we
ALI HADDAD AND YVES
286
MEYER
have I lx' - x ll oo = f while II RF(x') - Rdx) 11 = EJN. The constant 13 is obviously not optimal. The proof of Theorem 7 begins with a standard lemma
LEMMA 5 . Let F c H be a closed convex set containing O. Let Xo E II, Zo = PF (xo ) , Yo = Xo - PF (xo ) , and d = Ilyo l l · If X o = YI + Zl where Zl E F and IlyI iI < d + Ell xo ll , then we have ,,2
II Yl - Yo II < 2 E + 2 " xo l The proof is standard and left to the reader.
(34)
The set F is defined by II . 11* < 1 in the following discussion and the other notations of Lemma 5 are kept. We then have LEM MA 6 . If 0 < 1 + 1], then we have
f
< 1, 0 < 1] < 1 , Xo = Yl + Zl , II Yl l 1 < d + E l l x o ll, Il zI i I .
:/:r, the ROF algorithm yields f = u+ v where I l v l l = A and .r u(x)v(x) dx = Ilu l lBv II v I!.. Since v = f - u E BV, it implies that u is a cartoon image. Conversely if u is a cartoon image and if v is dual to u, then f = u + A 7J it; the ROF decomposition of f. We can conclude: .
3. A function u is a cartoon image if and only if there exists a function J of bounded variation and a tuning parameter A such that the RO F algorithm yields f = u + v. PROPOSITION
Here are some other examples and counter-examples. Let n be a bounded domain with a C3 -boundary. Then the indicator function XI! of n is a plain image. Indeed we consider the normal vector vex) at x E 8n and extend it to a C2 -vector field H (x) such that IIHlloo = 1 . We then define g= divH ami we have Il gl l. < 1 . But ( 54) It implies I Ig li . = 1 and f is a cartoon image. Similarly a piecewise mnstant func tion with jump diHcontinuities across C3-boundaries is a cartoon image. Here are two counter-examples. The indicator function of a polygon cannot be a cartoon image, as it is proved in [26] . A second (;Qunter-example is given by f(x) = exp ( - lx I 2 ) Then J fg dx = IIfllBV and I Igli. = 1 imply g(x) = I x l - l . This 9 cannot belong to BV. However the radial function :p defined by ip(x) = 1 when I x l < 1 , cp(x) = 1:1:1-2 if I xl > 1, is a cartoon image. We now have: .
10. Let 9 be a canoon image and let 9 be dual to g. If A > AO = IIjj I!BV, then the corresponding Osher- Vese decomposition of 9 is the trivial decom position given by u = g, v = o. THEOREM
We have AO > 27f by Lemma 10 and I lgll. = 1 . Therefore Theorem 10 is consis tent with Lemma 11. Theorem 10 implies uniqueness. We argue by contradiction. Let 9 = u + v be an optimal decomposition minimizing K (n) . We have
I lgll B v =
gg dx = J 1Lg dx +
J vjj dx
=
( 55)
It + h
Ih l < lI u llBV and I h l ::; II v ll. l Ig llll v by Lemma 2. It implies Il g llBv < lI u llBV + II v ll. lljjllBv < lI u linv + A l l v l l . unless v = O. The trivial decomposition
But 9
= 9 + 0 is winning against 9 = 'u + v.
Let us now treat a more involved situation where f = 9 + h with being a cartoon image. Let 9 be dual to 9 and AO = IIgI lB V·
I lh ll.
< E, 9
VAlUATIONAL METHODS IN IMAGE PROCESSING
293
THEOREM 11. If A > Ao, then the Osher- Vese algorithm yields a decomposition f = u + v 1JJheTf� 71. is close to 9 in L2 . More precisely 2€A (56) 1111, - gil. < A _ A ) 1 1 11, - 911 2 < G(A ) vC o
Here we do not know whether the optimal decomposition is unique. The con stant G(A) will be made explicit in the proof. We first write
1=
J f (x)g (.'E) dx = II + 12 = IlgllBV + J g (x) h (x) dx
We also have
1=
u(x)g (x) dx +
v ( x )g(x ) dx =
h
+ 14
(57)
(58)
We used the fact that 9 is plain to obtain h = IIgll BV. Then Lemma 2 yields the following bounds 12 > - ll gII Rv llhl l . and 13 + 14 < IlglI . llull Bv + IlglIBv llvl1 . Therefore
IIgll. 1 1911LJV - llgIIBv llhll. < I < IlglI . llullBv + 11911 Bvllvll. It suffiees to oLerve that Ilg li. = 1 to obtain I lgllBV - Go l l h l l . < I l u l l BV + Go llvll. But we also have
..
(59 )
(60)
(61) since f = 11, + v is assumed to be an optimal decomposition. Since A > Go) it suffices to combine (60) and (61) to obtain II vll. < ��gO('. We have u - 9 = h - v and Il u - g il . < 2 t A ACO by the triangle inequality. On the other hand f = 11, + v is an optimal Osher-Vese decomposition. It implies lIullB v < I lglIBV + Af which yields Ilu - gllBV < 2 1!911BV + >.c. Then J Ill, - 912 dx < Ilu - 911BVIlu - gil. < 2€ A ->CO (2 11 911BV + fA) . This ends the proof of the second assertion. Theorem 1 1 is raising the problem of the converse implication. Is it true that cartoon images are the only functions for which the Osher-Vese algorithm yields a trivial answer ? Theorem 1 1 is a first step which should be completed by more mathematics and more numerical experiments. We would like to thank Jerome Gilles for providing us with numerous examples. We express our gratitude to the referee ; his constructive remarkH were a valuable help.
References [1J
S.-1. Amari and J.-F. Cardoso. Blind source separation -semiparametric statistical approach.
IEEE Trans. on
Sig. Proc. 45
( 1 1 ) : 2692-2700,
November
1997,
Special issue on neural
networks.
[2]
L. A mbrosio. Corso introduttivo alla teoria geometrica della Misura ed aile Superjici Min ime. Scuola N ormalc Superiore Lecture Notes, Pisa,
[3]
(1997).
G. Aubert and J-F. Aujo!. Mudeling very oscillatory signals. Applications to image process� ing. INRIA, no.
[4] J-F. Aujol,
4878, (2003).
G . Aubert, L . Blanc-FcTal.l d, and
A. Chambolle.
Image decomposition; Applica
4704, (2003). [5) H . B . Barlow. Unsupervised learning. Neural computation 1 ( 1989) 295-31l. [6J G . Bellettini, V. Caselles and M. Novaga. The total variation flow in RN . J. Differential Eqs. 184 (2002) 475-525. [7J G . Bellettini, V. Caselles and M . Novaga. Explicit solutions of the eigenvalue problem -div(Du/IDuil = u. SIAM J. on Math. A n al . 36 4 (2005) 1095-1129. lion to lextu1-ed images and SAR images. INRIA no.
294
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ALI HADDAD AND YVES MEYER G . Bourdaud. Fonctions qui operent sur les espaces de Besov et de
Poincare, Analyse non limlaire
[9] [10] [11] [12]
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
Tri e bd. Ann. Inst. Henri
(1993) 413-422. G. Bourdaud and M.E.D. Kateb. Fonctions qui op erent sur les espace.' de Besov. Math. Ann. 303 (1995) 653-675. A. B raides Approximation of free discontinuity pro blem•. Lecture Notes in Mathematics 1694 Springer (1991). E. Candes and D. Donoho. Ridgelets: a key to higher-dimensional intermittency? Phil. 'Trans. R. Soc. Lond . A 1760 ( 1999) 2495-2509. E. Candes and D. Donoho. New tight frames uf cu" velels and op tim al representation of ob jects with piecewise C2 singulari ties. Communications on Pure and A pplied Mathematics, Vol . LVII, (2004) 0219-0226. J.-F. Cardoso. Blind signal separation: statistical p,inC'iples. Proceedings of the IEEE Spe cial issue on blind identification and estimation 9 (10): 2009-2025, October 1 99B. A. ChamboUe and P.L. Lions. Image recovery via total variation minimization and related problems. Numer. Math. 76 (1997) 167-178. A. Cohen, R. DeVore, P. Petrushev and H. Xu. Nonlinear approximation and the space BV(R2 ) . American Journal of Mathematics 121 (1999) 587-628. G . David. Singular sets of minimizers for the Mumford-Shah functional. B irkhiiuser (2004). R. DeVore an d B. J. Lucier. Fast wavelet techniques JOT' near optimal image compression. 1992 IEEE Military Communications Conference O ctober 1 1-14, (1992). D. Donoho and I. Johnstone. Wavelet shrinkage: Asymptopia? J.R.Statist. Soc. B 57 (1995) 301-369. D. Gabor. Theory of communicati on. J. lEE, 93 (1946) 429-457. J. G ill"". Ph. D. J une 22 (2006) available from Jerome . Gillesllietca . fro E. Giusti. On the equation of surfaces of prescribed mean curvature, Existence and unique ness without boundary conditions. Invent. Math. 46 (1978) 1 11-137. D. Hubel. DiscUT'so de inv es ti dura de Doctor Honoris Causa. (1997) Universi d ad Alltonoma 10
.
de Madrid.
[23] A. Hyviirinen , J Karhunen, and E. Oja. Independent Component Analysis. John Wiley & Sons (2001). [24] S . M allat . A Wavelet Tour of Signal Processing Academic Press (1998). [25] D. Marr. Vision, A computational investigation into the human representation and process ing of visual information. W.H. Freeman "'lid Co (1982). [26] Y. M eyer . Osc'iliating patterns in image processing and in s ome nonlinear evolution equa tions. (L ewis Memorial Lectures) AMS (2001 ) . [27] J-M. Morel and S. Solimini. Variational methods in image segmentation. Birkhiiuser, Boston (1995). [28] D. Mumford. Book review on [39] . Bulletin of the American Mathematical Society 33 n.2, April 1996. [29] D. Mumford and J. Shah. Boundary deteelion by w'inimizing functlOnals. Proc. IEEE Conf. Compo Vis. Pattern Recognition (1985). [30] D. Mumford and J. Shah. Optimal representations by piecewise smooth functions and asso ciated variational problems. Comm. Pure Applied Mathematics 42 (5) (1989) 577-685. [31] B. A. Olshausen and D. J . Field. l!)mergence of stmple-cell receptive field properties by learn ing a sparse code for natural images. N ature 381 (1996) 607-609. [32] S. Osher and L. Rudin. Total variation based image restoration with free local constraints. In Proc. IEEE ICIP, vol I, pages 31-35, Aus tin (Texas) USA, Nov. 1994. [33] S. Osher, L. Rudin and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D 60 (1992) 259-268. [34] S. Osher, A. Sole, and L. Vese. Image decomposition and restoration using total variation minimization and H - 1 -norm. Multiscale M odeling and Simulation 1 (3): (2003) 349-370. [35] P. Oswald. On the boundedness of the mllpping f t---> I f I in Besov spaces. Comment. Univ. Carolinae 33 (1992) 57-66. [36] J . Peetre. New thoughts on Besov Spaces. Duke Univ. M ath Series (1976). [37] L. Vese and S. Osher. Modeling textures with total variation minimization and oscillating patterns in image processing. Journal of Scientific Computi ng 19 (2003) 553-572. .
.
.
.
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VAlUATIONAL METHODS IN IMAGE PROCESSING
[38]
L. Vese and S . Osher. lmage denoi ing and decomposition with total variation minimization s
oscillatory functions. Special issue on Mathematics and Image Analysis, Journal of Mathematical Imaging and Vision 20 (2004) 7-18. and
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, Los ANGELES, CALIFORNIA 90095-1555 E-mail address: ahaddadlDmath . ucla. edu CMLA, ENS-CACHAN, 94235 CACHAN CEDEX, FRANCE E-mail add.·ess: ymeyer
,
Contempor:ny Mathema.tics Volume 446, 2007
Null hypersurfaces with finite curvature flux and a breakdown criterion in General Relativity Sergiu Klainerman
In honor of Haim Brezis, for his luminous energy, inspiration
and
friendship
1. Introduction ,
I report on recent work in collaboration with Igor Rodnianski concerning a geomet criterion for breakdown of solutions (M, g) of the vacuum Einstein equations,
ric
Rct,l3(g)
=
(1)
O.
Here Ro,13 denotes the Ricci curvature of the I 3 + 1 dimensional Lorentzian manifold (M, g). The main result discusser! here is stated and proved in [Kl-Ro6] ; the proof depends however on the results and methods of [KI-RolJ , [KI-Ro2J , [KI-Ro3] [KI-Ro4] which establish a lower bound for the radius of injectivity of null hy persurfaces with finite curvature flux as well as [KI-Ro5] in which we cunstruct a Kirchoff-Sobolev type parametrix for solutions to covariant wave equations. Assume that a part of space-time MJ C M is foliated by the level hypersurfaces �t of a time function t , monotonically increasing towards future in the interval I C JR, with lapse n and second fundamental form k defined by, k(X, Y)
=
n=
g(DxT, Y) ,
( _ g(Dt, Dt») -
1 /2
(2)
where T is the future unit normal to �t, D is the space-time covariant derivative associated with g, and X, Y are t:mgent to L:t . Let � o be a fixed leaf of the t foliation, corresponding to t = to E I, which we consider the initial slice. We assume that the space-time region M r is globally hyperbolic, i.e. every causal curve from a point p E MJ intersects La at precisely one point. Assume also that the initial slice verifies the assumption. 1991 Mathematics
Subject Classification.
The author is partially supported by
NSF
35J10
grant DMS-0070696. ©2007 American Mathema.tical Society
297
298
SERGlU KLAINERMAN
A 1. There exists a finite covering of I:u by a finite number of charts U such that for any fixed chart, the induccd metric g verifies
\;Ix E U with 6.0 a fixed positive number.
(3)
Though our work in [KI-Ro6] covers only the second of the following two situations below, it applies in principle to both. (1) The surfaces
2: t
are asymptotically flat and maximal. trk = o .
(2) The surfaces 2:t are compact, of Yamabe type - 1 , and of constant, nega tive mean curvature. They form what is called a (CMC) foliation . trk
=
t 0 denot es the value of t at p. Therefore, we deduce the following, �
LEMMA 1.2. If 0�/ 2 . fp is sufficiently small, then for any t > 0 ( 14) II F ( t) IIL= :s II F (t OO) IIL� + II DF (t oo ) IILoo . �
�
(3) Arguing recursively and using the standard local existence theorem for the Yang-Mills system one can find bounds for all components of the curvature tensor F(p) depending oIlly on the fact that fp is unifonnly bounded and the initial data F (O) is smooth. (4) One can show that ( 14) remains true even as we take into consideration the presence ofthe third term in (1:')) . This i� done by choosing a specified gauge condition for A called the Cronstrom gauge. That is, one asslImes that the connection I-form A satisfies (x y)QAa = 0, where XCi are the space-time coordinates of p and yO those of a point q E N- (p). One can use t.his condition to derive uniform estimates of 8A in terms of F . (5) In [KI-Ma] the global regularity result wa.'i reproved by strengthening the classical local existence result to A E III (IRa) and E E L2 (]R3 ) , which iH at the same regularity level as the energy norm. That required, instead of the pointwise estimates ( 14), a new generation of L4 type estimates, called bilinear. The premise of the [KI-Ma] approach was the fact that, once we have a local existence result which depends only on the energy norm of the initial data, global existence can be easily derived by a. simplc continuation argument. �
2
and in what follows c > 0 is a universal constant . Here
we denote
by A ;S
B any inequality of the form
A < cB, where
302
SERGJU KLAINERMAN
In what follows we give a short summary of how the mains ideas in the proof of the Eardley-Moncrief result for Yang-Mills can be adapted to General Relativity.
(1) One can easily show that the curvature tensor R of a 3 + 1 dimensional vacuum spacetime (M, g), see ( 1 ) , verifies a wave equat.ion of the form, ( 15)
DgR = R * R where Dg denotes the covariant wave operator Dg = D"D" . (2) Recall that the Bel-Robinson energy-momentum tensor has the form
Q[R] a{3'Y" = R",x'YIL R; J' + * R",x'Y1' * R; t· and verifies, D � Q,,{3'Y� = O. It can thus bc used to derive energy and flux estimates for the curvature tensor R. As opposed to the case of the
Yang-Mills theory, however, in General Relativity the background metric is dynamic and thus does not admit , in general, Killing fields (and in particular a time-like Killing field) . This means that we can not associate conserved quantities to a divergence free Bel-Robinson tensor. It is at this point where we need crucially our bounded deformation tensor condition A2. Indeed that condition suffices to derive bounds for both energy and flux associated to the curvature tensor R. Using the Bel-Robinson energy momentum tensor Q the energy associated to a slice 2:t is defined by the integral
Q [R](T, T, T, T),
E(t) =
(16)
while the flux, through the null boundary N- (P) of the domain of depen dence (or causal past) :7- (p) of a point p, is given by the integral
F- (p) =
(
1
N-
(p)
Q[R] (L, T, T, T), ) '
( 17)
where L is the null geodesic generator of N- (p) normalized at the vertex p by < L, T >= 1 . As in the case of the Yang-Mills equations, it is precisely the bound edness of the flux of curvature that plays a crucial role in our analysis. In General Relativity the flux takes on even more fundamental role as it is also needed to control the geometry of the very ohject it is defined on, i.e. the boundary of the causal past of p. This boundary, unlike in the case of Minkowski space, are not determined a-priori but depend in fact on the space-time we are trying t.o control. (3) In the construction of a parametrix for ( 1 5) we cannot, in any meaningful way, approximate Dg by the flat D'Alembertian D. To deduce a for mula analogous to (14) one might try to proceed by the geometric optics construction of parametrices for Dg, as developed in [Fried] _ Such an approach would require additional bounds on the background geometry, determined by the metric g, incompatible with the limited assumption A2 and the implied finiteness of the curvature flux. Vie rely instead on a geometric version, which we develop in [KI-Ro5], of the Kirchoff-Sobolev formula, similar to t,hat used by Sobolev in [Sob) and Y. C. Bruhat in [Brl , see also [M] . Roughly, this can be obtained by applying to (15)
303
CURVATURE
the measure Ao(u), where u is an optical function3 whose level set u = 0 coincides with N- (p) and A is a 4-covariant 4-contravariant tensor de fined as a solution of a transport equation along N- (p) with appropriate (blowing-up) initial data at the vertex p. After a careful integration by parts we arrive at the following analogue of the formula ( 14) :
R(p) = R° (P; 00)
A . (R * R) +
Err · R, r iN - (p;"o)
( 18)
N- (p;oo) where N- (p; 0o) denotes the portion of the null boundary N- (p) in the time interval [t (p) - ') W] + r2 [Va(A) FLa , W] . .
( 43)
Observe indeed that the last term on the right hand side of (43) contains covariant derivatives of F. Thus estimate (42) seems at first glance impossible. To avoid this problem we need to take into account the fact that the integral along a null geodesic of r2 [ViA) Ft, W] compensates for the loss of derivative. This can only be done by using once more the Bianchi identities.
References [And] [Br] [C-K] [EMl] [EM2] [Fried] [HE] [HKM]
M. Andersson, Regularity for Lorentz metrics under curvature bounds,arXiv:gr qc/020907 vI, Sept 20, 2 002 . Y. Choquet-Bruhat, Theoreme d'Existence pour certains systemes d 'equations aux derivees partielles nonlineaires., Acta Math. 88 (1952), 141-225. D. Christodoulou, S. Klainerman, The global nunlinear stability of the Minkowski space, Princeton Math. Series 4 1 , 1993. D. Eardley, V. Moncrief, The global exi.,tcnce of Yang-Mills-Higgs fields m 4dimensional Minkowslci space. I. L ocal existence and smoothness properties. Comm. Math. Phys.83 (1982) , no. 2, 171-191. The global existence of Yang-Mills-Higgs fields in 4D. Eardley, V. Moncrief, dimensional Minkowski space. II. Completion of proof. Comm. Mat h. Phys.83 (1982), no. 2, 193-212. H.G. Friedlander The Wuve Equat>on on a Curved Space-time, Cambridge University Press, 1976. Hawking, S. W. & Ellis, G. F. R. The Large Scale Structure of Space-time, Cambridge: Cambridge University Press, 1973 Hughes, T. J. R., T. Kato and J. E. Marsden Well-posed quasi-linear second-v'rder hy perbolic systems with applications to nonlinear elastodynam'ics and general relativity, Arch. Rational lvIech. Anal. 63, 1977, 273-394
fact the estimate for � (A)W is a lot more complicated divergence, but we should ignore this here. 14ln
as
it leads
to a
logarithmic
CURVATURE
(KIJ (KI-Ma) (KI-Ro1] (KI-Ro2]
(KI Ro3] -
(KI-Ro4]
311
S. Klainerman. PDE as a unified subject Special Volume GAFA 2000, 279-315 S. Klainerman and M. Machedon, Finite Energy Solutions for the Yang Mil ls Equa tions in JRl+3, Annals of Math., Vol. 142, (1995), 39-119. S. Klainerman and I. Rodnianski, Causal g eometry of Einstein- Vacuum space times with finite curvature flux Inventiones Math. 2005, vol 159 , No 3, pgs. 437-529. S. K lainerman and I. Rodnianski, A geometric approach to Littlewood-Paley theory, to appear in GAFA(Georn. and Punet. Anal) S. Klainerman and 1. Rodnianski, Sharp trace theorems for null hypersurfaces on Einst ein metrics with finite curvature flux, to appear in GAFA S. Klainerman and I. Rodnianski, Lower bounds for the radws of injectivity of null -
Rodnianski, A Kirchoff- So balev pammetrix for the wave equa, tions in a curved .pace-time. preprint. S. Klainerman and I. Rodnianski, A breakdown criterion in General Relativity preprint. V. Moncrief, Personal communication. S. Sobolev, Methodes nouvelle a 1-esoudre Ie probleme de Cauc hy pour les equations lineaires hyperboliques n orm ales Matematicheskii Sbornik, vol 1 (43) 1936, 31 -79. hypersurfaccs, preprint
lKI-Ro5) [KI-Ro6] [M] [Sob]
S.
Klainerman and
1.
,
DEPARTMENT OF
E-mail address:
MATHEMATICS,
PRlNCETON UNIVERSITY, PIUNCETO:-l NJ
serilDmath . princeton. adu
08544
Cuntemporary Mathema.tics
Volume 446: 2007
Some Liouville theorems and applications YanYan Li Dedicated to
Haim Brezis
,
with high respect and friendship
We give expo"ition of a Liouville t heorem established in [6] which is a novel extension of the classical Lionville theorem for harmonic fuuctions. To illustrat.e some ideas of the proof of the Liouville theorem, we present a new proof of the classical Liouville theorem for harmonic functions. Applications to gradient estimates of the Liouville theorem, as well as that of ear lier ones in [5], can be found in [6, 7] ami [10]. ABSTRACT.
The Laplacian operator Cl. is invariant under rigid motions: For any function u on JR.n and for any rigid motion T : JR.n -> JR." ,
Cl.(u 0 T )
=
(Cl.u) 0 T.
The following theorem is classical : (1)
u E C2, Cl.u = 0 and u > 0 in JR.n imply t.hat u
constant .
In this note we present a Liouville theorem in [6J which is a fully nonlinear version of the classical Liouville theorem ( 1 ) . Let u be a positive function in JR." , and let 1/J : JR.nU{ oo} JR.""U{ oo} be a Mobius transformation, i.e. a transformation generated by translations, multiplications by nonzero constants and the inversion x -> xllxl2. Set ->
1l.p : =
I J", I
On
y.-2
(u 0 1/;) ,
where J", is the Jacobian of 1/;. It is proved in [3J that an operator H(u, 'Vu, 'V 2 u) is confurmally invariant, i.e. H (ut/J , 'Vu", , 'V2u",)
=
lIeu, 'Vu, 'V2u) 0 1/; holds for all positive
u
and all Miibius 'Ij;,
if and only if H is of the form
H (u, 'Vu, 'V2u)
=
f(,\(AU))
1991 Mathematics Subject Classification. 35J60, 35J70, fi3A30. Key words and phrases. Liouville theorem, conformally invariant, elliptic, comparison prin ciple, gradient estimates. Partially supported by NSF grant DMS-0101118.
313
314
YA NYAN LI
where
I is the n x n identity matrix, ,\ (AU) = (AI ( AU), . . . , An (AU» denotes the eigenval ues of AU, and f is a function which is symmetric in A = (AI , . . . , An ) . Due to the above characterizing conformal invariance property, AU has been called in the literature the conformal Hessian of 'U , Since i= 1
Liouville theorem (1) is equivalent to
1£ E C2, A(AU) E Ofl and
(2) where
fl :=
u
> 0 in lR" imply that
1£
constant,
n
{,\ I L Ai > O}. i=1
Let (3 )
f
c
lRn be an open convex symmetric cone with vertex at the origin
satisfying
(4) Examples of such f include those given by elementary symmetric functions, For 1 < k S n, let I 3, let r satisfy (3) and (4), and let u be a positive locally Lipschitz viscosity solution of (6)
A(AU) E ar
Then u
_
'u(o) in JRn .
REMARK 2 . It was proved by Chang, Gursky and Yang in [1] that positive
C l ,l (JR4) solutions to A(AU.) E ar2 are constants. Aobing Li proved in [2] that positive C1,l (JR3 ) solutions to A(AU) E ar2 are constants, and, for all k and n, positive C3 (jRn ) solutions to A(AtL) E ark are constants. The latter, result for c3 (JRn) solutions is independently established by Sheng, Trudinger and Wang in [9] . Our proof is completely different.
In order to illustrate some of the ideas of our proof of Theorem 1 in [6] , we give a new proof of the classical Liouville theorem (1). We will derive (1) by using a .::l : Let
n be a bounded open subset of jRn, 2 > and v E C (f!) n 2, containing the origin O . Assume that u E Cl2oc(n \ {O}) satisfy and .::lu < 0, u > v, in n \ {OJ .::lv > ° in n, and on em. u>v Then inf (u - v) > O.
Weak Comparison Principle for
n\{o}
It is easy to see from t.his proof of the Liouville theorem ( 1 ) that the following Comparison Principle for locally Lipschitz viscosity solutions of (5) , established in [5 , 6] , is sufficient for a proof of Theorem 1 . PROPOSITION l . Let f! be a bounde� open subset of ]Rn containing the origin 0, and let u E cf�� (f! \ {O}) and v E CO, l (n). A ssume that u and v are respectively positive viscosity supe" 'soI1dion and subsolution of (5), and
u>v>0
on an.
Then
inf (u - v» D\{O}
the
O.
For the proof of Proposition 1 and Theorem 1 , see [5, 6] . In this note, we give
Proof of Liouville theorem (1) based on the Weak Comparison Principle for .::l . Let
1 vex) : = - [min u(y) J l x I 2-n, 2 Ivl= l Since Ul and Vl are still harmonic functions, an application of the Weak Comparison Principle for .::l on f! :=the unit ball yields
(7)
lim inf Ivln-2u(y) > O. Iy l�oo
316
YANYAN LI
LEMMA 1. For every x E Rn, there exists '\0 (x)
,\n -2 >.. 2 ( y - x) ) < u (V) ux,>, (y) : = I y _ ;"(; I n 2 u (x + I v 2 xl _
>
'if
0 s'uch that
0 < >.. < >"0 (x ) Iv ,
-
xl
> >.. ,
Proof. The proof is essentially the same as that of lemma 2.1 in [8] . Without loss
of generality we may take x = 0, and we use u), to denote 'uo , >, . By the positivity and the Lipschitz regularity of u, there exists TO > 0 such that '1'
7, - 2
"
'if
n-2
u( r, B) < s 2 1L ( s , B) ,
The above is equivalent to u>, (y )
cly I2- n ,
C
Let >"0 : = (
Then
(9)
maxl zl '1'0'
0 < ,\ < AO, Iyl > ,\ , o
Because of Lemma 1, we may deane, for any x E Rn and any 0 < 8 < 1, that A,, (X )
:=
SUP{11 > 0 I ux,)' (Y) < (1
+ 8)'u(y),
LEMMA 2 , For any x E Rn and any °
, and '\" to denote respectively 1£0 ,>, and '\,, (0) . Since the harmonicity is invariant under conformal transformations and multiplication by eOIlstants, and since -
-
an applicatioIl of (7) yields, using the fact that (1£), », inf [(1 + 8)U.\6 (V ) O O.
Namely, for some constant c > 0, ( 1 0)
-
(y ) > clvI 2 - n ,
317
SOME LIOUVILLE THEOREMS AND APPLICATIONS
By the uniform continuity of u on the ball {z I Izl < .A8}, there exists E > 0 such that for all A6 < A < A6 + € , and for all Iyl > .A, we have ( 1 + o)u(y) - U5.. (Y) + [U5., (y) - u>. (y)] -A26 Y 2y A 2u( > cly l2 -n _ I Y I 2 -n I An-2 u( ) An)1 6 Iv l 2 I v l2 This violates the definition of A6. Lemma 2 is established. ( 1 + o)u(y) - u>. (y)
>
_
>
-
By Lemma 2, Ao
00
,
for all 0 < 0 < 1 . Namely,
(1 + o)u(y) > ux,>. (Y),
'r/ 0 < 0" < 1 , x E ]Rn , Iy - x l >
Sending 0" to 0 in the above leads to
c l I 2 -n . 2 v
).
>
o
O.
n E x V ]R , Iy - xl > ). > O. Liouville theorem (1) is established.
u(y) > u r,>. (Y) ,
This easily implies 'U
u(O).
o References
[lJ S.Y.A. Chang, M. G ursky and P. Yang, A prior estimate [or a class of nonlinear equations on 4-manifolds, Journal D'Analyse Journal Mathematique 87 (2002), 151-186. [2J A. Li, Liouville type theorem for some degenerate conformally invariant fully nonlinear equa tion, in preparation. [3J A . Li and Y.Y. Li, On some conformally invariant fully non li near equations, COfllm. Pure App!. Math. 56 (2003), 14IG-1464. [1J A. Li and Y.Y. Li, On some conformally invariant fully lIonlinear equations, Part II: Liouville, Harnack and Yamabe, Acta Math. 195 (200.5 ), 117-154. [5J Y.Y. Li, Degenerate conforrnally invariant fully nonlinear elliptic equations , arXiv:math.AP /0.';04598 vI 29 Apr 2005; v2 24 May 2005; final version, to appe". I n Arch. Rational Mech. Anal. [6J Y.Y. Li, Local gradient estimates of solutions to some conform ally invariant fully nonlinear equations, arXiv:math.AP /0605559 vI 20 May; v2 7 Jul 2006. [7J Y. Y. Li, Lucal gradient estimatrn of solutions to some conformally invariant fully nonlinear equat ions, C . R. Math. Acad. Sci. Paris, Ser. J 343 (2006), 249-252. [8J Y.Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalitiesfor semilinear elliptic equations, Journal d'Analyse Mathematique 90 (2003), 27-87. [9J W. Sheng, N.S. 'I'rudillger and X.J. Wang, The Yamahe problem for higher order cllrvatures, arXiv:math.DG/0505463 vl 23 May 2005. [10J X.J. Wang, Apriori estimates and existence for a class of fully noulinear elliptic equations in con formal geometry, Chill. Ann. Math. 27B (2) (2006), 169-178.
OF MATHEMATICS, N J 08854, U SA
DEPARTMENT AWAY,
RUTGERS UNIVERSITY,
E-mail address: yyli0math. rutgers . edu
110
FHI>LI:'IGHUYSEN
ROAD,
PISCAT
Contemporary l\/Iathematics Volume 446, 2007
Analysis on Faddeev Knots and S kyrme Solitons: Recent Progress and Open Problems Fanghua Lin and Yisong Yang
It is
,
our great pleasure to dedicate this article to Haim Brezis
for his leadership, encouragement, and suppa,·t over many years.
We report some recent progress and discuss some related unsolved problems concerning the existence of the energy-minimizing configurations in the Faddeev quantum field theory model giving rise to knotted solitons and in the Skyrme model modeling elementary particles. Many issues related to the corresponding evolutionary systems for the Faddeev knots and Skyrme solitons, however, remain untouched. These are rather unusual and challenging systems of nonlinear partial differential equations of hyperbolic type. We also describe some simple but fundamental mathematical issues concerning these ABSTRACT.
models.
1. Introduction The area of quantum field theory is fascinating for analysts because it involves all types of analysis problems at both the classical levels and the quantum levels of diverse subjects of fundamental importance. According to quantum field theory, elementary particles and their interactions are elegantly described by continuous fields defined over spacetime and hosted within a vector bundle. Spacetime sym metry is related to special relativity and gravity and vector bundle symmetry is related to all other fundamental inter-particle forces. There are many interest ing and important mathematical issues in quantum field theory worthy of pursuit, which provide challenges as well as opportunities for mathematicians. For example, topological solitons known as domain walls, vortices, monopoles, and instantons in their simplest cases are often related to integrable systems and the construction of these solutions leads to further development of the solution methods including in verse scattering method [1 , 34, 48, 60, 95], the Backlund transformations [1, 72], 2000 Mathematics Subject Classification. 35J20, 35Q51, 58Z05, 81T45, 81V35. Key words and phrases. Quantum field theory, topological invariants, Faddeev model,
Skyrme model, calculus of variations, compactness, minimization, growth laws. The first author was supported in part by NSF Grant DMS 0201443. The second author was supported in part by NSF Grant DMS--0406446. ©2007
319
American Mathema.tical Society
320
FAKGHUA LIN AND YISONG YANG
the Penrose twistor method [8, 86] and ADHM [6], duality [45], group represen tations [46] ; when the systems encountered are not integrable, functional analysis methods are needed in order to understand the structure of the solutions [41, 93J. In this survey article, we report some progress in the mathematical understand ing of two import.ant but closely related quantum field theory models, the SkYrIne model and the Faddeev model, and we also discuss some unsolved problems. The central idea embedded in t.he Skyrme model [76, 77, 78, 79] is to use con tinuously extended , topologically characterized, relativistically invariant, locally concentrated, soliton-like fields to model elementary particles. There are two ways in order to achieve such a construdion in space dimension greater than one: (i ) introduce gauge fields, or (ii) introduce higher-order derivative terms, in the field theory Lagrangian. The first category models are various Yang-Mills gauge field theory models which allow static solitons characterized topologically by winding numbers, hOUlOt.Opy classes, and the Chern indices, and there often exists a beau t.iful self-dual reduction which gives rise to solutions as absolute energy minimizers among various topological classes. The second category models are the Skyrme type models where, although the field configurations arc ela.�sified by homotopy classes, no self-dual reduction exists for nontrivial solutions, and one has to study the minimization problcm for the original energy functional. It is interesting to not.e that , at the quantized level, the Skyrme model gives rise to two types of particles which are essential for nuclear interactions, namely, the mesons which are quantum fluctuations around t.he topologically trivial field configuration, and the baryons which are effectively realized as (topologically nontrivial) solitons. In this way, the baryon-meson and baryon-baryon scat.tering [40, 89] come naturally into the pic ture. More recently, the Skyrme model and its various variations have been applied to many areas including the quantum Hall effect [28, 65], Bose-Einstein conden sates [12, 70j, and cosmology [17, 61, 74] . In the classical Skyrme model, the field configuratioIlS are topologically represented by the homotopy group 71'3 (83) and the fundamental mathematical question is to ask whether there is a static Skyrme en ergy minimizer among each homotopy class. For some introductory reviews on the Skyrme model, see [35, 40, 94] . The Faddeev model [29, 30] was also proposed out. of the same motivation as that of Skynne [79], namely, to model elementary heavy particles by topological solitons, and may be viewed as a constrained or refined Skyrme model [22J . Al though, at a first glance, it seems that the Fadrleev model is a variant of the Skyrme model, it brings about a highly nontrivial twist and allows a stunningly new struc ture: the existenr:e of knotted solitons. It may be interesting to recall that, for mathematicians, knot theory has long been a theory of classification of knots by means of combinatorics and topology. For example, Tait [81] enumerated knots in terms of t.he crossing number of a plane projection; Alexander [31 discovered a knot invariant, known as the Alexander polynomial, arising in 3-dimensional ho mology; Jones [42, 43] found a new knot invariant, known as the Joncs polynomial, which enabled several conjectures of Tait to be proved [62, 63] ; based on a heuristic quantum-field theory argument, Witten [90J derived from the Chern-Simons action a family of knot invariants including the Jones invariant; finally came the Vassiliev invariants [85] which cover the Alexander polynomial and the .Jones polynomial and lay a general framework for the study of the combinatorial aspects of knots. In the Faddeev model, the field configurations are represented by the homotopy group
FADDEEV
7r3(82),
KNOTS AND SKYRME SOLITONS
the set of the Hopf classes from
83
to
82 ,
321
which is identical to the set of
integers, and the knotted solitons, or the Faddeev knots, are energy-minimizing field configurations among the Hopf classes. Using computer simulation, Faddeev
[32, 33]
and Niemi
first produced a ring-shaped (unknotted) Hopf charge (class)
one soliton. Shortly after the seminal work of Faddeev and Niemi, a more exten sive computer investigation was conducted by Battye and Sutcliffe
=
up t.o
=
[13, 14, 15]
who performed fully three-dimensional, highly convincing, computations for the
Q
Q
1
Q
8
= 1 , 2, 3, 4, 5, the energy-minimizing solitons are ring-shap ed and higher charges cause greater distortion, and for Q = 6, 7, 8, the solitons become knotted or linked. In particular, the trefoil knot appears at Q = 7. Thus, as in the case solution configurations of the Hopf charge that, for
from
Q
and found
of the Skyrme model, we are facing again a topologically constrained minimization problem and our ultimate question is to find out whether there is a static Faddeev
energy minimizer among each topological class defined by a nontrivial Hopf index (charge) . For a review, see
[31] .
In this survey article, we report some recent progress on the understanding of the minimization problems for the Skyrme and Faddeev models. In the next section, we review the related problems for the sine-Gordon model , the sigma model, and the Yang-Mills theory, and we emphasize the mathematical differences between these classical models and the Skyrme and Faddeev models we are facing.
3,
In Section
we review the Skyrme and Faddeev models and we introduce both the time
dependent governing equations and the static minimization problems associated with these models. In Section
4,
we present a series of existence results concerning
these static minimization problems. We emphasize the novelty and effectiveness of our techniques developed in treating these problems. In Section
5,
we review our
work on two-dimensional static Skyrme model which is of independent interest and serves as another illustrative comparison of the concentration-compactness principle and our method . Throughout this article, as we discuss various problems and technical iH..'iues, we also present some unsolved problems worthy of future study. In section
6,
we review other mathematical directions and developments concerning
the Skyrme and Faddeev models. In particular, we comment on some additional unsolved problems. In Section
7,
we conclude the article with a summary.
2. Topological solitons, energy lower bounds, and minimization (1 + 1)
dimensional sine-Gordon model, which is also the common origin of both the In order to describe the Skyrme model, we start with the classical
Skyrme and Faddeev models and has often been used as an illustrative labora is governed by a map cP from the Minkowski space
tory mathematical model in particle physics.
time RI, l of the time coordinate
of mesons of mass
'"
> 0
The effective meson field theory
Xo = t and space coordinate Xl = X with the circle 81 . Rewriting ¢ in terms of its two compo
(+-), into the unit nents, ¢o and cP l , we can represent these components by an as cPo(t, x) = cos a(t, x), ¢l (t, x) = sin a(t, x) . The relativistic signature
governing the evolution of the field
(2.1)
£=
�
a
angular variable a Lagrangian density
is
0,., ao'" a
- ",2(1 - cos a) ,
so that its corresponding Euler-Lagrange equation is of the form
(2.2)
FANGHUA LIN AND YISONG YANG
322
This equation, usually called the sine-Gordon equation, gained most of its popular
ity a bit later, after the discovery of the inverse scattering transformation method in the mid- 1960s. The equation (2.2) itself was known and investigated much earlier in
differential geometry with the problem of isometric embeddings of the hyperbolic planes (with negative constant Gaussian curvatures) into the Euclidian 3-space.
Though Skyrme did not seem to be aware of these results in geometry, he did
manage to obtain one-soliton solutions represented by the functions (2.3 )
a(x)
=
4 tan- 1 (e± I«
x-xo) ,
called the 21r-kinks. Since (2 . 1 ) is relativistic (or Lorentzian invariant) , we may use
a Lorentzian boost to switch on the time dependence of the solutions via replacing
(x - xo )
in (2.3) by
(x
-
xo
-
vt)/J1
-
v2 •
Besides these traveling wave solutions,
which Skyrme identified with mesonic excitations of small amplitudes, Skyrme also viewed these 21r-kink solitons as "particles" and he paid attention to the conserved normalized current (2.4) which automatically satisfies the conservation law phasized that this conserved current is
not
8J.tJJ.t = O.
It should be em
a Noether current, meaning it is not a
(2.1); it is not a conserved quantity by the field evolution either, meaning it does not follow from the equation of motion consequence of the symmet.ry of the Lagrangian
(2.2). Rather, it is a topological current which reflects the topological st.ructure of
the model, which will be discussed below.
First, note that quantization around the trivial solution
expected perturbative
meson particles
a = 0 gives rise to the
of mass '" as in the standard Klein-Gordon
model. However, in addition to the above well-known structure, the sine-Gordon
model also has a structure that allows the existence of non-perturbative heavier particles described by localized solutions such as those of the Skyrme 27r-kinks.
In order to see this more transparently, we recall that the associated Hamiltonian (energy) density of (2. 1 ) may be written as
1{
(2.5)
It is seen that the model
=
1 2
has
(a; + a;) + ,,2 ( 1 - cos a). count ably many potential wells which give us the
ground states a N = 2N1r,
(2.6)
N E Z,
and finite-energy nontrivial solutions interpolate between two distinct ground
Without loss of generality, we can malized boundary condition
consider
aCt, - (0) = 2n1r,
(2.7)
states.
the equation (2 . 2) subject to the nor
a(t, (0 ) = 2 ( n + N)1r.
field 4>(t, x) = eia(t,x) winds around the circle S1 as x goes from the left end to the right end of the space axis, and is directly relat.ed to the conserved current (2.4) by the formula It is clear that the integer N counts the number
(2 .8)
Q(a:)(t) =
where p =
JO
of times
J p(t, x) dx = 1: JO (t, x)dx = � 2
is the "charge" density which is seen
( 2 . 2 ) satisfying ( 2 . 7) or ( 2 . 8) ill called
an
to
N-soliton.
the
(a(t, 00) - a (t, -oo» be topological.
A
= N,
solution of
323
FADDEEV KNOTS AND SKYR.ME SOLITONS
It is well known [67] that (2.2) is integrable and one can use the inverse scat tering or the Biicklund transformation method to construct all N-soliton solutions of the equation [72] . An important characteristic of an integrable system is that it possesses infinitely many conservation laws, and our topological one, (2.8) , is just one of them. To evaluate the mass (energy ) of an N-soliton, we rewrite (2.5) as (2.9)
•
Integrating (2.9) over the full space line and using (2.7) , we obtain the topological energy lower bound (2. 10)
E=
00
- 00
for an N-soliton solution, where the sign is chosen to preserve the positiveness of the energy. For static solutions, the minimal bound is saturated if the equation O:x
(2. 1 1 )
± 2;,; sin
� =0
is satisfied, which is a self-dual or anti-self-dual reduction (in the sense of Bogo mol'nyi [19]) of the second-order static sine-Gordon equation O:xx
(2.12)
2 .
- ;,; sin 0: = O.
In other words, the solutions of (2. 1 1) are automatically the solutions of (2.12). We can also establish the converse. In fact, let 0: be a solution of (2. 12) satisfying the boundary condition (2 .7) and set p± = O:x ± 2;,; sin �. We can check that p± satisfies the separable equation
(x) a p± P±. x = ±K cos
(2.13)
2
whose solutions are represented as the integral (2. 14)
x Xu
cos a Cy l dy 2
=
p± (xo )R± (x) .
Using the boundary condition (2.7) , we see that either R+ (x) -+ 00 or R- (x) -+ 00 0 as as x -+ 00 . Since it follows from the boundary condition (2.7) that p± (x) x -+ 00 , we conclude from (2.14) that P+(xo) = 0 or P- (xo) = 0, which implies p+ = 0 or P- O. That is, a solution of (2.12) subject to the boundary condition (2.7) must be a solution of the self-dual or anti-self-dual equation (2. 1 1 ) . In this sense, (2.11) and (2.12) are equivalent . The equation (2.1 1) is elementary. Since a nontrivial solution of (2 . 1 1 ) ean never attain a value of the form 2m1l' (m E Z) at any point in (-00, 00), we see that, modulo translations of the form a f-> 2k1l' + 0: (k E Z) , the only static solutions are those one-soliton solutions of Skyrme given in (2.3), whose mass is the topological energy minimum determined in (2.10) with N = ±1, -+
_
(2.15) We have seen that the sine-Gordon model does not aJlow static N-soliton solu tions when the topologieal soliton charge N is not unit and higher-charge solitons
324
FANGHUA LIN AND YISONG YANG
must be nonstationary
[67] . From this,
we derive a valuable lesson that the topolog
ically cOIlstrained minimization problem for the static sine-Gordon model, namely, the innocently simple problem (2. 16)
EN = inf E(a) =
has no solution if
N i= 0, ± l .
J
OO
-
1 - a; + 1\;2( 1 _ cos a) dx Q ( a) 2
oc
The next example is the classical
0(3)
=
N ,
sigma model which is also the second
common origin of the Sky!lIIe and Faddeev models. The field configuration is a spin
¢ defined over the (2 + I )-dimensional Minkowski spacetime ]R2,1 and taking range in the unit sphere, S2 , of ]R3, namely, ¢ = ( 4)1 , ¢z , 4>3 ) , ¢I + ¢� + ¢� = l eW = l . The dynamics of the field
(t, · ) goes to
a
constant unit vect.or, which
¢(t, . ) a continuous map from S2 to S2 . Hence ¢(t, . ) represents a homotopy class in 1r (S2) = Z and is thus characterized by an integer N(t). This integer 2 N (t) is also the Brouwer degree deg(¢(t, . » , of ¢(t, ·) which measures the number of times SZ being covered by itself under the map ¢ (t, . ) . In fact, the integer N(t) makes
is again a topological "charge" . To see this, note that there is again an associated conserved topological current,
(2.20)
JI-I =
so that the total charge through the expression
Ji' ,
� El-l v,,/ ¢ 8
Q (¢)(t)
.
given by
(a,A> /\ o,¢),
/-t, v, 'Y
= 0, 1, 2,
may be calculated using the charge density p =
JO
1 r €jk ¢ ' (Oj ¢ /\ ok
(3.11)
E( rp ) =
1
( L laj rp l 2 + L laj¢ 1\ ak¢1 2 ) dx, 1It3 1<j
(5.2)
Q ( o) =
1
471"
1l!.2
o · (010 II 020) dx.
,
Like before, we are interested in the basic minimization problem
(5. 3 )
EN = inf{E(o) I E(o)
. and /1 satisftJ
(5.4)
then the minimization problem (5. 3) has a solution for N = ± l . Moreover, E1 < EN for all lNl > 2 if >'/1 < 12.
Roughly speaking, the condition (5.4) guarantees E1 < EN for all INI > 3. Note also that the condition (5.4) is crucial for the exclusion of dichotomy (the alternative (iii)) in t.he concentration-compactness principle. We show how to use our substantial inequality method to improve (5.4) signif icantly. First, we observe that the following substantial inequality holds for any >., J.I > 0:
5.2. Let N be a nonzero integer and {OJ } a minimizing sequence of the problem (5.3). Then either (i) holds (hence a subsequence of {OJ } converges weakly to a solution of (5. 3)) or there are nonzero integers N1 and N2 such that THEOREM
(5. 5) A s a consequence, if § denotes the subset of Z \ {O } for which every member N E § makes (5. 3) solvable, then § of 0. In particular, for any N E Z:: \ {OJ, there are integers N1 , . . . , Nt E § such that
(5.6)
N = N1 + . . . + Ne
and EN > E!V! + . . + EN£ ' .
Next we recall that we established [51] a topological lower bound which states that there is a positive constant C(>', J.I, N) (i.e., the constant only depends on the coupling parameters >. and J.I and the nonzero integer N) such that E(o) > 41T1 deg(ol l + c(>., /1, Q(o)) (Q(o) of 0). In particular, we have
(5.7)
EN > 441V1 ,
N of O.
If for N = 1 the compactness (the alternative (i)) for a minimizing sequence of (5.3) does not occur, then by Theorem 5 .2 there are nonzero integers N1 and N2 so that 1 = N1 + Nz and (5.8)
Since EN, > 0 and EN2 > 0, we �ee from (5.8) that Nl of ±1 and N2 of ± l . However, one of the Nl and N2 must be odd. Assume N1 is odd. Then I Nl l > 3 .
336
FANGHUA LIN AND YJSONG YANG
1N2 1 > 2.
Since N2 must be even, so get
Inserting these facts into
(5.8)
and
(5.7),
we
El > 47r (3 + 2).
(5.9)
On the other hand, llsing the stereographical projection as a test map, we may get an upper estimate for
Er
of the form
1 47r 1 + 2
(5. 10)
It i� clear that the inequalities
(5.9)
and
(5. 10)
are compatible only when
In other words, energy splitting (dichotomy (iii»
cannot occur when
(5. 1 1 ) Under this condition, the minimization problem
(5.3)
has
a
AIL > 192.
A, f.L
satisfy
solution for N
which gives a three-fold improvement upon the statement of Theorem
5. 1 .
See
for details.
= 1,
[52]
6. Other directions and developments In this section, we review some directions and developments not covered in the previous sections and we comment on some other unsolved problems. In the first two Hubsections, we are concerned about the Skyrme modeL \Ve return to the Faddeev model in the third subsection. vVe then disc1lsS HOrne additional mathematical issues on the Skyrme and Faddeev models in the last subsection.
6.1. Radially symmetric solutions in the Skyrme model.
We first con
sider radially symmetric solutions of the Skyrme model, also called the hedgehogs. In terms of the is given hy
group element and the radial variable
SU(2)
(3. 14)
in which
the Lagrangian density
(6. 1)
£(f)
=
nj (x)
(3. 15)
�2 (Dt !)2
_
=
xj /1'
and
becomes
1 ( D f) 2 r 2
wet, x )
H(f) =
� (Dd) 2
+
= 2/(1', t). Under this ansatz,
sin2 / _
1.2
with the associated Hamiltonian
(6 .2)
� (Or f)2 + Si�: /
1+
Si;:/ + (Orf)2
Finite-energy condition implies that the hedgehog angular variable the bounda.ry condition
lim fer, t) = (N + k)7r, T_O
(6 .3)
for some integers N and
k.
Q (I) (t)
21 1 "" 7r
(3.6)
satisfy
kiT
takes the reduced explicit form
0
Or/(r, t) sin2 /(7', t) d1'
- (1(0, t) - f(oo, t» = N, 7r
=
/ must
r CjklDjf(r, t) (n · [Dk n J\ Dt n] ) sin2 / (1', t) dx \ 47r JJlI.3
-(0.4)
lim fer, t) r---+ oo
.
On the other hand, we observe that, under the hedgehog
ansatz, the topological charge
-
a hedgehog
l' = l x i ,
FADDEEV KNOTS AND SKYRME SOLITONS
337
in terms of the boundary condition (6.3). The Euler-Lagrange equation of (6. 1 ) is 2
(6 . 5) Itt - frr - - fr r
=
2f . 2 j'; . 2 sm f + 2' sm(2 f) r r rr
-
sin(2f ) r2
sin2 I 2 1+ r
'
,
r
> 0,
which governs the dynamics of hedgehog solitons of the Skyrme model and whose well-posednesH has not yet even been studied. We now consider static hedgehogs only which are the solutions of the following two-point boundary value problem
(r2 + 2 sin2 f ) 1T>' + (1,.) 2 sin(2f) + 2r fr
(6.6)
f(O )
which makes the Skyrme energy (6.7)
E(f)
=
27l'
00
o
sin(2f) 1 +
N7r,
1 (00) 2
Si�: =
sin2 f + r2
f
,
r>
0,
0,
sin2 f dr
finite. Of course, (6.6) is the Euler-Lagrange equation of (6.7) . It can be shown that, for any integer N, (6.6) has a solution which may be obtained as the energy minimizer of (6.7) subject to the boundary condition stated in (6.6). See [26, 92] . In other words, among any topological class in 7l'3(S3 ) , the static Skyrme energy has a critical point which minimizes the energy functional among all radially symmetric field configurations. This result, although simple, is in sharp contrast with what we know for the sine-Gordon model where critical points of the energy of higher topo logical charges do not exist, as illustrated in Section 2. However, it is not known whether any of these critical points would be the absolute energy minimizers for the Skyrme model. In [56, 71] , the authors attempted to establish that the hedgehogs are absolute energy minimizen; at Q = N = ± l . Unfortunately, their proof appears to be problematic: their main idea is to use a procedure called Gelfand's Valley Method and make comparisons through the Steiner symmetrization and rearrange ment inequalities. However, the described minimization procedure leads to some incorrect orthogonality conditions (the expressions (4.5) and (4. 18) in [71]), which in fact may only be asserted as being obtained by maximization, contradicting the original expectation of these authors. More recently, a study of the interaction energy of widely separated Skyrme solitons, hence the configurations are not radially symmetric, was carried out [59] based on asymptotic analysis.
6.2. Geometrized Skyrme model and generalization. Motivated by con structing explicit or approximate static Skyrme solitons, Manton and Ruback [58]
geometrized the Skyrme model so that the energy governs maps from S� (the three sphere of radius R > 0) into the unit sphere S3. Note that, in this context, the classical Skyrme model may be recovered in the R ---7 00 limit. Lieb-Loss [49, 55] and Manton [57] showed that, modulo isometries, the "identity" map u ( x) = xI R is the unique absolute energy minimizer among the topological class defined by N = 1 for R < 1 . Lieb and Loss [49, 55] also showed that the identity map is a local minimizer for R < J2. Manton and Ruback [58] earlier showed that the identity map is unstable for R > J2. In fact, these results can be reformulated [91] for maps between two closed n-dimensional Riemannian manifolds, (M , g) and (N, h), where 9 and h are the metrics on M and N respectively, as follows.
338
FANGHUA LIN AND YISONG YANG
N be a differentiable map and u'h denote the pullback of the Let u : M metric h nnder u. Use dVg to denote the canonical volume element of (M , g) and aj (A) the elementary symmetric polynomials formed from the n eigenvalues of the symmetric matrix A. The generalized static Skyrme energy is then written ---+
(6.8)
E(u ) =
1M {al (g-lu'h) + an_ l (g-l u· h) } dvg•
Replacing the pair S'k and S3 , we consider the situation when (M , g) and (N, h) are homothetic [49, 55, 9 1] . That is, there is a diffeomorphism 'IjJ : M N such that 'IjJ* h = ,.,.2 g for some constant ,.,. > O. Then there holds the general result [91J that, up to isometries, the homothetic map with ,.,. > 1 is the unique minimizer of E among all maps of nontrivial topological degrees. Besides, this homothetic map is a (stable) local minimizer for the energy provided that [91J ,.,.2 + 21 T2( n- l ) (n - l) (4 - n) > 0 , ---+
(6.9)
,.,. 2 ..!. _ ! n 2
+ ,.,.2 (n- l)
� + n2 _ 2 n
(n - 1 )
>
0.
Note that, when n = 3, we have T2 > 1/2, which is the result of Lieb-Loss [49, 55] and Manton [57] ; for n = 4, we have T4 > 1/6; for n = 5, we have 1/2 > ,.,. 6 > 1/12; etc. More recently, Riviere proved in [68] that, for the geometric Skyrme model governing maps from S'k into S3 , there is an Ro E ( V3fi, J2] such that for R < Ro , the identity map x/ R remains to be the unique minimizer. Whether or not one can take Ro to be equal to J2 remains open. We also note that, in the geometric compact context here, there has been no study on the direct minimization problem for the energy functional (6.8).
6.3. Fractional-exponent growth law and knot energy. We noticed that one of the crucial facts that guarantees the existence of energy minimizing Faddeev knots in infinitely many Hopf classes is the sublinear energy growth property EN < C l NI3 /4 . That is, the minimum value of the Faddeev energy for maps from IR3 into 52 with Hopf invariant N grow sublillearly in N when N becomes large. This property implies that certain "particles" with large topological charges may be energetically preferred and that these large particles are prevented from splitting into particles with smaller topological charges. Such a feature may be a clue to the "stability of matter" problem and may also be relevant to the existence and stability of large molecular conformation in polymers and gel electrophoresis of DNA. In these problems, a crucial geometric quantity that measures the "energy" of a physical knot of knot (or link) type K (or simply knot) is the "rope length" L(K), of the knot K. To define it, we consider a uniform tube centered along a space curve r. The "rope length" L cr) of r is the ratio of the arclength of r over the radius of the largest uniform tube centered along r . Then (6. 10)
L (K) = inf{L(r) I r E K}.
A curve r achieving the infimum carries the minimum energy in K and gives rise to an "ideal" or "physically preferred" knot [44, 47], also ealled a tight knot [21]. Clearly, this ideal configuration determines the shortest piece of tube that can be closed to form the knot. Similarly, another crucial quantity that measures the geometric complexity of r is the average number of crossings in planar projections
FADDEEV KNOTS AND SJ 0 are two universal constants and the exponent p satisfies 3/4 < p < 1 so that in truly three-dimensional situations the preferred value of p is sharply at p = 3/4. This relation strikingly resembles the fractional-exponent growth law for the Faddeev knots just discussed and reminds us once more that a sublinear energy growth law with regard to the topological content involved is essential for knotted structures to occur. In our quantum field theory problem, it is the underlying property and struc ture of the homotopy group 1T3 (S2) and the Faddeev energy functional formula that guarantee such sublinear growth. Generally, it seems that such a property may be related to the notion of quantitative homotopy introduced by Gromov [36J . For ex ample, we may consider the Whitehead integral representation of the Hopf invariant and the "associated" knot energy ala Faddeev. More precisely, let u : R4n-1 -+ s2n (n > 1 ) be a differentiable map which approaches a constant sufficiently fast at infinite. Denote by f2 the volume element of s2n and IS2n l = Is'n f2. Then the integral representation of u in the homotopy group 1T4n _ l (s2n), say Q (u) , which is the Hopf invariant of u, is given by 1 (6.13) Q ( u) = I S2n l IR4n-l v 1\ u' (f2) , dv = u* (f2) . We can introduce a generalized Faddeev knot energy for such a map u as follows, (6. 12)
(6.14) For this energy functional, we are able to establish the following generalized sub linear energy growth estimate < G2 I NI (4n- 1 )/4n , < EN _ (6 . 1 5) G1 IN I (4n- 1 )/4n _ where EN - inf{E(u) I E (u) < 00, Q(u) = N} and GI , G2 > 0 are universal constants. It is seen clearly how the dimension number of the space comes into play. We have reported some of our preliminary results along this line of generalizations in [53J and detailed work will appear in a separate forthcoming paper.
6.4. Additional technical issues. In addition, we mention some other in
teresting technical questions which may be worth pursuing. Soholev spaces of mappings. Returning to the original Faddeev model and the minimization problem (3.24) , we have introduced the following Sobolev-space of mappings from R3 into S2 ,
X = {n : R3 -+ S2, E(n) < oo}. It was proved in [50J that for each n E X, dCn' CO » = 0, here f2 is the area form on S2 . Hence, by a result of Bethuel [18], n can be strongly approximated by a sequence nj E Goo (R3, SZ) such that nj -+ n in H l (R3 , S2) as j -+ 00. (6.16)
FANGHUA LIN AND YISONG YANG
340
81)(n) = 0 on �3, where
questions of smooth maps in the spaces of Sobolev maps. ->
Energy splitting.
imizers in every topological class of maps can be easily proved over a bounded For both Faddeev and Skyrme models, the existence of min
contractible smooth domain in
IR3
R,
(in particular, on a ball of radius
Understanding of these minimizers as
R -> 00
say
BR)'
remains to be a difficult question.
In fact, it is even unknown whether the minimum value
(6.18) satisfies
EJ.j = inf{E(n) I n : BR EfS -> EN
as
R ->
->
82 , n laB ,, = canst, Q(n) = N}
00.
We know that, if maps in a minimizing sequence of the Faddeev (or Skyrme) energy functional split as in the Substantial Inequality, then we have EN > + . . . + ENe - It is believable, in thi::; case, that, one would also have EN < EN, + . . . + Hence, EN = EN] +- . + EN, . This is the case for the two-dimensional Skyrme
EN1
EN, .
.
model due to the technical lemma
6. 1
in
[51]
(see also Lemma
7.4
in
[50] ) .
The
validity of this lemma in three dimensions for either the Faddeev or Skyrme model i::; unknown . On the other hand, if one has this lemma for the three-dimensional case, then one can show that E.� -> EN as -> 00 is true. Moreover, if the Skyrme would he valid.
model has only the cla::;s
Regularity.
N = ±1
R
being realized by minimizers, then
EN = IN I El
One of the most fundamental problems that remains open for both
Skyrme and Faddeev models is the regularity of solutions . It may be relatively easier to check the regularity of those specially constructed I:iOlutions (such as the hedgehogs) as described above. It is however a difficult problem for the absolute energy minimizers.
These questions are also closely related to similar questions
in nonlinear elasticity (see
[U]).
Indeed, the minimization problems here may be
compared with similar constrained minimization problems in nonlinear elasticity.
7. Conclusion Quantum field theory renderH to analysts a wide variety of mathematical prob lems of fundamental importance and a subclass of these problems are various con strained minimization problems giving rise to topological soliton::; modeling elemen tary particles and their interactions. As illustrated by the sine-Gordon model, there sometimes may not exist a critical point of the energy functional over a given topo logical class, and the existence of absolute energy minimizers is often a consequence of a self-dual or Bogomol'nyi structure in the problem and a direct minimization
FADDEEV KNOTS AND SKYRME SOLITONS
341
is likely to be impossible as witnessed by the earlier studies ranging from Lhe sine
Gordon model t.o the Yang-- Mills gauge field theory models. In the study of the Skyrme and Faddeev models, there is no useful self-dual or Bogomol'nyi structure.
Consequently, we have to consider the direct minimization problems of the corre
sponding energy functionals. In this situation, the well-established concentration
compactness principle cannot be used directly. Thus, we need 1.0 analy,,;e the be
havior of the splitting field configurations of a minimizing sequence. As a result,
we can establish an important inequality, called the substantial inequality, resem bling the process of a composed particle splitting into finitely many sub-particles during which Lhe total topological charge of the composed particle is exactly di vided into the sum of the topological charges of the sub-parti(:les and the mass
(i.e.,
the energy) of the composed particle is at least equal to the sum of the masses
of its sub-particles, with possible extra uncounted kinetic and bounding energies,
resulting in a topological charge conservation law and a mass or energy subaddiLive inequality, relating the charge and mass of the composed particle to the charges and masses of its sub-particles. As a consequence of this inequalit.y and some suit able energy estimates, it follows that splitting cannot occur at the unit topological charge for boLh the Skyrme model and the Faddeev model. In other words, the existence of a unit-charge Skyrme soliton and Faddeev knot is proved. In the case of the Faddeev model, we have in addition established a fractional-exponent growth
law for the energy minimum in terms of the topological charge. Such a growth law
ensures the existence of the Faddeev knots realizing an infinite class of topological charges and plays an e�Hential role in explaining why knotted structure is preferred
over isolated multiple-soliton configurations. In the Lwo-dirneIlt;ional Skyrme modcl case, although both the concentration-compactness method and the substantial in
equality method work, the latter provide� a much stronger existence result than the former. It is our hope that the described substantial inequality method may be use
ful to other diffieult topologically constrained minimization problems in quantum field theory.
Acknowledgment.
The authors would like to thank Xiaosong Lin for many help
ful conversations and communieations.
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[71J Y. P. Rybakov and V. 1. Sanyuk, Methods for studying 3+1 localized structures: the Skyrmion as the absolute mimmizer of energy, Internat. J. Mod. P hys. A 7 (1992) 3235-3261. [72J A. C . Scott , F. Y. F. Chu, and D . W. McLaughlin, The soliton: a new concept in apphed science, P roc. IEEE 61 (1973) 1443-1483. [73J J. Shatah, Weak solutions and development of singularities of the SU(2) (J'-model, Commun. Pure App\. Math. 41, 459--469 (1988). [74] N. Shiiki and N. Sawado, Regular and black hole solutions in the Einstein-Skyrme theory with negative cosmological constant, Class. Quant. Grav. 22 (2005) 3561-3574. [75J L. M. Sibner R. J . Sibner, and K. Uhlenbeck, Solutions to Yang-Mills equations that are not self-dual, Proc. Nat. Acad. Sci. USA 86 (1989) 8610-8613. [76J T. H. R. Skyrme, A nonlinear field theory, Pro c. Roy. Soc. A 260 (1961) 127-138. [77J T. H . R. Skyrme, Particle states of a quantized meson field, P roc . Roy. Soc. A 262 (1961) 237-245. [78] T. H. R. Skyrme, A unified field theory of mesons and baryons, Nucl. Phys. 31 ( 1962) 556569. [79] T. H. R. Skyrme, The origins of Skyrmions, Internat. J. Mod. Phys. A 3 (1988) 2745-275l. [80J W . A . St rauss, Nonlinear Wave Equations, Amer. Math. Soc., Providence, RI, 1989. [81J P. G. Tait Scientific Papers, Cambridge U niv. Press, Cambridge, 1900. (82) G . Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura App\. 110 (1976) 352-372. [83] C. H . Taubes, The existence of a non-minimal solution to the SU(2) Yang-Mills-Higgs equations on 1R3, Parts I, II, Commun. M at h Phys. 86 (1982) 257-320. [84J A. F. Vakulenko and L. V. Kapitanski, Stability of solitons in S2 nonlinear (J'-model, SOy. Phys. Dok!. 24 (1979) 433-434. 185] V. A. Vassiliev, Invariants of knots and complements of discriminants, in D evelopments in Mathematics: the Moscow School, Chapman & Hall, London, 1993. pp. 1 94-250. 186] R. S. Ward, On self-dual gauge fields, Phys. Lett. A 61 (1977) 81-82. 187] R. S. Ward, Hopf solitons on 83 and R3 , Nonlinearity 12 (1999) 241-246. [88] .1. H. C. Whitehead, An expression of Hop! 's invariant as an inte.qral, Proc. Nat. Acad. Sci. 33 (1947) 1 1 7-123. 189] E. W itten, Baryons in liN expansion, NucL P hys. B 160 (1979) 57-115. 190] E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351-399. 191] Y . Yang, Generalized Slr:yrme model on higher-dimensional Riemannian manifolds, J. Math. P hys. 30 (1989) 824--828. [92J Y. Yang, On the global behavior of symmetric Skyrmions, Lett. Math. Phys. 19 (1990) 25- 33. [93J Y. Yang, Solitons in Field Theory and Nonli near Analysis, Springer, New York 200l. [94] L Zahed and G . E. Brown, The Skyrme model, Phys. Reports 142 (1986) 1-102. 195] V. E. Zakharov and A. B. Shabat, Intemction between solitons in a stable medium, Sov. Phys. JETP 37, 823-828 ( 197::l). ,
,
.
,
COURANT INSTITUTE OF MATHEMATICAL SCIENCES, NEW YO RK 10021 E-mail address: l1nf«lcims . nyu . edu
NEW
YOUK UNIVERSITY, NEW YORK,
DEPARTMENT OF MATHEMATICS, POLYTECHNIC UNIVERSITY, BROOKLYN, E-mail address: yyang«lmat h . poly. edu
NEW
YORK 1 1 20 1
2007
Contemporary MFlthp.mA.tir:J:! Vulume 446, -
The precise boundary trace of positive solutions of the equation
b.u
=
uq
in the supercritical case.
Moshe Marcus and Laurent Veron
To Haim, with friendship and high esteem.
precise boundary trace of positive solutions of Au uq in a smooth bounded domain !1 C RoN , for q in t.he "uper-crit.ical case q > (N + 1 ) I (N 1 ) . The construction is performed in the framework of the fine topology associated with the Bessel capacity C2 /q , q ' on 8!1. We prove that the boundary trace is a Borel mellsure ( i n general unbounded) ,which is outer regular and essentially absolutely continuous relative to this capac ity. We provide a necessary and sufficient condition for such measures to be the boundary trace of .. pORit.ive solution and prove that the corresponding generalized boundary value problem is well-posed in the class of qc and every compact set K c an, the maximal solution of (1.1) vanishing outside K is a-moderate. Their proof was based on the derivation of sharp capacitary estimates for the maximal solution. In continuation, Dynkin [4J used Mselati's (probabilistic) approach and the results of Marcus and Veron [16J to show that, in the case q < 2, all positive solutions are a-moderate. For q > 2 the problem remains open. Our definition of boundary trace is based on the fine topology associated with the Bessel capacity C2/q,q' on Dn, denoted by 'Iq . The prcscntation requires some notation. ,
Notation 1 . 1 . a:
13 > 0 put p(x) dist (x , an) and n" = {x E n ; p (x ) < 13 } , n� = n \ n" , I:" = an{3'
For every x E IR N and every
b: There exists a positive number
Vx E n"o
( 1 .2)
3!
;=
130
such that,
a(x) E an : dist (x, a(x))
=
p(x).
If (as we assume) n is of class C2 and 130 is sufficiently small, the mapping x (p(x) , a(x)) is a C2 diffeomorphism of 0.130 onto (0, 130) x an. e : If Q c an put I:,,(Q ) = { x E I:" : a( x ) E Q}. d: If Q is a 'Iq-open subset of an and u E C (an) we denote by u� the solution of (1.1) in n� with boundary data h = UXE8(A) on I:" . f-t
Recall that a solution u is moderate if lui is dominated by a harmonic function. When this is the case, u possesses a boundary trace (denoted by tru) given by a bounded Borel measure. The boundary trace is attained in the sense of weak con vergence, as in the case of positive harmonic functions (see [13J and the references therein) . If tr u happens to be absolutely continuous relative to Hausdorff (N 1) dimensional measure on an we refer to its density f as the L1 boundary trace of u and write tr u = f (which should be seen as an abbreviation for tr u = jd lHIN -1) . A positive solution 'lL is a-moderate if there exists an increasing sequence of moderate solutions {un } such that Un r u. This notion was introduced by Dynkin and Kuznetsov [7J (see also [9J and [3] ) . If ,1 is a bounded Borel measure on an, the problem -
(1.3)
-Llu + uq = 0 in n, u = /l on an
possesses a (unique) solution if and only if /l vanishes on sets of C2/q,q,-capacity zero, (see [15J and the references therein) . Thc solution is denoted by Uw The set of positive solutions of ( 1 . 1) in n will be denoted by U(n). It is well known that this set is compact in the topology of C (n ) , i.e., relative to local uniform convergence in n.
PRECISE BOUNDARY TRACE
347
Our first result displays a dichotomy which is the ba.'iis for our definition of
boundary trace.
1.1.
Let u be a positive solution of ( 1 . 1 ) and let e either, for every 'rq-open neighborhood Q of �, we have THEOREM
u dS /' j3� 0 iE {3 (Q )
( 1 . 4) or
( 1 .5 )
lim
j3�O
The first case OCCUTS if and only if
E,,(Q)
l Uqp(X)dX = DO ,
( 1 . 6)
e
udS
an . Then,
= DO
lim
ther'e exists a 'rq-open neighborhood Q of
E
such that < DO.
D = (0, ,60 )
x
Q
f01' every 'rq -open neighborhood Q of e . A
point e
holds, and a
E an is called a singular point of u in regular point of u in the second case.
denoted by S(u) and its complement in
an
by R(u).
the first case, i.e. when ( 1 . 4)
The set of singular points is
Our next result provides additional information on the behavior of solutions
near the regular boundary set R ( u).
THEOREM 1 . 2 . The set of regular points R (u ) is 'rq -open and theTe exists a non-negative Borel measure p on an possessing the following properties.
(i) For every cr E R(u) there exist a 'rq-OJlt!1t neighborhood Q of cr and a moderate solution 11! s'lLch that Qc
( 1 . 7)
and ( 1 . 8)
u
�
-> 'UJ
R (u )
p( Q) < -
,
lucally uniformly in n
(ii) It i.� outer regular relative to 'rq •
Based on these results we define the tr Cu
( 1 .9)
=
,
DO ,
(tr w ) xQ
precise boundary trace of u
by
(fJ, S(u » .
Thus a trace is represented by a couple (p, S), where
P is an outer regular measure relative to
an \ S.
= PXQ'
'rq
S C an
is 'rq-closed and
which is 'rq-Iocally finite on R
However, not every couple of this type is a trace,
A necessary and sufficient
condition for such a couple to be a trace is provided in Theorem 5 . 1 6 . The trace can also be represented by a Borel measure
( 1 . 10) for every Borel set (1.11)
v(A)
A
c
=
an. We
put
It(A) DO
'rq .
otherwise,
tr u : = v.
This measure has the following properties: (i) It is outer regular relative to
if A c R(u),
=
v
defined as follows:
.'
MOSHE MARCUS AND LAURENT VERON
348
(ii) It is essentially absolutely contirmous relative to C2/q,q" i.e., for every '!'q-open set Q and every Borel set A such that C2/q,q' (A) = 0, v(Q) = v(Q \ A).
The second property will be denoted by v
j-
0,
= 1 2 .... ,
,
> O.
This is possible because our asRumption implies that there exists a compact. subset of E2 \ El of positive capacity. By induction we obtain E' m C E'm m cD m-l-, U Fe",.
(2.12) and consequently
m.
(2. 13) Since Fm C Ern , (2.13) implies that (2. 14)
- UOO G . - UOO k=1 E" k k=1 F'k k , '
-
Grn ;= Urn k = 1 Fe. k= 1 E"k = Urn k .
352
MOSHE MARCUS AND LAURENT VERON
{Em }
The sequence (2. 14) ,
constructed above satisfies
(2.15)
2.6.
Indeed, by
(2.5),
(2. 10) and
k=m+1 < 2 � C2/q,q' (Ek \ Ek l ) < z - m+1C2/q,q' (E). k=m+l ""
Next we show that the Het
(2.13), OO l Fk 0 there exists an open set 0. such that CZ/q,q' ( OE) < € and E \ 0. is covered by a finite subfamily of D. By [1, Sec. 6.5. 1 1) the (a, p)-fine topology possesses the quasi Lindelof property. Thus there exists a denumerable subfamily of D, say {Dn}, such that PROOF.
0 = U {D : D E D} :!. UDn . Let On be an open set containing Dn such that C2/q,q' (On \ Dn) < (/(2n 3). Let K be a compact subset of E n (Uj'" Dn ) such that C2/q ,q, (E \ K) < (/a. Then {On} is an open cover of K so that there exists a finite sub cover of K, say { 01 ' . . , Od. ,
It follows that
C2jq,q , (E \ U�=I Dn ) < C2jq,q, (E \ K) + L CZjq,q' (On \ Dn) < 2(/ 3. Let 0. be an open subset of an such that E \ U�=lDn c O. and C2/q,q'(0.) < €. This set has the properties stated in the lemma. 0
LEMMA 2.6. (a) Let E be a q-quasi closed set and {Em} a proper q-stratification for E. Then there exists a decreasing sequence of open sets {Qj } such that UEm := E' C Qj for every j E N and
(2 . 20)
q (i) n Qj = E', Qj +l C Qj , (ii) lim C2jq ,q, (Qj) = C2jq,q, (E) .
354
MOSHE MARCUS AND LAURENT VERON
(b) If A is a q-open set, there exists a decreasing sequence of open sets { Am } such that A c nAm =: A' , (2.21) A ,t A' . Furthermore there exists an increasing sequence of closed sets {Fj } such that Fj C A' and (2.22)
(i)
U FJ = A' , -
•
PROOF. (a) Let {em} be a sequence of positive numbers decreasing to zero satisfying (2 .6). Put Then E' = nQj and (2.23 )
q 00 Em jk' Therefore, /2 C2/q,q· (Qj \ D ) s C2/q,q. ( Qj \ U�= 1 (Emr� j ) S 2 - k +1C2/q,q. (E) Vj > jk. Hence
C2/q,q' (Qj \ D) --> 0 as j --+ 00 . (2 .24) Let { D;} be a decreasing sequence of open neighborhoods of E' such that C2/q,q' (D i) --+ C2/q,q' ( E'). By (2.24) , for every i there exists j(i) > i such that
(2.25)
C2/q,q. (Qj( i) \ Di) --> 0 as i -+ 00 .
lt follows that
C2/q, q' (E') < lim C2/q , q' ( Qj (i) < lim C2/q,q' (Di) = C2/q,q' (E' ) = C2/q,q' (E ) . This proves (2.20) (ii). (b) Put E = an \ A and let { Em } and {I'm} be as in (a). Then (2.21) holds with Am := an \ Em . In addition, (2.22)(i) with Fj := an \ Qj is a consequence of (2.20 ) (i ) . To verify (2.22) (ii) we observe that, if K is a compact subset of A' then, by (2.24), C2/q,q' (K \ Fj) --> O. Let, {Ki} be an increasing sequence of compact subsets of A' such that
C2/q,q· (Ki) i C2 /q q· (A ) = C2/q,q· (A ) . As in part (a), for every i there exists j(i) > i such that ,
(2.26) It follows that
C2/q,q' ( Ki \ Fj (i)
'
-+
0
as
i --+
00 .
C2/q,q' (A ' ) > lim C2/q,q' ( Fj(i) � lim C2/q,q' ( Ki ) = C2/q,q' ( A') = C2/q,q' ( A ) .
PRECISE BOUNDARY TRACE
355
o
This proves (2.22) (ii).
LEMMA 2.7. Let Q be a q-open set. Then, for every e E Q, there exists a q-open set Q� such that e E Q� c Qe c Q.
PROOF. By definition, every point in Q is a q-thin point of Eo = ao \ Q. Assume that diam Q < 1 and put: Tn = 2 -n ,
Kn = {O' : Tn+1 < 10' - el < Tn },
En := Eo n Kn n B1 (e ) · Thus En is a q-closed set; we denote E = U� oEn. Since e is a q-thin point of E , DC � (T - N +1+2/ ( q- l ) C / (B n E)) q - I < 00 2 q,q' n , � n o which is equivalent to
q l � (T -N+1+2/ Cq- I ) C 2/q,q' (En ) ) - < 00 L....o Let {Em , n}�=1 be a q-proper stratification of En. Let en := {Em,n }�=l be a q-proper sequence (relative to the above stratification) such that (I , n E (0, Tn+2) and C2/q,q' (Vn ) < 2C2/ q ,q' (UEn) where Vn := U� I E:"rr:;;, n Bl (0· Then Vn C Kn- 2 \ Kn +2 ' t; is a q-thin point of the set G = UO"Vn and e rt G. Consequently e rt G. Put 2 / rr Zn : = U�= l E:" :n,, n BI/2 (0 , Fo := Ug"Zw 00
•
11
Since Zn C Vn it follows that t; is a q-thin point of Fo and e rt Fo . Consequently Qo := (Q n B1 /2 (e) ) \ Fo is a q-open subset of Q such that -
-
e E Qo ,
Q o c (Q n Bl/2 (�) ) \ Fo C (Q n B1 / 2 (0 ) \ E c Q .
o
3. Maximal solutions We consider positive solutions of the equation ( 1 . 1 ) with q > qc, in a bounded domain n c ]R N of class C2 A function u E Lioc(n) is a sllbsollltion (resp. supersolution) of the equaLion if -Au + lul q- I u < 0 (resp. > 0) in the distribution sense. If u E Lioc(n) is a !iubsolution of the equation then (by Kato's inequality [8]) A l u l > lul q · Thus l u i is subharmonic and consequently u E L�c(n). If u E Lioc (O) is a solution then u E C2 (0) . An increasing sequence of bounded domains of class C2 , {On}, such that On T 0 and On C 0,,+ ] is called an exhaustive sequence relative to O. PROPOSITION 3.1. Let u be a non-negative function in L�c(n) . (i) If u is a subsolution of ( 1 .1), there exists a minimal solution v dominating u, i. e., u < v < U fOT any solution U ? u. (ii) If u is a supersolution of (1.1), there exists a maximal solution w dominated by u, i. e., V < w < u for any solution V < u. All the inequalities above are a. e . .
MOSHE MA
356
Reus AND LAURENT VERON
PROOF. Let u. = J.u where J, is a smoothing operator and u is extended by zero outside O. Put fi = lim€-+o u. (the limit exists a.e. in 0 and fi u a.e.) . Let ii in L 1 ( 0 ) it follows that (:In, 0iJ' �iJ etc. be as in Notation 1 . 1 . Since u. =
u. I E(j
--+
'ij, 1
--+
in L1 (2:B)
E(j
for a.e.{3 E (0, (30) . Choose a sequence {{3,, } decrea.sing to zero such that the above convergence holds for each surface 2:n := 2:iJ,, ' Put D" : = O�n ' Assuming that u is a subsolution of ( 1 . 1) ill 0, U. is a subsolution of the boundary value problem for (1.1) in Dn with boundary data u. En ' Consequently ii is a subsolution of the boundary value problem for ( 1 . 1 ) in D" with boundary data fil E n E L 1 (2:n). (Here ii q in L}oc ( O) . ) we use the assumption 'U E L�c(O) in order to ensure that u� Let 'Un denote the solution of this boundary value problem in thc L1 senHe: --+
vn = iL on 2:"..
Then Vn E C2(Dn) n L ""' (Dn ) , Vn < lI u IlL�(Dn) and the boundary data is assumed in the L 1 sense. Clearly iL < Vn in Dn, n= 1 ,2 . . . . In particular, 'Un < Vn+l on �n . This implies Vn :s Vn+l in Dn. In addition, by the Keller-Osserman inequality the scquem:e {vn } is eventually bounded in every compact subset of O. Therefore v = lim Vn is the solution with the properties stated in (i). Next assume that u is a snpersolution and let {Dn } be as above. Since u E Lq (Dn) there exists a positive solution Wn of the boundary value problem q u Aw ' D U In ,
-
-
10
Hence U + Wn is superha.rmonic and its boundary trace is precisely ii i . ConseEn quently U + Wn > Zn where Zn is the harmonic function in Dn with boundary data ill . Thus Un := ZM. Wn is the smallest solution of ( 1 . 1 ) in Dn dominating u. This implies that {un } dccrease;; and the limiting solution U is the smallest solution of 0 (1.1) dominating U in O. En
-
PROPOSITION a.2. Let u, v be non-negative, locally bounded functions in O . (i) If tt , V are subso/utions (resp. superso/utions) then max(u, v) is a subsolution (re$p. mine u, v) is a supersolution) . (ii) If u, v are supersolutions then u + v is a superso/ution. (iii) If u is a subsolution and v a superso/ution then (u v)+ is a subsolution. -
PROOF. The first two statements are well known; they can be verified by an application of Kato's inequality. The third statement is verified in a similar way: q 6.(u - v) + = sign + (u - v)6.(u v) > (u - vq)+ > (u - v)� . o
-
Notation 3. 1 . Let u, v be non-negative, locally bounded fUllctions in O. (a) If u is a subsolution, [ul t denotes the smallest solution dominating u. (b) If 'U is a supersolution, [uP denotes the largest solution dominated by 7L. (c) If u, v are subsolutions then u V v : = [m ax(u v)Jt. (d) If u, v are sllpersolutions then u /\ v : = [inf( u, v)] l and u EB v := [u + v) t . (e) If 'IL is a subsolution and v a supersolution then u e v := [(u v) + I t . ,
-
357
PRECISE BOUNDARY TRACE
The following result was proved in [9] (see also [3, Sec. 8.5] ) . PROPOSITION 3.3. (i) Let {ud be a sequence of positive, continuous subsolu tions of (1.1) . Then U := sup Uk is a subsolution. The statement remains valid if subsolution is replaced by supersolution and sup by inf. (i'i) Let T be a family of positive solutions of ( 1 . 1 ) . Suppose that, for every pair UI , U2 E T, ther'e exists v E T such that
max(uI ' U2) < v, resp. min( u I ' U2 ) > v.
Then there e:r:ists a monotone sequence {un} in T such that Un j sup T, resp. Un 1 sup T. Thus sup T (resp. inf T) is a solution. DEFINITION :3.4. A solution u of (1.1) vanishes on a relatively open set Q c 812 if u E C(n U Q) and u 0 on Q. A positive solution u vanishes on a q-open set A c 80 if u = sup{v E U(n) : v < u, v = 0 on some relatively open neighborhood of A } . When this is the case we write u r::; O. =
A
3.5. Lp.t A be a q-open subset of 812 and u I , 1[2 E U(n). (a) If both solutions vanish on A then Ut V U2 r::; O. If U2 r::; 0 and 'Ul ::; U2 then LEMMA
UI
r::;
A
A
O.
A
(b) If u E U (n) and u r::; 0 then there exists an increasing sequence of solutions A {Un} C U(n ) , each a/ which vanishes on a relatively open neighborhood of A (which may depend on n) such that Un j u . (c) If A, A' are q-open sets, A ,!!., A' and u r::; 0 then u r::; O. A
A'
PllOOF. The first assertion follows easily from the definition. Thus the set of solutions {11} described in the definition is closed with respect to the binary operator V . Therefore, by Proposition 3.3, the supremum of this set is the limit of an increasing sequence of elements of thiR set. 0 The last statement is obvious. DEFINITION 3.6. (a) Let u E U (n) and let A denote the union of all q-open sets on which 'U vanishes. Then 812 \ A is called the fine boundary support of u, to be denoted by supp �!1u. (b) For any Borel set E we denote UE = sup{u E U (n) : n � 0, eo = 812 \ E}, Thus UE
=
Uii;'
Ee
3.7. (i) Let A be a q-open subset 0/ an and {un} C U (12) a sequence of solutions vanishing on A. If {un } converges- then U = lim Un vanishes on A . In particular, if E is Borel, UE vanishes outside E. (ii) Let E be a Borel set such that CNq,q' (E) = 0, If U E U(I.1) and U vanishes on every q-open subset of EC = 812 \ E then U = O. In particular, UE = O . (iii) If { An } is a sequence of Borel subsets of 812 such that C2/q,q, (An) 0 then UAn O. LEMM A
->
->
MOSHE MARCUS AND LAURENT VERON
358
(i) Using Lemma 3.5 we find that, in proving the first assertion, we may assume that {un} is increasing. Now we can produce an increasing sequence of solutions {wn} such that, for each n, Wn vanishes on some (open) neighborhood of A and lim Wn = lim Un' By definition lim Wn vanishes on A. Let E be a q-c1osed set. By Lemma 3.5(a) and Proposition 3.3, there exists an increasing sequence of solutions {un} vanishing outside E such that UE = lim Un Therefore UE vanishes outside E. O. The (ii) Let An be open sets such that E C An , An 1 and C2/ q , q' (An) sets An have the same properties and, by assumption, u vanishes in (An)C : = 80. \ An . Therefore, for each n , there exists a solution Wn which vanishes on an open neighborhood Bn of (An)C such that Wn < '11. and Wn --+ u. Hence Wn < UKn 0, the capacitary where Kn = B� is compact and Kn C An· Since C2/q,q' (Kn) estimates of [16J imply that lim UKn = 0 and hence u = O. (iii) By definition UA n = U,4n ' Therefore, in view of Proposition 2.2(iv), it is enough to prove the assertion when each set An is q-c1osed. As before, for each n, there exists a solution Wn which vanishes on an open neighborhood Bn of (An Y such that Wn < UAn and UAn Wn O. Thus w'" < UKn where Kn = B�, is 0 it follows that UKn --> 0, which compact and Kn C An . Since C2/q,q, (Kn ) implies the assertion. 0 PROO F .
-
--t
-
-
-->
-
-->
-
-->
3.8. Let E, F be Borel subsets of an. (i) If E, F are q-closed then UE A UF = UEr, F ' (ii) If E, F are q-closed then LEMMA
(3.1 )
UE < UF UE = UF
-{=;=}} .:
q
[ E C F and C2/ q, q, (F \ E) > O J,
;. E :!.. F.
(iii) If {Fn} is a decreasing sequ.ence of q- closed sets then lim UFn = UF where F = nFn .
(3.2)
(iv) Let A C an be a q-open set and let U E U(0.) . Suppose that u vanishes q-locally in A, i. e., for every point (J E A there exists a q-open set ACT such that (J E Au
c
A,
Then u vanishes on A . In parlicu/ar each solution u E U(0.) vanishes on an \
SUPP�!1 '11. . (i) UE A UF is the largest solution under inf(UE ' Up) and therefore, by Definition 3.6, it is the largest solution which vanishes outside E n F. (ii) Obviously PROOF.
(3 .3)
E :!.. F
;· UE = Up ,
In addition, (3 .4) Indeed, if K is a compact subset of F \ E of positive capacity, then UK > 0 and UK < UF but UK i UE · Therefore Up = UE implies F :!.. E. (iii) If V : = lim UPn then UF < V. If Up < V then C2/q ,q' (suPPlm V \ F) > O. But
359
PRECISE BOUNDARY TRACE
sUPPbn V C Fn so that sUPPbn V c F and consequently V < Up .
(iv) First asl)ume that A is a countable union of q-open sets {An } such that u � 0 An for each n. Then u vanishes on U� Ai for each i. Therefore we may assume that the sequence { An} is increasing. Put Fn = Of! \ An . Then u < Up" and, by (iii) , UFn 1 UF where F = an \ A. Thus u ::; UF , i.e., which is equivalent to u :;t O. We turn to the general case. It is known that the (0, p)-fine topology possesses the quasi-Lindelof property (1)00 [1, Sec. 6.5.1 1) ) . Therefore A is covered, up to a set of capacity zero, by a countable subcover of { AQ' : a E A}. Therefore the previous argument implies that u � 0 0 A
THEOREM 3.9. (a) Let E be a q-closed set. Then, UE
(3.5)
=
inf{ UD : E c D c an, D open}
= sup{UK : K c E, K compact} .
(b) If E, F are two Borel subsets of an then (3.6)
(c) Let E, Fn, n = 1 , 2, . . be Borel subsets of Of! and let u be a positive solution of (1.1). If either C2/ q,q' (Et::.Fn ) --+ 0 or Fn 1 E then .
(3.7)
UPn
--+
UE .
PROOF. (a) Let {Qj} be a I>equence of open sets, decreasing to a set E' ,t E, which satisfies (2.20) . Then Qj ! E' and, by Lemma 3.8 (iii) UOj 1 UE . This implies the first equality in (3.5) . The second equality follows directly from Definition 3.4 (see also Lemma 3.5). -
(b) Let D, D' be open sets such that E n F e D and E \ F e D' and let K be a compact subset of E. Then �
�
-
(3.8)
To verify this inequality, let v be a positive solution such that sUPPbn v C K and let {fJn} be a sequence decreasing to zero such that the following limits exist: D . W = 1I· m vfjD , W, = hm vfj ' . n_oo n
'I,.-CO
n
(See Notation 1.1 for the definition of vf {) Then Since, by [16) UK
v ::;
=
W+
w' ::; UD + UD' .
VK , this inequality implies (3.8). Further (3.8) and (3.5) imply UE
Upn E EEl UE\F .
This implies (3.6). (c) The previous statement implies, UB
0 E then, by Lemma
.....
w.
(ii) If A is a q-open subset of an,
-
(4.10)
VQ q-open ; Q
(iii) Finally,
q
C A.
[u] A = 0
(4.11)
U Rj 0 :
PROOF. Case 1 : E is closed. u E C(O U A) and u = 0 on A. E then
Since u vanishes in A ;= an \ E, it follows that If, in addition, D e an is an open neighborhood of
A
).
udS � 0 { J�jl ( D')
so that
(4.12) Since
it follows that (4.1 3 )
U=
1I·
m ufJ
1)
'
If we assume only that D is q-open and E C. D then, for every E > 0, there exists
an open set 0< such that D It follows that
C
0 . Assertion (4.10) in the
opposite direction is a consequence of Lemma 2.7 and Lemma 3.8 (iv). Case 2. We consider the general case when E is q-closed. Let {En} be a stratifi cation of E so that C2/q,q' (E \ En) -+ O. If D i� q-open and E !!:. D then, by the first parL of the pToof, (4.14) By (4.5) (4.1 5 ) Let {Ih} be a sequence decre�ing to zero such that the following limits exist 'UJ
'UJn := l� ([U]E\EJK ,
: = l� ugk ,
Then, by (4. 14) and (4.1 5),
[ulE" Further, by (3.7) ,
<w
. The assertion in the opposite direction is proved as in Ca�e l . This completes the proof of (i) and (ii). Finally we prove (iii). First assume that U � O. If F ii:i a q-closed set such that A q - q q F c A then there exists a q-opcn set Q such that F c Q c A. Therefore, applying (4.9) to v : = [ulF and using (4. 10) we obtaiu k
= lim v� < limu� = O. In view of Definition 4.3 this implies that [U]A = O. v
3fl4
MOSHE MARCUS AND LAURENT VERON -
q
Secondly assume that [u] = O. Then [u]Q = 0 whenever Q C A. If Q is a - q q-open set such that Q c A then [u]Q = 0 and hence u � O. Applying once again Q Lerruna 2 . 7 and Lemma 3.8 (iv) we conclude that U � O. 0 A
A
DEFINITION 4.6. Let u, v be positive solutions of ( 1 . 1 ) in n and let A he a q-open subset of an. We say that u = v on A if u e v and v e u vanish on A (see Notation 3.1). This relation is denoted by u � V . A
THEOREM 4.7. Let u, v E U(n) and let A be a q-open subset of an . Then, u � v ·: ;. lim lu - vl � = 0,
(4.16)
A
-
{3--+0
q
for every q-open set Q s1J.ch that Q C A and
u ::f v
(4. 1 7)
·: :. [u]p = [vIp, q
for every q-closed set F such that F C A .
PROOF. By definition, U � v is equivalent to u e v � 0 and v e u � O. Hence , A
A
by Lemma 1.5 (specifically (4. 11» ,
[u 8 v]p
(4. 18)
for every q-closed set F
� A.
=
0,
A
[v e uJp = 0,
� A, Lemma 4.5 implies that ( ( v - n)+ ) � -> O.
Therefore, if Q
( (u - v)+) �
->
0,
(Recall that u6v is the smallest solution which dominates the subsolution (u - v>+ -l This implies (4.16) in the direction > ; the oppm;it.e direction is a consequence of Lemma 3.7. \Ve turn to the proof of (4. 1 7) . For any two positive solutions u, v we have (4.19)
u + (v - u) + < v + (u - v)+ < v + n e v.
If F is a q-ciosed set and Q a q-open set such that F ( 4.20)
[u]./>' < [vJQ + [u e v]Q .
� Q then,
To verify this inequality we observe that, by (4.19),
[U] F ::; [v] Q + [v]Q ' + [u e v]Q + [u e v] Qc.
The subsolution w := ( [U] F - ( [v]Q + [u e v] Q » + is dominated by the supersolution [v]Qc + [u e vlQc which vanishes on Q . Therefore w vanishes on Q. Since the boundary support of [wi t is contained in F it follows that [wi t 0 so that w O. q - q If U ::f v and F C Q c Q c A then (4.20) and (4.18) imply, [U] F < [v]Q .
Choosing a decreasing sequence of q-open sets {Qn} such that nQn :!., F we obtain [U]F < lim[v]Qn = [V]F ' Similarly, [V] F < [U]F and hence equality. q Next assume that [VIF = [U] F for every q-closed set F C A. If Q is a q-open - q set such that Q c A we have, u 6 v < ( [ulo $ [u] oc ) e [vlQ < [ulQc,
PRECISE BOUNDARY TRACE
365
because [u] Q [v]Q . This implies that u e v vanishes on Q. Since this holds for every Q all above it follows that u e v vanishes on A. Similarly v e u vanishes on D A. =
COROLLARY 4.8. If A is a q-open subset of an, the relation :::; is an equivalence A
relation in U (D.) .
PROOF. This is an immediate consequence of (4. 16).
D
5. The precise boundary trace 5.1. The regular boundary set. We define the regular boundary set of a positive solution of (1.1) and present some conditions for the regularity of a q-open
set.
DEFINITION 5 . 1 . Let u be a positive solution of (1.1). a: Let D e aD. be a q-open set such that C2/q , q, (D) > O . D is pre-regular with respect to u if
(5. 1)
10 [u] '}pdx < 00
VF
t D,
F q-closed.
b: An arbitrary Borel set E is n;gular' if there exists a pre-regular set D such q that E C D. c: A set D e aD. is CJ-regular if it is the union of a countable family of pre-regular sets. d: The union of all q-open regular sets is called the regular' boundary set of 'U , and is denoted by R(u) . The set 5(u) = aD. \ R(u) is called the singular boundary set of u . A point P E R(u) is called a regular boundary point of 'll. ; a point P E 5 (u) is called a singular boundary point of u.
Remark. The property of regularity of a set is preserved under the equivalence relation :t. However note that a point is regular if and only if it has a q-open regular neighborhood. LEMMA 5.2. If D is a q-open pre-regular set then every pO'int � point. Furthermore there exists a q-open regular set Q such that
(5.2)
-
E
D is a TI�gular
e E Q c Q e D.
If F is a regular q-closed set then there exists a regular q-open set Q such that q F e Q, PROOF, By Lemma 2,7, for every e E D , there exists a q-open set Q such that (5.2) holds, Therefore Q is a regular set and e is a regular point . The last assertion is a consequence of Lemma 2.4. D DEFINITION 5.3. Let u be a positive solution of (1.1) and let {Qn} be an increasing sequence of regular q-open sets. If On t. Qn+ l we say that {Qn} is a regular sequence relative to u. If Q is a q-opcn set, {Qn } is a regular sequence relative to u, Qn C Q and Qu := u� 1 Qn � Q we refer to Qo as a proper representation of Q and to {Qn } as a r'egular decomposition of Q, relative to u .
366
MOSHE MARCUS AND LAURENT VERON
LEMMA 5.4. Let u E U(n) . A q-open set Q C an is a-regular if and only if it has a proper representation relative to u. In particular every pre-regular set has a proper representation. PROOF. The 'if' direction follows immediately from the definition. Now sup pose that Q is a-regular. Then Q = Ul" En where En is q-open and pre-regular, n = 1 , 2, ' " . By Lemma 2.6, each set En can be represented (up to a set of capacity q q zero) as a cOlmtahle union of q-open sets {An,j}j I such that An,j e An,HI C En . We may assume that An,j e En; otherwise we replace it by An,j n En . Put -
If k + j = n then Ak,j
q
C
Qn = Uk +j=n Ak,j '
Ak,j+l C Qn+ l . Hence q - q Qn e Qn+1, Qo : = UQn � Q.
THEOREM 5.5. Let D be a q-open set such that C2/q ,q ' (D) (i) Suppose that (5.3)
liminf f3 O , .....
I (ufl)q(p " Ii �,
.
-
o
> 0,
f3) dx < 00.
Then D is pre-regular, (ii) Suppose that D is a pre-regular set. Then there exists a Borel measure J1 on D q such that, for every q-closed set E e D, (5.4) PROOF. (i) Let {f3n} be a sequence decreasing to zero such that (5.5) By extracting a subsequence if necessary we may assume that {ufn } converges locally uniformly in n to a solution w . Then, by Lemma 4.5, if E is q-closed and q E e D, (5.6) By (5.5) and Fatou's lemma,
Hence, by (5.6), (5.7) Thus D is pre-regular. (ii) By Lemma 5.4, D possesses a regular decomposition {Dj}. Put Wj = [UJ D, .
367
PRECISE BOUNDARY TRACE
Then {Wj} is increasing and its limit is a solution Wo < w with w as defined in (5.6). Thus Wo is a moderate solution. If E � D is a q-closed set then, by (4.6) ,
By Lemma 2.5, for every k E N there exists an open set Ok and a natural number jk such that C2jq ,q, (Ok) < 11k and E \ Ok � Djk • By Theorem 3.9 Since [ulo.
-+
0 we conclude that
q [wol E = rul E VE e D : E q-closed.
(5.8)
If Wo is moderate then tr [wolE = {lXEtr wo, which implies (5.4) . We turn to the case where Wo is not moderate. The solution and we denote {lj = tr wj
w)
is moderate
{I = lim J.Lj .
,
By (4.6), Wj = [wj+klDj ' Therefore J.Lj = J.Lj+ kX oj = J.LXD{ Therefore if E is q closed and E � Dj for some j, (5.4) holds with /-t as defined above. If E is q-closed
q and E C D then E � E' := U(E n Dj ) = U(E n Dj). Put Ej := E n Dj . It follows q
-
that
tr rul E; =
J.LXEj
i /-tX E"
Since D is pre-regular, rulE is moderate. Put Ej = E \ Dj and observe that nj'" Ej is a set of capacity zero so that (by Lemma 3.8) UE� ! 0 and hence lim[ul Ej ! O. Since
rul E < rul E, + rul E' and rul E' 1 0 1
we conclude that
1
On the other hand tr rul E > tr rul E;
-+
WX E'
= /-tXE'
This implies (5.4) .
0
COROLLARY 5.6. Let D be a q-open set such that C2/q , q' (D) > O. Suppose that, q for every q-open set Q such that Q e D, -
(5.9)
Then D is pre-regular. PROOF. This is an immediate consequence of Lemma 2.4 and Theorem 5.5.
0
368
MOSHE MARCUS AND LAURENT VI
O.
Eo (Q) (ii) If � E R(u), there exists a q-open regular set D s'uch that � E D , Further there exists a q-open set Q such that � E Q c Q e D. Consequently
(5. 1 1 )
sup udS < 00, 00 Q so that (5.3) holds. In view of this fact, Theorem 5.5 and the arguments in its proof imply assertion (ii). 0
r
5.3. q-perfect measures.
DEFINITION 5.8. Let J..t be a positive Borel measure, not necessarily bounded,
on an. (i) We say that J..t is essentially absolutely continuous relative to C2/q,q' if the fol lowing condition holds: If Q is a q-open set and A is a Borel set such that C2/q,q' (A) = a then
J..t ( Q \ A) = j..t(Q).
This relation will be denoted by J..t
j-< C2/q,q"
369
PRECISE BOUNDARY TRAUE
(ii) /-I is regular relative to q-topology if, for every Borel set E C an, /-I(E) = inf{tL C D) : E c D c an , D q-open} (5.13) = sup{Ji(K) : K c E, K compact}.
/-I is outer regular relative to q-topology if the first equality in (5. 13) holds. (iii) A positive Borel measure is called q-perfect if it is essentially absolutely con tinuous relative to C2/q,q' and outer regular relative to q-topology, The space of q-perfect Borel measures is denoted by Mq(on). LEMMA 5.9. If /-I C2/q,q' (A) = 0 then
(5.14)
/-I C A)
=
E
Mq (an) and A c an is a non-empty Borel set such that
00
o
if /-I ( Q \ A) = 00 VQ q-open neighborhood of A, otherwise .
If /-10 is an essentially absolutely continuous positive Borel measure on 812 and Q is a q-open set such that J.Lo(Q) < 00 then /-Io l Q is absolutely continuous with respect to C2/q,q' in the strong sense, i. e., if {An} 'is a sequence of Borel subsets of an, C2/q,q ' (An ) 0 :. /-Io (Q n An) O. Let /-10 be an essentially absolutely continuous positive Borel measure on 812. Put -+
-+
/lo(E) := inf{/-Io(D) : E c D c 812, D q-open},
(5.1 5)
for every Borel set E
C 812. Then
Jio < Ji, /-Io (Q) = Ji(Q) VQ q-open (b) /-tI Q = /-LoI Q for every q-open set Q such that /-Lo(Q) < 00
(a) (5 , 16)
.
Finally /-L is q-perfect; thus Ji is the smallest measure in MIq which dominates Ito ·
PROOF. The first assertion follows immediately from the definition of Mq. We
turn to the second assertion. If tLo is an essentially absolutely continuous positive Borel measure on an and Q is a q-open set such that Jio (Q) < 00 then /l,o 'X.Q is a bounded Borel measure which vanishes on sets of C2/q,q,-capacity zero. If {An} is a sequence of Borel sets such that C2/q,q' (An) -+ 0 and /-tn : = /-Io 'X.QnAn then Hence u!'n -+ 0 locally uniformly and /-In --' 0 weakly with respect to C(8n) . /-to (Q n An) -+ O. Thus /-to is absolutely continuous in the strong sense relative to C2/q, q' . Assertion (5.16) (a) follows from (5.15). It is also clear that /-t, as defined by (5. 15), is a measure. Now if Q is a q-open set such that /-Io(Q) < 00 then /-I(Q) < 00 and both /-to l Q and /-II are regular relative to the induced Euclidean topology on Q an. Since they agree on open sets, the regularity implies (5. 16) (b). If A is a Borel set such that C2/q,q' (A) = 0 and Q is a q-open set then Q \ A is q-open and consequently /-I(Q)
J.Lo(Q) = ,lO (Q \ A)
/-I(Q \ A). Thus /-I is eSHentially absolutely continuous. It is obvious by its definition that /-I is 0 outer regular with respect to C2/q,q" Thus /-I E Mq( an). =
=
MOSHE MARCUS AND LAURENT VERON
370
5.4. The boundary trace on the regular set. First we describe some prop erties of moderate solutions. In this connection it is convenient to introduce a related term: a solution is strictly moderate if
lu i < v ,
(5.17)
v harmonic, n
vqp dx
80 be the mapping given by II(x) = a(x) (see Notation 1.1) and put II!3 := fI E".
1 : If cP is a function defined on 80 put cP* := ¢ a II. This function is called the normal lifting of ¢ to 0!3o ' Similarly, if ¢ is defined on a set Q c 80, ¢* is the normal lifting of ¢ to O!3o (Q) . 2: If 'P is a function defined on E!3 we define the normal projection of 'P onto
ao by
'P�(O = 'P (II13 1 (0),
Vf ..
E
ao,
If v is a function defined on O!3o then v� denotes the normal projection of v(/3, ') onto 80, for /3 E (0, /30 )' PROPOSITION 5.10. Let u be a moderate solution of (1.1), not necessarily pos
itive. Then:
(i) u E L1 (0) n Lq (O; p) and u possesses a boundary trace tr n given by a bounded Borel measure /1 which is attained in the sense of weak convergence of measures: (5.18)
r n�¢dS = f3-u .Jan
ulji*dS = lim
lim
lji d/1,
13-0 Ell an for every ¢ E C(80) . (ii) A bounded Borel measure /1 is the boundary trace of a solution of (1.1) if and only if it is absolutely continuous relative to C2/ Q, Q" When this is the case, there exists a sequence {/1n} C W-2/q q (80) such that /1n -> JL in total variation norm. If /1 is positive, the sequence can be chosen to be increasing. Note that these facts imply that /1 is a trace if and only if 1 /1 1 is a trace. Q 2 (iii) u is strictly moderate if and only if Itr ul E W- / ,q(80) . In this case the boundary trace is also attained in the sense of weak convergence in W-2/q,q (80) of {u� : f3 E (0, f3o) } as /3 O. In particular (5. 18) holds for every Iji E W 2/ q q (80) U C(80) . (iv) If JL : = t.r u and {JLn} is as in (ii) then u = lim ul'n ' In particular, if u > 0 then u is the limit of an increaSing sequence of strictly moderate solutions. (v) The measure /1 = tr u is regular relative to the q-topology. (vi) If u is positive (not necessarily strictly moderate), (5. 18) is valid for every ' , / q rp E ( W� Q n Loo ) (80) . ,
--->
,
'
Remark. Assertions (i)-{iv) are well known. For proofs see [15] which also contains further relevant citations. Proof of (v). If JL is a trace then 1-1+ and /1- are t.races of solutions of (1.1). Therefore it is enough to prove (v) in the case that /1 is a positive measure.
37 1
PRECISE BOUNDARY TRACE
Every bounded Borel measure on 00 is regular in the usual sense: p,(E) = inf{p,(0) : E c 0, 0 relatively open} =
sup{p,(K) : K c E, K compact}
for every Borel set E c oO. Since J.L(E) < inf{p,(D) : E c D c 00, D q-open} < inf{J.L(O) : E c O c oO, 0 relatively open} it follows that such a measure is also regular with respect to the q-topology. Proof of (vi). By (ii) there exists an increasing sequence of strictly moderate solutions {vn} such that Vn i u. If fJn : = tr Vn then
r vncP*dS = ��O JE� lim
cP dfJn,
an
for every ¢ E W2/q,q' (00). Since {P,n } increases and converges weakly to p, J.L in total variation. Hence it follows that P,n .......
an
an
¢ dp,n .......
=
tr u
¢ dp,
for every bounded ¢ E W2/q , q' (00) . If, in addition, 1> ;;: 0, we obtain lim inf u¢*dS > ¢ dp,. ��o Ep an On the other hand, since u� fJ in the sense of weak convergence of measures, it follows that .......
(5.19)
lim �mp
u�dS < p"
lim inf
i u�dS
;;:
J.L
E for any closed set E c oO, respectively, open set A c oO) . It is easily seen that, in our case, this extends to any q-c1osed set E (resp . q-open set A) . Therefore if A is q-open and then (5.20)
If ¢ E W2/q,q' (00) n LOO (an) and I e 1R is a bounded open interval then, by [1 , Prop. 6.1.2, Prop. 6.4. 10] , A : = ¢-1 ( 1) is quasi open. Without loss of generality we may assume that ¢ < 1 . Given k E N and m = 0, . . . , 2k - 1 choose a number am, k in the interval (m2- k , (rn + 1) 2- k ) such that P,n( r/> - I ( { am,d) = 0. Put k A71l,k = ¢ - l ( (am ,k , am+1,k]) m = 1 , . . . , 2 - 1 , AO,k = ¢- l ( (ao,k' al,k ] ) and 2k _ l !k = L m2- k XAm,k 71l=0 Then fk ....... ¢ uniformly and, by (5 .20), lim fk u�dS /3-0 an This implies assertion (vi).
=
an
fk dfJR'
o
372
MOSIIE MARCUS AND LAURENT VERON
THEOREM 5.1 1 . Let u E U(D) .
The regular set R(u) is a-regular and consequently it has a regular decomposi tion { Q } . (ii) Let
(i)
n
v1'. := sup{ [uJQ : Q q-open and regular } . (5.21) Then there e:cisf.� an increasing sequence of moderate solutions {wn} such that (5.22) SUPPbn 1J)n (Thus V1'. is a-moderate.)
� R(u),
Wn r V"-'
q (iii) Let F be a q-closed set such that F c R(u) . Then, for everll t > 0, there exists a q-open regular set Q€ such that C2/q,q' (F \ Q.) < e . If, in addition, [ull" is
moderate then F is Tegl1.1ar; consequently there exists a q-open regular set Q such that F � Q. (iv) With {Qn} as in (i), denote
(5.23) vn := [ulQn ' J1.n := tr vn , v := lim vn , J1.1'. := lim J1.n' Then, ( 5 24 ) Furthermore, for every q-open Teg'ltlar set Q, (5.25) J1.RX Q = tr [uJQ = tr [vnl Q · .
Finally, J.1.1'. is q-locally finite on R(u) and a-finite on Ro(u) := UQn . (v ) If {wn} is a sequence of moderate solutions satisfying conditions (5 . 2 2 ) then, ( 5 26 )
J1.n. = lim tr Wn (vi) The regularized measure J1.n. given by (5.27) J1.n.(E) : = inf{J1.,,(Q) : E c Q, Q q-open VE c aD, E Bore l} is q-perfcct. u � v". (vii) .
(viii)
n(u)
q
Fm' every q-closed set F c R(u):
[ujp = [V,J F . (5.28) If, in addition, J.1.1'.(F) < 00 then (UJF is moderate and tr ('lLl F = J1.RXF' (5.29) (ix) If F is a q-closed set then J1.R(F) < 00 .: :. [U]F is moderate .: :. F is regular. (5 . 30)
PROOF. (i) By [1 , Sec. 6.5.11] the (a:,p)-finc topology possesses the quasi Lindelof property. This implies that R(u) is a-regular. By Lemma 5.4 R(u) has a regular decomposition { Qn}· Recall that Qn C Qn+l and C2/q ,q' (R(u)\Ru(u)) = O. (ii) This assertion is an immediate consequence of (5.21) and Proposition 3.3. (iii) By definition, every point in R(u) possesses a q-open regular neighborhood. Therefore, the existence of a set Q., as in the first part of this assertion, is an
373
PRIWISbJ BOUNDARY TRACE
immediate consequence of Lemma 2.5. Let O. be an open set containing F \ Q, such that C2/q ,q' (0,) < 2f. Put FE := F \ 0, . Then F, is a q-closed set, F, C F,
q
C2/q,q' (F \ F. ) < 2 € and FE C Q,. q Assertion 1. Let E be a q-closed set, D a q-open regular set and E c D . Then there exists a decreasing sequenee of q-open sets {Gn};:O 1 such that (5.31) and [ul en
(5.32)
->
rulE in U (n, p) .
By Lemma 2.6 and Theorem 4.4, there exists a decreasing sequence of q-open sets {Gn} satisfying (5.31) and, in addition, such that [ul Gn 1 [ul E locally uniformly in n. Since [ulGn :::; [ulD and the latter is a moderate solution we obtain (5.32). Put
En
:=
U�= l Fl /m' Dn := U�= l Q l/m'
Then En is q-closed, Dn is q-open and regular and En /!:. Dn . Therefore, by Assertion 1, it is possible to choose a sequence of q-open regular sets { Vn } such that
(5.3 3) By Theorem 4.4,
[ulF < [Ul En + [ul E'\En and [UJF\En 1 O. Therefore [ulEn i [UJ F. If, in addition, [UJF is moderate then and consequently, by (5.33),
[U,J En i [1tlF in U (0, p)
[ulvn -> [ulF in Lq (n, p). Let {Vn k } be a subsequence such that
(5.3 4) q
Vn n F and that C2Iq,q, (F \ En) 1 0. Therefore C2/q ,q, (F \ v,..) -> O . Consequently F /!:. w := n);" 1 Vnk and, in view of (5.34), [uJw is moderate. q Obviously this implies that W is pre-regular (any q-closed set E C W has the property that [ltlE is moderate) and F is regular. Finally, by Lemma 2.4, every Recall that En
C
q-closed regular set is contained in a q-open regular set. (iv) Let Q be a q-open regular set and put IJ.Q = tr [uJQ. If F is a q-closed set such that F
(5.35)
q
C
Q then, by Theorem 5.5,
[uJE' = IJ.QXr
In particular the compatibility condition holds: if Q, Q' are q-open regular sets then
(5 .3 6 )
- IJ.QnQ' = IJ.QXQnQ - - , = IJ.Q'XQnQ
,'
374
MOSHE MARCUS AND LAURENT VERON
With the notation of (5.23), [vn+kJQk = Vk and hence ftn+kXQ· k = ftk for every k E N. 0 it Let F be an arbitrary q-closed subset of R(u). Since C2/q,q, (F \ Qu) follows that ->
(5.37)
In addition, [VJ F > lim [vn l F
=
[UJ F and v < u lead to,
(5.88)
[UJ F
=
[vl F .
If Q is a q-open regular set, [u)Q = lim[vnlQ < lim v" = : v and so vn < v. On the other hand it is obvious that v < vn. Thus (5.24) holds. By (5.35) and (5.37) , if F is a q-closed subset of R(1L) and [UJ F is moderate, (5 .39)
which implies (5.25). This also shows that ftnXF is independent of the choice of the sequence {ftn} used in its definition. This remains valid for any q-closed set F t. R(u) because C2/q' v so that R(lI) c R(v). On the other hand, since T is q-Iocally finite on R(v) = 80 \ F, it follows that S(lI) c F. Thus R(v) c R(u) and we conclude that R(v) = R(u) and F = stu). This also implies that v = ureg Finally stu) = S (v) U S(UF) = FT U bq (F), so that F satisfies (5.61 ). The fact that, for v E Mq(80), the couple (7, F) defined by (5.71) is the only one in (t(80) satisfying v = '['(T, F) follows immediately from the definition of these spaces. D Statements A-D imply (i)-(iv). as
Remark. If v E Mq(80) then G and v alternative representation: v := sup { u"xQ :
(5.72)
F" : =
(5.73)
defined in (5.63) have the following
G = U Q = U E,
Q E F,, },
Tv
{Q : Q q-open,
£v
v( Q) < oo}.
To verify this remark we first observe that Lemma 2.6 implies that if A is a q-open set then there exists an increasing sequence of q-quasi closed sets {En} such that A = Uj'" En . In fact, in the notation of (2.22) , we may choose En = Fn \ L where L = A' \ A is a set of capacity zero. Therefore
U D C U Q C U E =: H. -
On the other hand, -if E E £" then p'R( u) ( E) = Pn(u) (E) = v (E ) < 00 and, by Theorem 5.11 (ix), E is regular, i.e., there exists a q-open regular set Q such that q E C Q. Thus H = UD D. If D is a q-open regular set then D = Uj'" En, where {En} is an increasing sequence of q-quasi closed sets. Consequently, v
U"X = lim u"x D
Therefore
En
.
sup{ U"XQ Q E V,, } < sup{ UVXQ Q E Fv} < sup{ :
:
UVXE :
E E ev}.
PRBCISE BOUNDARY TRACE
383
On the other hand, if E E E" then there exists a q-open regular set Q such that E t Q. Consequently we have equality. Note that, in view of this remark, Theorem 1.3 is an immediate consequence of Theorem 5.16.
Acknowledgment. Both authors were partially sponsored by an EC grant through the RTN Program Front-Singularities, HPRN-CT-2002-00274 and by the French Israeli cooperation program through grant No. 3-1:352. The first author (MM) also wishes to acknowledge the support of the Israeli Science Foundation through grant No . 145-05.
References [IJ
Adams
D . R. and Hedberg L. r . ,
Fu nction spac,,_, and
potential
theory, Grundlehren Math.
Wissen. 314, Springer [2J
[3J [4J [5J [6J [7J
[8J
[9J
(1996). Beni/an Ph. and Brezis H., Nonl inear preoblems related to the Thomas-Fermi equation, J. Evolution Eq. 3 , 673-770 (2003). Dynkin E. B. Diffusions, Superdiffusions and Partial Differential Equations, American Math. Soc., Providence, Rhode Island, Colloquium Publications 50, 2002. Dynkin E. B . Superdiffu8ions and Positive Solutions of Nonlmear Partial Differential EqtLa tions, American Math. Soc. , Providence, Rhode Island, Colloquium Publications 34, 2004. Dynkin E . B. and Kuznetsov S . E. Superdiffusions and removable singularities for quasilinear partial differential equations, Comm. Pure App!. Math. 49, 125-176 (1996). 1/.'" dominated by harmonic junctions, Dynkin E. B. and Kuznetsov S. E. Solutions of Lu J . Analyse Math. 68, 15-37 (1996) . Dynkin E. B. and Kuznetsov S. E. Fine topology and fine trace on the boundary as.,ociated with a dass of quasilinear differential equations, Comm. Pure App\. Math. 51, 897-936 ( 1998). Kato T . , Shrodinger operators with Ringular potentials, Israel J. Math. 13, 135-148 (1972). =
Kuznetsov S.E. a-moderate solutions of Lu
Sc. Serie I [lOJ
Labutin D.
326,
(2003) . [l1J [12J [13J [14J [15] [16J [17J
1189-1 194 ( 1998).
=
u'" and fine trace on the boundary, C.R. Acad.
A., Wiener regularity for large solutions of nonlinear equations, Archiv
Legall J. F . , The Brownian snake and solutions of tl.u Fields 102,
393-432 (1995).
=
fiir Math.
u2 in a domain, Probab. Th . ReI.
J. F . , Spatial branching processes, random snakes and partial differ-entiat equations, Birkh1iuser, Basel/Boston/Berlin, 1999. Marcus M. and Veron L., The boundary trace of positive solutions of semilinear elliptic eAJuations: t he subcritical case, Arch. rat. Mech. Anal. 144, 201-231 ( 1998). Marcus M. and Veron L . , The boundary trace of positive solutions of semilinear elliptic equations: the supercritical case, J. Math. Pures Appl. 11, 481-524 (1998). Marcus M . and Veron L . , Removable singularities and boundary trace, J. Math. Pures Appl. 80, 879-900 (2000). Marcus M . and Veron L., Capacitary estimates of positive solutions of semilinear elliptic eAJuations with absorption, J. European Math. Soc. 6, 483-527 (2004). (2001). M selati B . , Classification and probabilistic representation of the positive solutions of a semi linear elliptic equation, Mem. Am. Math. Soc. 168 (2004). Legall
DEPARTMENT OF
MATHEMATICS,
TEcm-aON, HAIFA
32000,
ISRAEL
E-mail address: marcusmlDmath. technion . ac . i1 LABORATOIRE DE MATHEMATIQUES,
FACULTE
TOURS, FRANCE
E-mail address: veronllDlmpt . univ-tours . fr
UES
SCIBNCES, PARC DE GRANDMONT,
37200
Contemporary Mathematics Volume 446, 2007
Blow-up In Nonlinear Heat Equations with Supercritical •
Power Nonlinearity Hiroshi Matano
Dedicated to Prof.
Haim
Brezis on the occasion oJ his sixtieth birthday
ABSTRAGr. In this art icle we study blow-up of solutions of the nonlinear heat equation Ut = �u + luI P - 1u. We focus on the phenomena that are character istic to the supercritical range p > Ps : = N > 3, where N is the space dimension. We assume radial symmetry of solutions. What we present here is for the most part an overview of the author's recent joint work with Frank Merle [301 on supercritical blow-up and the author's recent paper [281 on the new application of the braid group theory to blow-up problems. Among other things we discuss (a) classification of blow-up profiles; (b) nonexistence of a 6-fundion type singularity; (c) continuation beyond the blow-up time; (d) determining type II blow-up rates via the braid group theory.
�+�,
1. Introduction
Blow-up in nonlinear heat equations has been a subject of extensive mathemat ical studies. The motivation comes from various fields of science such as plasma physics, combustion theory and population dynamics, a� well as in connection with some geometrical problems. See, for example, [4) for some of the physical back grounds. In the present article we consider the problem
(Ll)
Ut
=
(x E D, t > 0)
�u + lulp- 1 u
u(x, O)
=
uo(x)
(x E D),
where either 0 = RN or D = BR : = {x E RN I lx l < R}. In the latter case, we impose the Dirichlet boundary condition ( 1 . 2)
u(x, t)
=
0
(x E aD, t > 0).
Primary 35K55, 35B40, 74H35; Secondary 20F36. Key words and phrases. blow-up, nonlinear heat equation, singularity, braid group. This work was partly initiated during the aulhoc'. visit to Cergy-Pontoise UniverS ity and to U niversity of Paris-sud. 2000 Mathematics Subject Classification.
385
386
HIROSHI MATANO
The exponent p is supercritical in the Sobolev sense, that is,
P>P
(1 . 3)
E
..
N+2 ;= N 2 '
N
_
?.
3
and we assume Uu LOO (!1) n C(!1) . Throughout this article we mainly deal with radially symmetric solutions. In other words, u is expressed in the form
u(x, t) = U ( l x l t), ,
where the function U (r, t ) satisfies the equation
( 1.4 )
Ut
=
U'"7" +
N- 1
Ur + 1 U 11'- 1 U.
r Problem ( 1 . 1 ) posseset; a classical solution at least locally in time. We say that a solution u blows up in finite time if, for some T > 0, lim sup II u (·, t ) IIu'" = + 00. t�T Here T is called the blow-up time. The simplest example of a blow-up solution is that of the ordinary differential equation du/dt = l u l ,,-1 U, in which case u(t) = ±K ( T-t) -1/(p- l ) with K ; = (p _ 1)-1 /(P- l) . Another simple example is a self-similar blow-up solution, which is given in the form
( x-a ) u(x, t) = (T _ t) - P-l tjJ 1
(1.5 )
v'T - t '
where a is any point in RN and tjJ(y) is a bounded solution of the equation 1 for y E RN l:l.tjJ - � y . 'VtjJ tjJ + 1 1jJlp- l tjJ = 0 ( 1.6)
2
In both cases we have
p-1
)
II u (- , t) II L� = O (T - t) - P:, .
(
More generally, blow-up in ( 1 . 1) is categorized into the following two types; Type I
:
Type II :
lim sup (T - t ) p ' II u (- , t ) IIL Ps is the existence of "incomplete blow-up" . In late 1980's, Baras and Cohen [3] showed that every blow-up is "complete" if 1 < p < Ps ' which roughly meaIls that there is no way to continue the solution beyond the blow-up time. On the other hand, in the supercritical range P > P S ' some solutions can be continued beyond the blow-up time in a certain weak sense. In a joint work with M. Fila and P. Polacik [9] , the author has shown that such weak solutions become classical immediately after the blow-up time. We will discuss further properties of complete and incomplete blow-up. The organization of the article is as follows: In Section 2, we present funda mental estimates for supercritical blow-up. The highlight is the "no-needle lemma" , which shows that a 8-funetion type singularity never occurs if p > Ps ' This is in marked contrast with the critical case p Pa , for which such singularities are ob served in the formal analysis of [11] . This lemma plays a key role in deriving various properties of supercritical blow-up. In Section 3, we classify blow-up profiles and characterize type I and type II blow-ups in Lerms of the profiles. In Section 4, we discuss continuation of solutions beyond the blow-up time. Among other things we discuss the relation between the rat,e of blow-up and the rate of regularization after the blow-up time. In Section 5, we present an intriguing application of the braid group theory to blow-up probems. The goal is to determine all type II blow-up rates by analyzing the topological properties of certain braids. The braid group method for scalar par abolic equations was introduced by Ghrist and Vandervorst [17] as generalization of the zero-number argument. We further develop their theory to make it applicable to our blow-up problem. Sections 2 to 4 are based on the author's joint work with F. Merle [30], while Section 5 is based on the author's paper [28] . Note that Mizoguchi [36] has also obtained a result somewhat similar to [28] via a braid-group method, by partly borrowing the idea of [28] and partly using a different approach. =
2. Fundamental estimates Early studies of blow-up, including the pioneering work of Kaplan [25] and Fujita [13] , were mainly concerned with the question as to what conditions cause blow-up and what conditions guarantee global existence. In the mid 1 980's, Weissler [40] showed an example of single-point blow-up, which lead people to realize that blow-up iii generally a highly localized phenomenon. This triggered a flow of re search on the local and global structure of singularities that appear at the time of blow-up; see for instance [12, 6, 19, 20, 14, 39] for some of the early results. We define the blow-up set of a solution by
B ( uo )
:=
{x E n I :lxn
--4
x , tn / T such that I u(xn , tn ) 1
--4
oo},
where Uo denotes the initial data of solution u and T is the blow-up time. Each element of B(uo ) is called a blow-up point of u.
HIROSHI MATANO
388
2 . 1 . Rescaled equation. In studying the local structure of singularities in ( 1 . 1) , it is useful to introduce the following rescaled coordinates as in [18J . Given an arbitrary point (a, T) E D x (0, (0 ) , we set ;=
(2 . 1 )
=
where
y=
(T - l) v
1
( l)
l U X,
(T - t) v'l u(a + v'T - t v, t)
x-a
s = - log(T - t) .
v'T -t ' REMARK 2.1. The point (a, T) is the center of rescaling. Usually T is chosen
to be the blow-up time of u, but sometimes it is set slightly off the actual blow-up time in order to obtain extra \l�eful e�tirnates Ilear the blow-up time. Examples of such techniques can be found in Subsections 2.2, 2.3 and 4.2. The function Wa ,1' (y, s) satisfies the rescaled equation 1 ow 1 w + IwIP-1w, = �w - - y . \lw (2 . 2) 2 as p- 1
and allY solution of equation (1 .6) is a stationary solution of (2.2). In other words, a solution u(x, t) is self-similar if Wa,T (Y, s) is independent of s. Equation (2.2) is defined on the domain if 0 = { I x l < R } , {y E RN I l y + eS/2al < R es/2 } 0" ;= ,
,
We associate with (2.2) the following energy functional;
1 1 1 E ( w) = ( 2.3 ) -2 1 \lw I2 + 2 (p - l ) I w l2 - p + l IwIP+ ! p(y) dy , JRN where p (y) = (47r) - N/ 2 e- l y I2/4 . Here and in what follows it will be understood that Wa,T is defined for all y E RN even if 0 =1= RN , by setting Wa,T = 0 outside Oa. s . For any solution w (y, s) of (2.2) , we have
r
d
ds E(w(-, s»)
(2.4)
00 . It then follows from standard parabolic estimates and the existence of the Lyapunov functional E that w( y, s ) approaches a set of stationary solutions as s -> 00 . This roughly implies that any type I blow-up solution is asymptotically self-similar. In the case where 1 < p < Ps ' the boundedness of w follows from the fact that W is defined for all large 8 , as shown in [19, 21J; hence every blow-up is of Type I. Here radial symmetry is not required. To see this, we first observe that I d 2 p-l d -Iw l P+ 1 p dy w p y = - 2E ( w) + 2 ds RN P + 1 RN (2.5) p+ l 2 p- l > for s > So , - - 2 E ( w) +
£
.
p+l .
389
BLOW-UP IN NONLINEAR HEAT EQUATIONS
where So = - log T and W = Wa ,T . Since W is defined for all s > So , the quantity J w2 p dy docs not blow up in finite time. From this and the fact that E(we, s)) is nonincreasing in s, we obtain -2E(w) + ��� Cf w2 P fly) p�' < 0, hence (2. 6)
for s >
(2. 7)
80 ,
for
where G is some constant and Eo(a) p-1
:=
8
>
80,
E(Wa , T ( - . so) ) . From (2.5) one can also get
wwsp dy + 2E( m) �
1
2 P dy w J ) ( J w; p dy) + 2En ( a) ( < (G Eo"i1 (a)) ( � E (w ) ) 1 + 2Eo (a) ,
1 without the restriction of subcriticality nor radial symmetry, thus extending the earlier result in [20, Theorem 2.1] (for 1 < P < Ps and for small 0) with a much simpler proof. See [30, Section 2] for details. 2.2. Loo bounds for the supercritical range. If p > p" , an Loo bound on W does not follow directly from (2.7) and (2.9). However, since we are dealing with radially symmetric solutions, the equation is essentially one-dimensional in the region away from the origin, which means that any power p is subcritical (or even satisfies 1 < p < (N + 2 + 27"- 1 )/(N 2 + 2r- 1 ) with r 2, hence (2.9) follows directly without bootstrap argument). This yields a bound of the form for lyl > 1, 8 � 80 + 15 (2 . 13) I WO,T (Y, 8 ) 1 < CEr' for any small constant 15 > O. Now we apply the same argument to WO,T, , where Tl is a parameter ranging in [15 T, T], and T is the blow-up time of Then we obtain IWO,T, (y, 8) 1 < CEr' for Iyl > 1, 8 > log Tl + 15, ( 2.14 ) where Eo : = sup sup E (Wa,T, (., log T1 » ) . T, E [O T, Tl aER" Since Wa,T, (y , - log Tl) = A6 ' Wa,T( ,jAQy, - log T ) with Ao = TIT1 , it is easily seen that Eo < provided that both uo and V'uo are bounded, Observe also that 1 " T t - =-A( 2.15) WO.T (y, 8) = A p ' WO,T, ( V A y, s + log A) , P
,
'
"
1
"
=
-
a,
,
u.
_
,
-
-
,
-
00
Tl
-
t'
Now we let t vary over [o T, T) and Tl over (t, T] , Then we see that (2.15) holds for any 8 � - log((l - (5)T), A E [1, ) In particular, for each y with 0 < Iyl < 1, we can choose A = 1/1 Y l2 and apply (2.1 4 ), to obtain for 0 < ly l < 1 , 8 > - log (l - (5)1' + 0, I WO,T(Y, 8) 1 < CT I y l - p where CT = o (EO'''' ) . Combining this and (2.13), we get (2. 16 ) IWO.T(Y, 8) 1 < CT ( l + I Y I - :' ) for l y l > O, 8 > - log(1 - 15)T + 15, This and standard parabolic estimates imply that the derivatives of w are bounded in the region Iyl � 1 . Tlms the same rescaling method as (2.15) yields j , / < Iyl (2 ,1 7) 1 + W ( ) (j = 1 , 2, 3). 1 , 8) CT (Y lV' i Q,T Next we derive a global 1/)0 bound from the above estimate. Recall that the singular stationary solution of (2.2) is given by 'P* (y) = �* ( Iyl), where with (C· ) p-l = 2 1 (,"1' - 2 - p -l 2 ). (2.18) (0 .
2
1
,
'P
:l!ll
I:lLOW-UP IN NONLINEAR HEAT EQUATIONS
-
Eo is small so that CT < c* , then (2. 16) implies that WO,T (y, s) stays below CT + /tcp' (y), where It := CT (C* ) - l < 1 . Then one can construct a ' supersolution with initial data CT + ItCP that becomes bounded immediately. This gives an LOO bound for W in the range S 2: - log( l - o)T + 0 + 0 ( � So + 30). Rewriting 30 as 0, we obtain the following proposition: If the initial energy
( [30] ) . Let Ps
< P
0 there exist positive constants 10 , M depending only on p, N and 0 such that if w(y , s) is u radially symmetric solution of (2.2) defined on some interval So < s < ex:: and if
PROPOSITION
2.2
Eo
sup
:=
aERN sup
1 Ps '
Despite its simple proof, this lemma is exceedingly
useful in the study of supercritical blow-up. For example, it easily follows from this
lemma that, in any type II blow-up, the rescaled solution WO,T(y, s) converges to the singular stationary solution cp* (y) or
-cp* (y)
as
s
-t
ex::
(Theorem 3 . 1 ) .
LEMMA 2 . 3 (No-needle lemma for w). Let Ps < p < ex:: and 1 , 2 , 3, · . . ) be a family of radially symmetric classical solutions
defined for y E
RN ,
So
PS '
4
where
Since
p(y)
Lebesgue convergence theorem yields (2 . 22)
E (w,, (. + a,
for some constant
' s )
)
.......
M > O.
is
E (1jJ(. + a))
I w,, (y , s' W+1
where
11
=
1 follows
c ( 1 + I y l - 2i,"!.') ) 4
0, we denote by ZI (V) the number of zeros of V(1') that lie in an interval I C (0, 00). Then ZI (U(·, t) - cp*) denotes the number of intersections between the graphs of U (r, t) and .p* (1' ) in the region r E I. If n = B R , then Z(O. R) (U(-, t) - .p*) is finite for every 0 < t < T and is non-increasing in t. This follows from the result of Chen and Polacik [7] , which is a variation of the earlier result of Angenent [1] . See also [29, Subsection 2.3] . In the case where n = RN , we assume Z(0, 00 ) (U( -, 0) - cp*) < 00 for simplicity. Since Z(o. 00 ) (U(" t) - .p * ) is finite and non-increasing in t , the following limit exists: m(U) := limT ZI(U( · , t) - cp O ) , t...
where I = (0, R) if n = BR and J = (0, 00) if n = RN . In other words, ZI (U(" t) .p*) = m(U) for all t sufficiently close to T. It follows from [7] that the zeros of U(r, t) - .p*(1') are all simple when t is close to T, since otherwise t he number of zeros would drop further. Thus they are expressed by smooth functions (3.4)
(to < t < T) ,
395
BLOW-UP IN NONLINEAR HEAT EQUATIONS
where to is some number sufficiently close to T. One em,ily sees that lim infhT rl (t) = O. In fact, if this were not true, U Cr, t) must stay below P JL and let u(x, t) = U(lxl, t) be a solution of ( 1 . 1 ) that blows up al t = T. Suppose that the blow-up is of type II. Then moCU) 2:: m* Cp, N), where m*(p, N) ( > 2 ) is the "Morse index" of 'P' to be defined in (3.11) below. Now let us define the number m* (p, N). We linearize ( 1 .6) around 'P' and consider the following eigenvalue problem in the space H� : l I 1 1 p \ I I ( c· )pi:!..'ljJ - -y . "il7jJ 1p = - A1p ( 3.7) , 1p + 2 2 p 1 lyl where c· is as in (2.18), pry) = (47r )-N/�e-IYI2/4 is llli in (2.3) and -
H� = { v I
RN
P ( l "ilvI 2 + v2)dy
O.
'um ( . , t) =
et � ,
eM U.o +
Im(u)
=
(4.2)
we can let
globally defined for x Since each
10t e(t-T)� 1", (1l", ( - , T)) dT
In view of the monotonicity of the sequences the operator
O.
m --+
for 0
Te.
the blow-up time of solution
incomplete if T
Te.
Ps i see
it is known that every blow-up is complete if
As regards 1('lL) =
P u ,
and
for
< 00. As one eilliily sees
complete
if
u
n },
We say that the blow-up is
[3] .
of thc
and is thus uniquely determined by UQ . It is clear that u = Sec [15] for a more general treatment of proper extension.
{1m}
t < T.
E n,
However, in the supercriticai range Ps
< P
Tc(uo ) , (4.4)
wherc T(uo) and Tc(uo) denote, respectively, the blow-up time and the complete blow-up time (possibly 00) of a solution of (4. 1 ) with initial data uo. There is another approach to define continuation beyond blow-up based on the notion of limit L1 solutions. The following definition is a slightly modified version of what is found in [9]. In what follows n is a bounded domain in RN . DE�'INI'l'ION 4.3 (limit L 1 solution). By a limit Ll _solution on the interval o < t < T* « 00 ) we mean a function u(x, t ) that can be approximated by a sequence of classical solutions un (x, t) of (4. 1) in the following way:
UO,n
( 4.5)
Un( · , t)
(4.6)
f (un)
-->
-->
:=
Un c, 0)
ue , t)
fe u)
-->
as n -->
Uo in C(n)
in Ll (n)
for every
in L1 ( n x (0, t »
tE
00,
[0, T* ) ,
for every t E [O, T· ) .
We call a limit L l -solution a minimal L l _solution if it has an approximating se quence that is monotone increasing in n. By the strong comparison principle, we can show that a minimal L l solution is indeed the minimal element of all limit £1 solutions for a given initial data uo, hence it is independent of the choice of the approximat.ing sequence Un. Since each Un is a classical solution, it is easily seen that a limit £1 solution U satisfies (4.1) in the sense of distributions. Similarly, it is also a mild solution: u(·, t)
(4.7)
=
etA Uo +
t
°
e(t-r)A f(ue, T)) dT for ° < t < T* .
Consequently any limit L1 solution U belongs to C([O, T* ) , £1 (n» . It is clear that U = u for 0 < t < T(uo ) , where u denotes the classical solution of (4. 1 ) and T(uo) its blow-up time. Thus we may call u a limit £1 continuation of u. Note that, if UO ,n is increasing in n, then (4.6) follows automatically from (4.5) by virtue of the Kaplan estimate [25] and the monotone convergence theorem. The minimality of the proper extension u and (4.7) imply u < U for any limit £1 solution U. Hence T* < Tc (uo ) , since u(·, t ) < 00 for a.e x E n, t E [0, T* ) . Conversely, if UO,n < Uo (n = 1, 2, 3" , . ) the comparison principle yields Un < u, hence U < u. Thus the minimal L1 continuation U and the proper extension u coincide on the interval 0 ::; t < T* . Moreover, since (4.4) implies T(uo.,,) > Tc (uo), we can choose T* = T (uo) if U is minimal. If the solutions are radially symmetric, (2. 16) yields an estimaLe of the form ,
c
(4.8)
(
l un (x, t ) I < C lxl - p'l + (T* - t) - /l
)
for X E n, t E [bT* , T* )
for n = 1 , 2 , 3, . . . , where the constant C depends on T* , b but is independent of n . Thus the same estimate holds for a limit £ 1 solution U. Combining this estimate and the integral identity (4.7) , we see that U E C ( (0, T*) ; H1 (n) n U (!"!) for any q < N(p 1 )/2 ; see Proposition 2 .15 of [9]. Here is an example of a minimal L1 solution of ( 1 . 1) . Let n = RH, P > Ps and uA be the classical solution of ( 1 . 1) with initial data Uo = AV, where vex) = V(lxj) > ° is a given smooth function. Then, as is easily seen, there exists A * > 0 -
BLOW-UP IN NONLINEAR HEAT EQl1ATTONS
399
such that uA -> 0 as t -> 00 if A < A', while uA blows up in finite time if A > A· . Choosing a Requence Al < A2 < . -> A' and letting u : = lillln�oo 'uAn , we obtain a minimal Ll solution that is defined for all t 2:: O. We can show that u blows up in finite time and decays to 0 as t -> 00. The same is true for n = RN , provided that v E HI n Loo. See [30] for details. The above idea of constructing an unbounded global weak solution is due to [37] . Initially it was not known whether u blows up in finite time or remains smooth for all time. Later [15] confirmed blow-up for Ps < P < 1 + IV 6 10 , The upper restriction on p was removed in [34, 30] . 4.2. Speed of regularization after blow-up. In a joint work with M. Fila and P. Polacik [9] , the author has proved that any minimal £1 solution of ( 1 . 1) becomes classical immediately after the blow-up time, provided P s < P < P The paper abo deals with the equation Ut = �u + A e u with 3 < ]II < 9. The following theorem shows how fast regularization occurs after blow-up. It also extends the above result of [9] on (1.1) to all P > PB and all (possibly non-minimal) L l solutions under the as!:>umption that the blow-up is of type 1. .
.
JL '
THEOREM 4.4 (Type I regularization [30] ) . Let Ps < P < 00 and let u be a solution of ( 1 . 1 ) that blows up at t = T . Assume that the blow-up is of type I and let u be a limit Ll continuation of u defined on [0, T*) with T* > T . Then lim sup (t - T) /' llu( . , t) IILoc < 00.
(4.9)
t ",.T
OF PROOF. Since the blow-up is of type I, the local blow-up profile w* (y) = W* (lyl) is a bounded solution of ( 1 .6). Put 11 : = limr�oo W* (r)j* (r). As mentioned in (3.3), we have 11 f. 1. Now suppose that (4.9) docs not hold. Then there exists a sequence tl > t2 > t 3 > . . . -> T such that O U TLINE
(tn - T) P':l l l u(- , tn ) IIL= -+ OO
Define
as
n -> oo .
1
un (X, t) := A:; ' u( A x, A n t + T), An = tn T. Then Un is a minimal L l solution and satisfies Ilun e 1 ) IILo< -> 00. Since a pointwise bound similar to (4.8) , we have, for some constant C > 0, -
en =
o )..;; - '
(
1
u satisfies
).
Thus, by parabolic estimates, we ean choose a subsequence of {un} converging to some function u outside the origin. This function u is again a minimal £ 1 solution of ( 1 . 1 ) and is defined for all x E RN \ {O}, t E R, since A"fL O. Clearly we have lu(x, t)1 < CixlMoreover it is easily seen that p
(4.10)
,
-+
l .
u(x, t )
=
(-t)- p':'r
w* (�)
for t < 0,
u(x , O )
=
IlCP* (x) .
By Remark 2.4 (no needle lemma for 11. ) , we see that lIu(', 1 8) 11£'", = 00, where 0 < 15 < 1 is arbitrary. Therefore u(x, t) is singular for every 0 < t < 1. Given 0 < Tl « 1, we rescale u as i n (2. 1 ) and denote it by WO,T, (y, s) . Since u is a limit of a sequence of classical solutioIlH in HI n Lq with q > p + 1, we see that the energy E(WO,T1 ( s» is nondecreasing in s, and that Lemma 2.3 (no-needle lemma) holds for WO,T, . Therefore limB�OO WO.Tl = cp' , since Wo,T, is unbounded. On the other hand, (4. 10) implies wo.r, (y , - log Tl ) = IlCP*. Thus we must have E(IlCP*) > E(cp* ), but this is impossible since E(llcp*) = (if ) J p(cp')P+ldy achieves its strict maximum at /.L = 1 . This contradiction proves the theorem. 0 -
-,
;:;:
HTR.OSHI MATANO
400
5. Type II blow-up and the braid group If a blow-up is of type TI, its exact blow-up rate is not easy to determine. This is because type II blow-up solutions, by definition, do not obey the standard scaling law. Herrero and Velazquez [22, 23] constructed examples of type II blow-up for the range p > pJL and computed their blow-up rates explicitly. However, the problem of determining all type II blow-up rates has long remained open. Recently the problem was partially solved by Mizoguchi [35], who proved that any type II rate coincides with one of the Herrero-Velazquez rates, provided p > PL : = 1 + N�1O ( > P.IL ) · The proof of [35J uses a three-step argument : first to show the validity of the eigenfunction expansion away from the origin by using integral estimates, second to give an upper bound for the growth rate at the origin by the zero-number argument, and finally to use a matching argument as in [22, 23J . In thiH section we show how the braid group theory can improve the result of [35J while significantly simplitying the proof at the same time. More precisely we extend the above result to the range p > P J L under a weaker atlsumption. Our proof is bll.'led on a remarkably simple topological argument and is less reliant on heavy technical estimates. This flection is based on the paper [28J . As mentioned in Introduction, a somewhat similar result has been obtained in [36] by partly borrowing the idea of [28J and partly using a different argument. For simplicity, hereafter we deal with only positive solutions.
5.1. Herrero-Velazquez solutions. RecaU that a blow-up is of type II if alld only if the rescaled solution w := 'IliO ,T ( Y , s ) of (2.2) converges to the singular
Htationary solution :p* (y) : = c* lyl -2/(p- 1) as s -> 00 (Theorem 3 . 1 ) . In a naive view, such solutions lie on the stable manifold of :p* , so an eigenfunction expansion around :p* may give a good first-order approximation. However, since :p* does not belong to the space where ( 1 . 1 ) is well-posed (such as Lq(l1) with q > N(p - l)j2), the standard linearization technique does not work. The idea of Herrero and Velazquez [22, 23J is based on a matched asymptotic method. They used an ansatz that the behavior of the rescaled solution w away from the origin is well described by the linearized equation, while its profile near the origin is approximated by a family of stationary solutions of (2.25). Assume P > PJL and let Aj (j = 0, 1, 2" , . ) be the eigenvalues given in (8.9), and 11<j ('I') be the corresponding eigenfunctions. Then it can be shown that (5.1 ) for some positive constant Cj , where a is the constant defined in (3. 10) . Next let <Pa (T) be the stationary solution of (2.25). Then <Pa (T) is increasing in a > 0 and (5.2)
* ('1')
- k(a) T-In: + o(r- Ial )
'I'
->
00
k(a) > O. Now for each positive integer m, we set 17m
=
( Am la l -
2 p
_
1
) -1
'
It is shown in [22 , 23] that, for each Tn such that (5.3)
as
Am > 0
BLOW-UP IN NONLINEAR HEAT EQUATIONS
401
there existR a classical solution Wmer, s ) of (2.2) that is decreasing in r and possesses the following asymptotics for some positive constants K, *)
(5.5)
for r E [0, Ke - 7Jm S J ,
'1m S , e""s ] , e K r fo r E [ = m.
Here the parameter a is determined by the condition k(a) = em , with k (a) and em being the constants in (5.2) and (5.1) with j = m. This is a matching condition for the inner and outer asymptotics. The growth rate of Wm and the blow-up rate of the corresponding solution (5.6)
Um (r, t) := (T - t ) - P-' 1
arc thus given as follows, where (3 (5 . 7)
=
Wm ( (T - t)- 2 r, - log(T - t » 1
1 and C is some constant: � (Ial ) p 1 P 1 2
The existence of such a solution Wm has been proved rigorously by using a fixed point theorem; see also [33] , which restates the argument of [23] .
5.2. Braid group. The braid-group method for one-dimensional parabolic equations was introduced recently by Ghrist and Vandervorst [17] as generalization of the so-called zero-number argument that has been widely used as a powerful tool for qualitative analysis of various parabolic equations. While the standard zero number argument counts the number of intersections between the graphs of two solutions, the braid-group method of [17) keeps track of the entanglement among an arbitrary number of solutions. Let us briefly illut:)trate this approach. Given a C l function vex) on an interval [0, L], we define a continuous curve in R3 by ,[v]
(5.8)
:=
{ (x, v (x), v'(x» ; ° < x < L}.
Now let VI , 112 , . . . , Vk be C l functions on [0, L] whose graphs are nowhere mutually tangential. Then the curves , [VI ] , ,[112], ' . . , ,[11k] do not intersect with each other, hence they form a "braid" on k strands (see Figure 2). Naively, a braid is understood to be a topological type of mutually disjoint strands whose endpoints are aligned on two parallel lines. A precise defiuition will be given later. Hereafter we will expret:)s a braid by a "braid diagram" shown at the right end of Figure 2 .
o
L
�------�---. x
FIGURE
2 . Graphs and their braid expression (the case k = 2)
In what follows we only consider the case where k = 3. In this case, any braid diagram can be realized by joining the elements 0 0< , ... . -
- - - -
_
...
... ..
. -
CD
-
-
_
-
- . -
--- - - _ ...
----
---
-
6 . Simple parabolic reduction (the dotted areas remain unchanged )
5.3 (Parabolic reduction) . We say that B E B:1" is a simple par abolic reduction of A E Bt if there exist C, D E Bt and i E {1, 2} such that A = Cal D, B = CD. We denote this relation by DEFINITION
A I>, B . We say that B E Bt is a parabolic reduction of A E Bt if there exist AI , . . . , Aj E Bt such that A 1>, AI l>, . . . 1>, Aj = B. We denote this relation by A I> B. The above reductioIl process itself is also called parabolic reduction. We use the notation A I> B to mean either A I> B or A = B . Note that A 1>, B implies e(A) -f(B) = 2 . Therefore, if A I> B, then £(A.) -feB) is a positive even integer. Now we apply the above idea to equation ( 1 04) in the (r, t) coorditanes:
N-1 -l p Ur + lUI U for r > 0, t > to. Ut = Urr + ( 5. 10) r Given a solution triple U1 , U2 , U3 , we consider a braid f3[ U1 ( - , t ) , U2 (·, t) , U:l (-, t) 1 on some interval fro, r1l with 0 < ro < rj . Then the following proposition holck
Let U1 , lh, U3 be solutions of (5. 10) defined for to < t < t1 and suppose that the values Uj(ro, t) (j = 1, 2, 3) remain mutually distinct for every t E [to, ttl and that the same is true of the values Uj (rl , t) (j = 1, 2, 3) . Then P ROPOSITION SA .
(5 . 1 1)
f3 [ Ul e-. to) , U2 ( -, to ) , [h ( - , to ) 1
I> f3[ UI ( · , t t), U2 (-, t t l , U3 ( - , t I ) J .
REMARK 5.5. By (5. 1 1 ) , the length £(f3[ Ud· , t), U2 ( - , t), U3 ( - , t) J ) is non-increas ing in t, hence it is a Lyapunov function for (5. 10) . However, (5. 1 1 ) contains far richer information on the dynamics of solutions than what e(f3) alone can tell us.
The following proposition provides a useful tool to check parabolic reducibility of braids. An outline of its proof will be given in Subsection 5.5.
PROPOSITION 5.6 ([28) ) . Let A, B E Bt . Then for any P conditions are equivalent:
(a)
A I> B
(b)
PA
I>
PB
(c )
AP
E
I>
Bt , the following
BP.
405
BLOW-UP IN NONLINEAR HEAT EQUATIONS
5.4. Main theorem and its proof. Let u(x, t) = U ( l x l , t) be a positive �olution of ( l . 1 ) that blows up at t = T. In the case where n = RiV , we as,mme: (5.12) Clearly the Herrero-Velazquez solutions (5.6) ( "HV-soluLions" for Hhort) satisfy this condition. IL can be shown that if p > P JL then (5.12) implies
T'j (t)
(5. 13)
-->
as
0
t
-->
T
(j = 1 , 2, · " , mo (U» ,
where rj is as in (3.4) and mo (U) is the number of vanishing intersections defined in (3.6) . See [28] for a proof based on a )lera-number argument . The following is the main theorem of this section. It states that the blow-up rate of a type II blow-up solution is uniquely determined by mu (U):
THBOREM 5.7. Let P JL < P < 00 and let U(lxl, t), U ( l x l , t) be positive solutions of ( 1 . 1 ) that blow up at t = T with a type II rate. If n = RiV, assume also (5. 12). Denote f,y W(lyl, s), W(lyl, s) the corresponding rescaled solutions of (2.2 ) . Then -
�
-
(i) if mo ( U) < mo(U) , there exist T > 0 and So > - log T such that
II W e , s) I I £O'" < I I W ( " s + T ) IIL =
(5. 14)
fo r s E [so , 00 ) ;
(ii) if mo(U) = mo ( U) there e.rist T > 0 and So > - log T such that -
,
(5. 15)
I I W e , s - T ) IIL= :S II W( · , s ) IIL= < II W ( · , s + T ) I I L= --
�
for s E [So, 00) .
Recall that if U ( r, t ) is a type II blow-up solution, then mo (U) 2: m* (p, N) by Theorem 3.4, hence Amo ( U ) > O. Therefore, if >"mo(U) '" 0, we see from (5.3) and (5.5) that there exists an HV-solution Urn (and its IeHcaled solution Wm ) with � mo (Um) = mo(U) . Putting W = Wm in (5.15), and considering that the growth rate of WTn is exponential, we see that II W ( s) II L� / I I WTn ( . , s) II L= remains bounded boLh from ahove and below. Thus we obtain the foHowing corollary: .,
COROLLARY 5.8. Assume that 0 is not an eigenvalue of (3.8) . Let u (x, t) : = U(lxl, t) be a positive solution of ( 1 . 1 ) that blows up at t = T with a type II blow-71p rate. If 0. = RN , ass'ume also (5. 12) . Then I IW(·, s)II L= and II U( ·, t) IILoo satisfy the same estimates as in (5.7) with m = mo (U). PROOF OF THEOREM 5 . 7 . Since the assertion (ii) follows from (i) by exchanging the roles of W, W, it suffices to prove (i) . Now for p,ach A > 0 , we define �
tP'- (r, t) := >.. /1 fj ( .,,0. r, T - ).. (T - t» . Then fj>. i� again a solution of ( 1 . 4) and it blows up at t = T with a type II rate. Note that
(5 . 16)
where y, s are as in (2 . 1 ) with a = 0. Therefore, proving (5. 14) is equivalent to showing the following estimate for some t o E [0, T) and a sufficiently small >.. > 0: for t
E
[to , T) .
By Lemma 5.10 below, we have II U(-' t) IIL� = U(O, t) for t sufficiently close to T, therefore what we have to show is the following estimate for some to E [0, T) :
(5.17)
U(O, t) < fj>.(O, t)
for t E [to, T) and 0 < >.. «
1.
HIROSHI MATANO
406
Choose to sufficiently close to T so that the number of intersections between U(r, t) and * (r) remains constant in to ::; t < T as in (3.4). Then the intersections between U(r, t) and * (r) are all transverse for every t E [to T) . As in (3.4) , we denote the zeros of U(r, t) - *(r) by r1 (t) < r2(t) < . . . < rm(U) (t) . Similarly, the zeros of U(r, t) - * (r) will be denoted by i\ (t) < T2 (t) < . . . < 1'm(U) (t) . By virtue of (5.13) , there exist ro, c > ° such that ,
(5. 18)
for t sufficiently close to T. (Here we understand that rmo ( U) -H (t) = 00 if mo (U) = m(U» . By redefining to if necessary, we may assume that (5.18) holds for all t > to · Then a simple comarison argument yields that, for some constant t5 > 0, (5. 19)
IU(ro, t)/* (ro) - 1 1 > t5
for to < t < T. I
Since * (r) is invariant under the transformation VCr) zeros of [f:" (r, t) - * (T) are given by
'ft (t) < r� (t) < . . < _
f�(U) (t),
�
,\ P
"
V ( J>: r') , the
fJ (t) := v'>..- lrj (T - '\(T - t» .
Therefore, if ,\ is chosen sufficiently small, we have (5.20)
fJ(t) >
for mo(U) + 1 < j < m(U), to < t < T. -
TO
-
By Theorem 3.1, W(lyl, s ) -> *(lyl) as S 00, hence the right side of (5.16) converges to (T - t)-/ 1 * « T - t)-1/ 2 Ixl) = * (Ixl) as '\ 0, Consequently -
-4
-4
(5.21)
fY' (r, to)
�
tfJ* (r)
in C1 «0, ro])
for 0 < ,\ «
1.
Since U(r, to) and * (r) intersect mo (U) times transversely in the interval (O, ro] , we see from (5.21) that, for ,\ is sufficiently small, (5.22 )
Z(O,Toi (U( - , to) - iP'(- , to » ) = mo CU) .
One can also show that, if ,\ is chosen sufficiently small, then - A (T , t)/4.>* Cro ) - 1 1 < t5 for to < t < T. (5.23 ) IU O
This follows from [30, Lemma 4.6] , which is used t.o prove (3.2). Hereafter we fix to , ro , '\ such that (5. 1 8) to (5.23) all hold. Figure 7 gives a schematic description of this situation, where U (5 .24 )
407
BLOW-UP IN NONLINEAR HEAT EQUATIONS
r=O I
I 00
!
./
I I
,
I I
1\
V
V
f\
1\
\
r
"
- TO
I I
!
:I
�
FIGURE 7. A schematic graph at t
I I
,
=
u(-, to)
to (for mo (U) = mo (U'),")
=
6)
Consequently there is some t2 E (to, tl) such that U(O, t 2 ) = UA(O, t 2 ) ' This means that the graphs of U and UA become tangential at r = 0, hence they lose at least one intersection at this moment, by the result of [7]. 'Without loss of generality, we may assume that the intersections between U and UA are transverse at t = tl, since degeneracy occurs for at most finitely many values of t. Now we choose a sufficiently small e > ° such that the two braids O [ U( " t l ) , fjA ( . , tl), *] and O [ U(" tl), fjA(., tl +e), * ] are topologically equivalent. Since UA ( . , t + c ) blows up at t = T e , there is t3 E (tl ' T) such that U(O, t3) = UA (O, t3 + c ) . This means that an intersection is again lost at T = O. As t approaches T e , the mo (UA ) inten>ections between UA ( - , t + c) and * are all swept away to the left end (i.e. r = 0). Thus, for an appropriate t* E (t3, T c), they all lie on the left side of the intersections between U and *, as illustrated in Figure 8. , U(r, to) , U(T, to) , U (T, tl) -
-
-
--+)
r
r=U I I I
- • - - - • -
.
I ' ",
\ "
=
I I I
ro
-.�_I___+___,I_+--I----"\_-_+_....J,._I__4_1___4-_I_
. . . . .. .\ . . . ,' �
fjA (r, t* + e) .
I I
�
'---4----+---;- if). (
�------------�
'
-,
t' + c)
U(" t')
I
,
FIGURE 8 . A schematic graph at t
=
t* (with trapped crossings)
As mentioned earlier, we lose at least one intersection between U and UA at r = ° as t passes t2 and at least one between U and U A (', t + c ) as t passes t3 . Since both losses occur at T = 0, we can interprct these losses as being "trapped" behind r = ° rather than being completely lost. This creates "trapped crossings" between the two strands, as indicated by the broken line at the left end of Figure 8. More intersections may be lost at r = 0, but the number is even since fjA(O, to) > U(O, to) and fjA (O, t* + e) > U(O, t*) . Therefore, we can reduce the number to two by an artificial parabolic reduction of the braid. Note also that U(r, to) and fjA (r, t* + c) may have other intersections in the right region of Figure 8, but we can again delete them by an artificial parabolic reduction. Thus a combination of the natural PDE based parabolic reduction and an additional artificial parabolic reduction convert the graph in Figure 7 to that in Figure 8.
HIROSHI MATANO
408
The braid diagrams corresponding to the graphs in Figures 7 and 8 are shown in Figure 9 ( for the case mo (U) = mo (U) = 6). They represent the braids -
A : = f3[ U (·, to), fPc-, to ) , 1>0 ] ,
B : = f3[ u(-, to), UA(-, t o + E) , 1>* ] .
To clarify the dependence of A , B on m = mo (U) , 'iT! mo (UA ) , we write them as A m,in , Em , in ' Their exrpessions differ depending on whether m, m are even or odd: 2 a) k oT, 2n (aT2 ay, 2n T (aT T = = A2k,2n A2k+l,2n B2k+l ,2n = �2�2n��2k+l ( 5 . 25 ) 2 k 2n+l 2n+ l (iT iT 2 k ( a) T T T aT, a) T A2k,2n+l A2k+l,2n+l , l 2k l 2 + 2n 2 + n k 2 tl 2 = T T, a l a a = a r B2k,2n+l B 2 k + 1,2n+ , =
v
=
.
v .
,
=
Here and in what follows the symbols a, r stand for 171 , 172 for notational simplicity. Conditions ( 5 . 1 9 ) (5.23) guarantee that no crossing between U and UA enten; nor exits from the right end as t varies in [to , to]. Thus, by Proposition 5 .4, we have A m,in t> Bm,in. However, this contradicts Lemma 5.11 below. This contradiction 0 disproves the hypothesis (5 . 24), establishing (5. 17) . The theorem is proved. ,
A B
FIGURE 9 . Corresponding braid diagrams (for t = to and t
=
to)
REMARK 5.9. The above idea of replacing UA (T, t l ) by U>' (r, tl + c) has its prototype in [35] ' where a result similar to but weaker than our Corollary 5.8 is proved by using the zero-munber arglUllent aided by more technical estimates. Our braid-group method gives a much more transparent argument. 5.lD. Let U be as in Theorem 5. 7. Then there exist ro > 0 and to E [0, T) such that Ur (r, t) < 0 for every 0 < r < ro, to < t < T. LEMMA
OUTLINE OF THE PROOF. This can be shown by counting the number of intersections between U(r, t) and the solutions of (1 .4) with initial data I\;(T ± E)0 where E is a small parameter. See [30, Remark 3. 15J for details. P
1
"
5 . 1 1 (Main lemma) . Let Am,in, Em,in be as in (5.25) with m < m. Then A m,in if Bm ,in ' LEMMA
PROOF. We first note that the following identities hold:
A2k-I,2n- l TaT = A2k_2,2n_l ar2 CTT, = B2k-l,2n-lrar B2k,2n B2k_2,2n _ l ar2 CTT. They can be derived by using the identities (ar2iT)T = T(ar2 a) , aj (Ta) = (rCT )Tj . A2k,2n
= A2k- l,2n Ta = B2k- l , 2nra
=
=
409
BLOW-UP IN NONLINEAR HEAT EQUATIONS
Thi� and Proposition 5.6 imply .: )- A2k-1 2n-1
I> B2k-l 2n- l
': ;- A2k-2 ,2n-l
I>
B2k-2 , 2n- l ·
Therefore, it suffice� to prove the lemma when m and m are both even integers. Moreover, since A2k,2nT(l 2 T = A2 k+ 2,2n and B2k, 2n T(l2 T I> B2k+2,2n , we have !
,
A2k,2n I> n2k,2n
)- A2k+2,2n
I>
HZk '· 2,2n .
Thus it suffices to consider the case where m m = 2n. Observe that T2 and (lT2 rr arc commutative, and that the braid T2 (1T2 (1 = T(lT(lTa commutes with every element of Bt . Consequently A2n,2n can be written as A2n,2n = (T2 aT2 a)" and it commutes with every element of Bt . Hence =
A2n,2nO"2 = (72 A2n�2n ':::::::' (J2 T2n (J"_2 (a2T 2 ) "-1 (1. j
Using tills observation, one can show that
A2n,2n(l = (I2 T2n aT2 A ' , B2n,2nO' = a2 TZn aT2 n' , A' (a2 T2 )n-1 , where B ' = T 2( n-1 l a2 By Proposition 5.6, A2n,2n I> B2n ,2n if and only if A' I> B' , so it, suffices to show that the latter does not hold. Observe that A' is written as a product of (12 and T 2 . This means that A' is rigid, and that its simple parabolic reduction has the same property. Consequently any parabolic reduction of A' i s achieved by �imply deleting the term (12 or T2 one after another without changing the order of letters. =
However, the number of the letter T on the left side of (I in 8' is 2(n 1), while no letter a in A' has that many T'S on its left side. This shows that B' is not a parabolic reduction of A' . This contradiction proves the lemma. 0 -
5.5. Proof of Proposition 5.6. We only give a brief outline. We start with the following lemma:
LEMMA 5.12. Let A, B E Bt and suppose that (lA 1>, aB. Then A PROOF. By
the
1>,
B.
a�sumption" there exist C, D E Bt such that
(lA = Cp2 D, aB = CD
where p = (I or p = T.
Case 1: Suppose that C is written as C = aCI for some C1 E Bt . Then A = C1P2 D, B = C1D, so the conclusion holds. Case 2: Suppose that C is non-rigid. Then by Lemma 5.2, the situation reduces to Case 1 , so the conclusion holds.
Case 3: Suppose that D is non-rigid. Then by Lemma 5.2, D is written as D = TaTDI for some Dl E Bt . Note that, by (5.9), we have p2 (TaT) = (TO'T)p2 , where p T if P = a and f; = a if p = T. Consequently, =
where
aA = C/(T(lT)Dl = C(TO'T)p2 D1 = C1 P2 D l ,
aB = GlD1 ,
C1 : = C(TaT). Since C1 is non-rigid, the situation reduces to Case 2.
Case 4: Suppose C, D are rigid, and that C = TG\ for some C1 E Bt . Then O'A = TC1p2 D,
aB = TCID.
Therefore, both a A, aB are non-rigid. Since TCI , D are bolh rigid, the non-rigidity of aA, O'B holds if and only if Cl = C2 pp, D = pD1 or C1 = C2 p, D = PpDl for
1 10
HIROSHI MATANO
some C2 , V 1 E Bt . In either case we can argue as in Case 3 above, and reduce the 0 situation to Case 2. This proves the lemma. Ot:TLINE OF THE PROOF OF PROPOSITION 5 . 6 . Since the assertions (a) � (b) and (a) � (c) are trivial, we only need to prove (b) � (a) and (c) � (a) . Let us show (b) � (a) ; the other assertion is essentially the same. It suffices to consider the case where P = (7 or P = T , since the general case follows by induction. \Vithout loss of generality we may assume P = (7 . Thus, what. we have to show is
(7A [> (7B �:. A [> B. We argue by induction on j := (f(A) - f(B))/2. (5.26)
[Step 1] If j
1 , the assertion is true by virtue of Lemma 5.12 above. [Step 2] Let the assertion be true for j = 1, . . . , m and set j = m + 1 . Then there exist A 1 , " . , Am E Bt such that =
(5.27)
Case 1 : Suppose that some Ai is written as Ai = (7Ai for some Ai E Bt. Then (7 A [> (7 Ai , (7 Ai [> (7B. The assumption of the induction argument then implies A [> Ai , Ai [> B, hence A [> B. Cillie 2: Suppose that no Ai can be written as A i = aA i . In view of Lemma 5.2, thi� means that every Ai is rigid and is written as Ai = TAi , i = 1, · . . , m. \Ve 0 can derive A I> B by induction on feB) . See [28] for details. -
-
-
-
-
-
6. Open problems A needle-like singularity in the critical case: As we have stated in Corollary 3.3, a needle-like singluarity does not appear at the time of blow-up if P > Ps ' The same is true in the suhcritieal range 1 < p < Vs ' However, the formal analysis of [11] indicates that a needle-like singularity may appear for sign-changing solutions if V = Vs ' Thus the critical case P = Ps seeIllS to be highly exceptional. So far, there is no rigorous study which fully explains this phenomenon. Uniqueness of limit L1 continuation: A limit L 1 continuation u of a solution u of (1.1) is defined as the limit of a sequence of approximating classical solutions Un �ati�fying (4. 5), (4.6). Limit L1 continuations form a subclass of "L 1 continuations" . The latter consist of all continuations of u beyond the blow-up time as an U solution. It is known that an L 1 continuation of a. given solution is not necessarily unique if Ps < P < P ( [8, 10] ) . However the question still remains open as to whether a limit L1 continuation of a given solution is unique or not. JL
Reversed blow-up profile: As shown in Theorem 4.4, an incomplete type I blow-up yields a type I regularization. One may then wonder if such a solution is asymptotically self-similar as t ',. T. This amounts to asking if the limit w· (y) := lim (t _ T) .�' u( Jt - T y, t) t'"T
always exists. The question is still open. The difficulty comes from the fact that the energy for the forward rescaled equation Ws = 6w + � y . 'Vw + P�1 W + l·iVIP-1W, where y = x/ Jt T, s = log(t T), is finite only for a. narrow class of funcitions. -
-
411
BLOW-UP IN NONLINEAR HEAT EQUATIONS
Non-radial problems: Because of lack of adequate a priori estemates, the nature of blow-up for equation ( 1 . 1) is largely unknown in the supercritical range without the assumption of radial symmetry. For example, it is known ([29, 30]) that no type II blow-up occurs for radially symmetric solutions in the range Ps < P < P J L ' but we do not know if the same is true for non-symmetric solutions. Even in the subcritical range 1 < P < P the situation is not so simple if the domain n is not convex. Thi� iH because the energy E(we, s ) ) may no longer be monotonically decreasing in s due to the contribution from the boundary of n. s '
Exponential nonlinearity: The equation (6. 1 )
is another important model equation for the study of blow-up. Intriguingly, the range 3 < N < 9 for (6. 1 ) looks much like the range Ps < P < P JL (the lower supercritical range) for (l.1), as far as the behavior of stationary solutions is con cerned. In fact, any radially symmetric stationary solution of (6. 1 ) on RN inter sects with the singular stationary solution log 2(�r22) infinitely many times, just as a (r ) does with * (r) (see Table 1 in Subsection 2.4) . In view of this, we are lead to speculate that every (radially symmetric) blow-up for (6.1) in the range 3 < N < 9 is of type 1. Here, a type I blow-up for equation (6.1) means that lim SuPt �T (u(x, t) + 10g(T t) < 00 . This question is still open. -
References
[I]
S. Angencnt,
The zero 8et of a solution of a parabo lic equatlOn, J. reine
angew. Math., 390
(1988), 79-96. [2J A. Artin, Theorie der Zapfe, Hamburg Abh. 4 (1925), '17-72. [3] P. Baras and L. Cohen, Comple t e blow-up after Tml>x for the solution of a sernilinear heat eq'"a ti u'll, J. Funet. A nalysis, 11 (1987), 142-174. [4J J. Bebernes and D. Eberly, Mathem a lical Problems from Combustion Theory, Springer Verlag, New York, 1 989. [5] J.8. Birman, Braids, Links, and Mapping Class Groups, Ann. Math. Studies, Princeto n UP ' 82, 1974. [6J X.-Y. Chen and H. Mat ano, Convergence, asymp t otic periodici ty and finite point bluw-up in one-dimensional semilinear heat equati ons, J. Differential Equations 78 (1989), 160--190. [7] X.-Y. Chen and P. Pol3i:ik, Asymptotic periodicity of positive solutions of reaction-diffusion equation.. o n a ball, J. reine angew. Math., 472 (1996), 17-51. [8] M. Fila, Blow-up of solutions of supercritical pambolic equations, Handbook of Evolution Equations, Elsevier (2005), 105-158. 19J M. Fila, H. Matano and P. PolaCik, Immedi a t e regularization after' blo'w-up, SIAM J. Math. Anal., 37 (2005), 752-776. [10J M. Fila and N . Mizoguchi , Multiple continuation beyond blow up , prcprint. [1 1 J F. Filippas, M.A. Herrero and J.J.L. Velazquez, Fast blow-up mechanisms for sign-changing solutions of a semilinear parabolic equatio n with critical nonlinearity, R. Soc. Lond. Pro c . Ser. A M ath. Phys. Eng. Sci., 456 (2000) , 2957-2982. [1 2J Friedman and J.B. McLeod, Blow up of posil1ve solutions of semilineur heat equations, In diana Univ. Math. J., 34 (1985), 425 -447. 113J H . FUjita, On the blowing up of solutions of the Cauchy problem for Ut c,,11 + ul+ o , J . Fac. Sci. Univ. Tokyo Sect. A . Math., 16 (1966), 109-124. [14J V . A . Galaktionov and S.A. Posashkov, Application of new comparison Theorems in the 'in vestigation uf unbounded solutions of nonlinear parabolic equations, Differential Equ at ions, 22 (1986), 1 16-183. [15J V.A. Galaktionov and J.L. Vazquez, Continua.ti on of blow-up solutions of nonlinear heat equati ons i.'fl seveml space dimensions, Comm. Pure Applied M ath . 50 (1997), 1-{)7. .
-
=
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HIROSHI MATANO
Garside, The braid group and other groups, Quart. J. Math. Oxford Ser., 20, No. 78 ( 1 969) , 235-254. [17J R. Ghrist and R. Vandervorst, Scalar parabolic PDE's and braids, preprint [18] Y. Giga and R. V. Kohn , A.symptotically sdf-.•imilar blowup of semilinear heat equations, Comm. Pure Appl. Math. , 38 (1985), 297-319. [ 1 9] Y . Gig" and R.V. Kohn, Chamcterizing blowup using similarity variables, Indiana Univ. Math. J . , 36 (1987), 1-40. [20] Y . Giga and RV. Kohn, Nondp.generacy of blow up for semilinear heat eq'uulions, Comm. Pure Appl. Math . , 42 (1989), 845-884. [21J Y. Giga , S . Matsui and S. Sasayama, Blow up rate for semi/inear heat equation with sub critical nonlinearity, Indiana Univ. Math. J., 53 (2004) , 483-5 14. [22] M . A . Herrero and J . .J.L. Velazquez, Explosion de solutions des equations paraboliques umilineaires supercritique" C.R. Acad. Sci. Paris, t . 319 ( 1 994) , 141-145. [23] M.A. Herrero and J.J.L. Velazquez, .4. blow up result for scmilinear heat equations in the supercritical case, (1994) unpuhlished. [24] D.O. Joeeph and T.S. Lundgren, Quasi/inear' DiTichlet problems driven by positive source_., Arch. Rat. Mech. Anal. 49 (1973), 241-269. [25] S. Kaplan, On the growth of solutions of qua8ilinear parabolic equations, Comm. Pure Appl. Math. 16 ( 1963), 327-330. [26] A.A. Lacey and D. Tzanetis, Complete blow-up for a semilinear diffU8ion equatinn with a sufficienlly large initial cond,tion, IMA J. Appl. M at h . , 41 (1988), 207-215. [27] O.A. Ladyzhenskaya, V.A. Solonnikov, and N.N. Ural'ceva, Linear' and Qu asilinear Equation of Parabolic Type, Translations of M at hemati cal Monographs 23, AMS, 1968. [28J H . MaLano, Determining type II blow-up rates for nonlinear hmt equations via the braid Y',vup theo,'y, preprint. [29] H. Matano and F. Merle, On nonexistence of type II blow up for a supercritica/ nonlinear heat equation, Comm. Pure Appl. Math . , 57 (2004), 1494-1541. [30] H. MaLano and F. Merle , Classification of Type I and 7'iJpe II behaviors for a super'critical nonlinear heat equatio n, preprint. [31J J . Matos, Unfocused blow up solution, oj semilinear pambolic equations, Discrete and Con tinuous Dynamical Systems, 5 (1999), 905-928. [32] F. Merle and II. Zaag, Optimal estimates for blow-up rate and behavior for nonhnear heat equations, Comm. P ure Appl. Math., 5 1 (1998), 139-196. [33J N. Mizoguchi, Type II blowup for a semilinear heat equation, Adv. Different.ial Equat ions 9 (2004) , 1279-1316. [34] N , M izoguchi, Boundedness of global solutions for a supern,licaL heat equation and its application, Indiana U niv. Math. J . , 54 (200ri), 1047-1059. [35] N. Mizoguchi, Rale of type II blowup for a semilinear hea.t equation, preprint. [36] N. Mizoguchi, Blowup rate of type II and the braid group theory, preprint. [37] W.-M. Ni, P.E. Sacks and J. Tavantzis, On th e u'Y1nplotic behavior of solutions of certain quasilinear parabolic equati o ns, J. Differential Equations, 54 ( 1984), 97-120. [38] P. Quittner, A. priori bounds for global solutions of a semilincar parabolic problem, Acta Math. Univ. Comenian"", 68 (1999), 195-203. [39J J.J.L. Velazquez, Estimates on the (n - l) -dimensional lIausdorff mea.mre of the blow-up sct for a semilinear heat equati on, Indiana Univ. Math . .1., 42 (1993), 445-476. [40] F. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equa tions, 55 (1984), 204-224. [16]
F.A.
'
GRADL.'
SOBOLEV MAPS ON MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
415
is topoLogically like a ball. By composing v with an appropriate diffeomorphism, we are in the situation where the map is JRn-valued, and then approximation is standard. It remains to prove that, if we pick appropriately c � 0 and x = x(c) , then the corre,sponding v's converge to u. Here it is where the hypothesis sp < n comes into the picture: it implies that
r !lVX,E - ullf-v" p dx isn
--
0 as
f: �
O. This
leads immediately to the desired conclusion. When sp = n, the idea of the proof goes back to Schoen and Uhlenbeck [SUJ; their proof works for w 1 ,n (§n ; §n ) maps. The general case is due to Boutet de Monvel and Gabber (Appendix to [BGP]). The starting point is that a map u in X belongs to VMO (=vanishing mean oscillation=the closure of smooth maps in BMO). AnaxESn
lytically, this means that, with h (u) = sup we have
f
f
B(x,Ii)'lsn .
lim Io (u) = O.
(2 . 1 )
lu(y) - u(z) l dy dz,
B(x,5) 'lsn
O�O
(For a proof of the Sobolev embedding w s,p(lR!n) � VMO when sp = n , see, e. g., [BNI] , Section 1.2). Next, the key ingredient is that, for VMo(§n; §n) maps, we have (2 .2)
lu. 1
-->
1 uniformly as E
->
O.
Indeed, if y E B (x, c:) n §n , then lu(y) - 'u. (x) 1 < c 1- l u. ( x ) 1 =
B(x . )n§n ,
( l u (y ) l - I u£ (x ) l) dy
1/2 2u, if lui < 1/2
extension of u and set v =
(2.6)
lu(x) I
. When u EVMO, we have
1 uniformly as I x l
-4
-4
1.
This is proved in [BNII] , Appendix 3. Though the rigorous proof is delicate, the result is intuitively clear: if Pr is the Poisson kernel (so that u = u * Pr), then in some sense Pro is close to a mollifier P l -r, so that morally this result is similar to (2.2). This implies that the right-hand side of (2.5) is well-defined. Indeed, near §n , V is §ll-valued, and thus its jacobian determinant vanishes. On the other hand, far away from §n, v is Lipschitz. \Ve may now try to define deg u as the right-hand side of (2.5). Note that, when u is smooth, v is Lipschitz, so that we fall back to the classical degree. The following result completes the proof of Theorcm 2.2 and incidentally gives, for §n-valued maps, an alternative proof of Theorem 1 in [BNI] .
PROPOSITION 2 . 1 . The map u I->
f
Bn +l
Jac
v
is continuous in VMO(13n; 13" ) .
PROOF. Consider a. sequence {ud converging in VMO to some u. Then there is a fixed r < 1 such that the corresponding harmonic extensions satisfy IUk(x)1 > 1/2 and l u (x) 1 > 1/2 whenever Ixl > r . To prove thh;, it. suffices to check that the .
.
argument leading to (2.0) yields uniform estimates when applied to a convergent sequcnce. Thus, with obvious notations, we have
B(O,r)
since
Uk
-4
U in C1 (B(O, 'f'» .
J ac vk -->
1 ,.----,-,:--; IBn +l l
B (O,r)
Jac v = deg u,
o
SOBOLEV
MAPS ON MANIFOLDS: DEGREE, APPROXlMATION, LIFTING
417
2.3. Estimates for the degree. Here, X = ws,p (§n ; §n) and sp > n (so that the degree exiHts) . A natural question is whether it is possible to estimate this degree in terms of the WS'P-norm. Before giving the answer, let us consider the more familiar situation where maps are continuous or better. If 1.1. is merely continuous, then there is no possible estimate, since the sup norm of 1.1. is always 1, while its degree can be any integer; the salIle argument shows that there is no estimate for the degree of VMO maps. However, if 1.1. is slightly better, then there is a control; for example, if 1.1. is Holder continuous, then its degree is controlled by its Holder semi-norm ( this can be shown as in the proof of Theorem 2.3 below ) . In view of the Sobolev embeddings, we would thus expect the following: if sp > n , there is a control, while, if sp = n , there isn't. Surprisingly, the answer is [BBM3] THEOREM 2.3. If sp =
n,
then
I deg 1.1. 1 < C i u l fv" ,p · Consequently, if sp > n there is an estimate of the degree in terms of l u lw'.p •
(2.7)
Second assertion follows simply from the first one and Sobolev.
PROOF. The case s may assume p > 1 -
=
n and p
=
1 is easy to treat ( using
(2.4)), so that we
We start with a simple remark: if we know how to prove this result when s is small, then we know how to prove it for all s. This follows from the Gagliardo-Nirenberg type embedding l-V" P Loo '-> wr, q if sp = rq and 0 < r < s,
(2.8)
n
valid except when s is an integer, p = 1 and r > s 1 is not an integer. The case p = 1 being settled, it thus suffices to treat the case s < 1 . We rely on (2.5). Let, for x E §n , r = rx E (0, 1) be �he smallest p such that lu(tx ) 1 > 1/2 for t E (p, l). Thus the set where the jacobian of v does not vanish is contained in U = {px ; x E §n , p < rx } . Since u is the harmonic extension of a map of modulus 1 , we have 1 Du(y) 1 < C(l - l y l ) - l and thus l Jac v(y) 1 < C(l - I y l ) -(n+l). -
Integration of this inequality over U yields I deg 1.1.1 < Ix denoting the 1 /2 Thus
(2.10)
deg u =
L nlan l 2 ;
this was first noted by Brezis. On the other hand, as we saw right after Open Problem 1 , lu l�'/2 L Inl l an l 2 . Using this remark and the continuity of the '"
degree in H1/2 (Theorem 2.2), we find
PROPOSITION 2.2. Degree formula (2, 1 0) is valid when u E H 1 /2 (§ I ; § l ) . Consequently, (2. 1 0) holds if u E w1/p,p for 1 < p < 2.
Last statement is simply a consequence of the Gagliardo-Nirenberg inequality. In a survey paper [BI], Brczis asked several challenging questions about formula (2. 10). Question 1 : since degree makes sense when u is merely continuous (or even VMO ) , can one give a meaning to the right-hand side of (2.10) in order to recover deg u from lanl? Soon after, this answer revealed to be (presumably) negative (Korevaar [Ko]). This suggested some more "modest" questions. Question 2: if u = L a em9 , v = L b",emo are continuous maps such that l anl = I bnl , is it true that deg u = deg v? Answer: no ( Bourgain and Kozma) . Question 3: same as Question 2 if u, v E W1/p,p and p > 2. This is partially open. Back to Question 1 : the absolute convergence of the series L nlanl2 is equivalent to n
u E H 1/2 Nevertheless, one may still hope give a meaning to its sum. Commonly k used summation procedures consist in taking either S = lim '""' nlanl2 or T = k---1' OCl � k .
-
lim '""' rlnlnlanl2. Korevaar's result concerns these two procedures.
r-+l- �
THEOREM 2.4. If U E CO (§1 ; § 1 ) then S even not exist. ,
or
T could be any real number, or
The proof is explicit: given a E JR, Korevaar exhibits a map u such that S = a (or T a). Of course, one may imagine some other summation procedure, but Korevaar's r:Dnstruction will probably take care of it. =
SOBOLEV MAPS ON MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
419
Concerning Question 3, the first answer was obtained for Holder maps in an un published work of Kahane. In the setting of Sobolev spaces, Brezis [B2] proved the following variant of Kahane's result.
THEOREM 2.5. If U E W1/3,3, then deg u = lim
(2.11)
£-->0
L
sin2 nE . la l2 n nE2
n,oO Consequently, the answer to Question 3 is positive when p < 3.
PROOF. \Ve assume u continuous; the general ease requires some more subtle consideration on lifting, developed in Section 3. Write u = ZdelW, where d =deg u and 1f; is continuous. It is easy to see that t/J E W1/3,3. The starting point is the
identity 2� u(e'(O+h) u(e ,o)d B = 21T L l an l2 sin nh 1m
1. o
2rr =
sin(dh+1f;(e, (/!+h) -1)!(e ,O» dB.
0
A second order Taylor expansion of sin(dh + 1f;(e'(8+h) - 1f; (e,e» yields
L l an l2 si
nn
h - dh
::;
Cl hl 2 + C
o
� 2
11f;(e'(8+h) - 1f;(e'/!)13 dB .
Integrating this inequality over h E (0, 210) and dividing the result by 2102 lead to sin2 nE C l an l 2 - d < CE + L nE2 102 n,oO
12 12rr 11)!(e'(8+h) £
0
0
'
- t/J(e,O) 1 3dO dh.
Using the fact that 1f; E W1/S,s , it is easy to see that the right-hand side of the 0 above inequality tends to 0 with E. In some sense, the above result is optimal: Kahane [Ka] proved that, if u E WI/p,p for some p > 3, then the limit in (2. 1 1 ) may be any real number. However, this still leaves the following
OPEN PROBLEM 2. For p > 3, can one compute the degree of u W I /p,p in terms of I an I ?
=
L aneme E
2.5. Another degree. As we saw in Sections 2. 1-2.2, one may prove existence of a degree first by establishing density of smooth maps, next by using (2.5) . Yet there is another natural way to do it: assume that the integrals in (2.4) or (2.5)
make sense, take this as the definition of the degree, and then prove that the result is an integer. This approach was taken by Esteban and MUller [EM] . In what follows, it is convenient to consider u not as a map from sn into lRn+l, but rather as an Sn-valued map. With tills in mind, the jacobian matrix of u is n x n, and (2.4) rewrites (2. 12)
deg u
=
Jac u.
THEOREM 2.6. Assume that u E wl,n-l (sn; §n ) is such that all the (n - 1 ) x (n - 1) minors of its jacobian matrix are in Ln/(n - 1) . Then the right-hand side of (2. 12) is an integer.
420
PETRU MIRONESCU
Note that the hypotheses imply that Jac u E LI . The argument relies essen tially on the area formula of Federer: if Jac u E Ll , then there is an integer-valued f 0 u(x) Jac u(x)dx = L l -function d on §n such that f (y)d(y)dy whenever
f : §n
�
J
� is smooth. The theorem amounts then to proving that d is constant.
The VMO degree and the degree defined in the above theorem are not related: if u E VMO, we need not have u E w 1 ,n- J . Conversely, a map that satisfies the assump tions ofthe theorem need not belong to VMO: pick a map 'l/' E W l, 1 (§2 ; ( - 1 /2, 1/2» which does not belong to VMO, and set u = (1/J, /1 _ 1/J2 , 0), which is §2-valued. Then the first order minors of u are in I} , since 1/J E W l l , while its jacobian de terminant vanishes, since u is §l-valued. Clearly, u does not belong to VMO. This leaves us with the following very vague question. ,
OPEN PROBLEM 3. Is there a "unified" degree theory?
The above theorem was generalized by Giaquinta, Modica and Soucek [GMSl]. Vie do not quote here their result, which needs notions of cartesian currents to be stated.
2.6. Degree beyond VMO. Let u be a map from §n x (0, 1) k into §n If u is
continuous, then one may define a degree of u as follows: fix any A E (0, l)k and set deg u =deg u(·, A). By homotopical invariance of the degree, this definition does not depend on A and yields a degree which is continuous for the sup norm. The same can be done if u E ws,P, with sp > n + k. Indeed, by Gagliardo-Nirenberg we may assume s < 1. By trace theory, >. u(·, A) is continuous from (0, 1/ into ws- k /p,p, and thu� into VMO. The degree being continuous for the BMO norm, we derive that deg u ( · , >. ) does not depend on A. Thus we may define a degree in WS,P if ws,p embeds into VMO (which is t.he same as sp > n + k). It turns out that the condition sp > n + k can be relaxed. In special cases, the following result was obtained by White [W] ; see also Rubinstein and Sternberg [RSJ . The general case is taken from [BLMNJ. ......
2.7. A ssume that sp > n + 1 . Let u E ws,p (§n x (0, 1) k ; §n ) . Then there is an integer d such that deg u(-, A) = d for a. e. A E (0, l)k. Thus, we may define the degree of u as this integer. In addition, the condition sp > n + 1 is optimal and the degree is continuous for the W·,p-norm. THEOREM
=
1 is settled by the discussion at the beginning at this section. Assume thus k > 2. Note that, for a. e. >., u ( · , >.) E WS,P cVMO, so that the map 1/J given by 'rjJ(A) =deg u(·, A) is defined a. e. With some work, olle may prove that 'r/J is measurable. For a. e. t l , . . . , ti- l , t i+ 1 , ' " tk E (0, 1), the map u( . , t l , " " ti - 1 , ' , tHI , · . t k ) is in W" p . If this is the case, then 1/J(tI , . . . , ti-1 , ' , tHJ, ' " t k ) is constant a. e. (cf the case k = 1). Existence of the degrcc follows from the following elementary
PROOF. The case k
.
1/J : (0, l)k � lR be a measurable function such that for a. e. t k ) is constant a. tl , . . . , ti- l , ti +l > ' " tk E (0, 1 ) , the map 1/J(t1 , . . . , ti- I , ', ti + l , e. Then 1/J is constant a. e. LEMMA 2.1 .
1,et
.
.
·
To prove continuity of degree, let Uk � u in W s ,p . Possibly after passing to a u ( - ' >') in W8,p for a. e. A. We conclude subsequence, we then have Uk (', A) -+
SOBOLEV MAPS ON MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
421
using continuity of the degree in VMO. Incidentally, this proves continuity for the Wr,q-norm as soon as rq > n. To prove optimality, let k = 1 , e E §n and u(x, A) = (x - 2Ae)/lx - 2Ae l . Then 1 , if oX < 1/2 . u E ws,p if sp < n + 1, while deg u ( , A) = D 0, if oX > 1 /2 •
2,7. General manifolds. One may define the Brouwer degree for continuous maps from M into N, provided these manifolds have same dimension, are compact, oriented and without boundary. This degree may be extended to VMO maps as follows: density of smooth maps in VMO(M; N) follows from (2.3). Existence of degree was proved by Brezis and Nirenberg [BNI] . THEOREM 2.8. The map GOO (M; N)
norm.
:1 7L
>->
deg
n
is ClJntinnous for the BMO
Their proof does not use integral formulae for the degree; yet, this can probably be done this way. The above result allows to state, e. g., Theorem 2.7 for maps from M x (0, I)k into N. It is also plausible that Theorems 2.3 and 2.6 are still valid for maps from M into N. For a map u from §n into itself one of the interests of the degree is that it describes the homotopy class of n : HopE's theorem asserts that if two continuous maps have the same degree, then they are humot.opic; t h is holds also for VMO [BNI] . In general, it is more natural to replace the degree with the homotopy class. This can be done indeed: one can associate to V MO maps a homotopy class, which is continuous with respect to BMO convergence ( this is obtained by copying the proof of Theorem 1 in [BNI]) . By mimicking the proof of Theorem 2 .7, one may prove the following result, essentially dne to White [W] ( see also [HaL 2]) : ,
THEOREM 2.9. Let
x
(0, 1)k; N), with M, N compact (but not necessarily of the same dimension) and sp >dim M + 1 . Let, for A E (0, I)k, d(A) be the homotopy class of u(· , A) in G(M; N) (this is well-defined when n ( · , A) E W',P, thus a. e.). Then d is constant a. e. We may thus define in this way [u] , the homotopy class of u . In addition, u >-> [u] is continuous for the WS'P-convergence. U
E
WS,P(M
3. §l-valued maps: lifting The question we address here is: given an §l -valued map u, can one find a lifting of u as smooth as u? In thc well-known case of continuous maps on domains in ]Rn , the answer is positive locally (i. e. on balls) , while globally the answer may be negative, due to the topology of the domain. In the context of Sobolev spaces, the local problem is already interesting. We aHHume throughout this section that u : C §l , where G is the unit cube in ]Rn . ---+
3.1. Lifting of Sobolev maps. Here u E W',P, and we look for a real function 'P E WS,p such that u = et'P. We start with the question of the uniqueness: if 'Pi , 'P2 E W" p lift u, is it true that 'Pi - 'P2 is a constant multiple of 27r7 This amounts to proving that, if 'P : G Z is in �Vs ,P , then 'P is constant . ,
---+
422
PETRU MIRONESCU PROPOSITION 3 . 1 . The only functions in WS,P( C; Z) are constants if and only
if sp > 1 .
In special cases, this was proved by Hardt, Kinderlehrer and Lin [HKL2] and Bethuel and Demengel [BD] . The general case is from [BLMN].
PROOF. Let Q 1 . Suppose now sp > 1 . Fix some i E { I , . . . , n} . For almost every E (0, 1), j of i, the map v given by vet) t, belongs to WS,P«O, 1 ) ; Z). If we mollify v , implies that the maps VE are close to Z; since these maps are smooth, they have to be close to a fixed integer when e: is small. Passing to the limits, we find that is constant a.e. Lemma 2.1 implies that 'U is constant a. e. D
Xj
= u(xt, . . . ,Xi-!, Xi+b . . . , xn)
(2.3) v
We next give an example of map with no lifting. Let : B2 §I , which belongs to WI,I. We claim that TL has no lifting in Argue by with r.p E For a. e. l' E (0, 1), restricted to the contradiction: circle C(O, 1' ) has a continuous representative, still denoted r.p, such that everywhere on C(O, 1') . This is impossible, since on any such circle the winding number of is 1 , while the one of e'CP is 0. In general, the question of existence of r.p was settled in [BBM1].
u
z/Izl,
u = e'CP,
WI,I.
u(z) =
-->
WI,I.
r.p
u = e'CP
u
a
3 . 1 . There is 2 and sp E [1, 2)
a) n ::::: or b) n > 2, s < 1 and sp E [2 , n). THEOREM
lifting r.p E WS,P for each
'u
E WS,P except when:
PROOF. Several cases are to be considered:
(i) When sp > n, is continuous. We then take r.p to be any continuous lifting of u; it is easy to see that r.p E W" p . (ii) When s > 1 and sp > 2, the idea of the proof goes back to a paper of Carbou [C] . Assume that r.p exists. Since u = e''P , we find that so that here it is where the hypothesis s > 1 plays a role. with The idea is then to solve the equation Dr.p and to prove that the solution is essentially the one needed. One may prove that E ws-I,,,; tills relics on multiplication properties of Sobolev spaces. On the other hand, is a closed then vector field, i. e., Formally, this is clear: if = = + Therefore (at least if is smooth) =
u
Dr.p = F,
Du = wDr.p,
F = -zuDu;
=
F,
BFi/Bxj BFj/ox;. F u I Du2 - U2 DuI. u (3. 1) of;jBxj - OFj/OXi = 2(out/OXjOU2/0Xi
F
F u U I W2 ,
-
out /8x/Jud8xj ) .
The rigorous justification of this equality is obtained by approximating u with smooth maps and requires sp > Next the right-hand side of (3. 1 ) vanishes. Indeed, is §l -valued, so that DU l and are collinear. A variant of Poincare's lemma allows then to write for some 'lj; E Finally, the map ue-'.p is constant (it is easy to check that its gradip.nt vanishes) , so that + C, with C appropriut,e constant, is a lifting of (iii) The case sp = n. In order to keep the presentation simple, we consider the special case of H 1 /2 maps on an interval; the general case follows the same lines. set v(x, e:) We regularize By trace theory, v E Hl ( (O, 1)2). Since EVMO, v has modulus close to 1 for small c . Thus w = is HI and of modulus 1 in (0, 1 ) x (0, 8) for small 8. In view of case (ii) , we may write
2.
DU2
u
W8,p.
u.
r.p = t/J
u:
U
F = D'lj;
= u * Pe:(x).
v/lvl
w = e'.p,
SOROLEV MAPS ON MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
with 1jJ E HI . By taking traces, we have U = e''P , with cp =tr 1jJ E H l /2 . ( iv) Cases a) or b) . Explicit examples show non existence. (v) The case sp < 1. It is the delicate one, and we refer to [BBMI] for details.
423 D
3.2. Estimates for the lifting. Once existence of lifting is established, the The proof natural question is whether we may estimate Icpl w in terms of I nlw of the above theorem is constructive, and yields estimates except when we are in the critical case sp = n. Actually, in this case there is no estimate. Here is an example. Let n = 1 ( so that C = (0, 1)) and let 1 < p < 00. Let cpdx) = if 0 < x < 1 /2 0, 2k1r(x - 1/2), if 1/2 < x < 1/2 + l/k and set Uk = e''I'k . It is easy to see that otherwise 211" , {Uk} is bounded in W1 /p,p . Since CPk helongs to W l /p ,p, any lifting of Uk in W I /p,p is CPk + C, by Proposition 3.1. It is easy to see that { lcpklw1 /p,p } is not bounded. Thus, in this limiting case there is no control of 'P in terms of u . However, the above CPk'S are bounded in W1, 1 . This suggests that there is a control of the phase, if not in Wl /p,p, then in a space containing WI /p,p and WI,I. For some values of p, this was proved by Bourgain and Brezis [BB] . • .p .
• .p
as
THEOREM 3.2. Let 1 < p < 2. Then each u E W I /p,P ( (O, 1); §1) may be wr'itten
(3 2 ) U = e C'P l + 'P 2 )
where I CP1 1w1,1 < C I'IL I �! 1 I1'" , and I cpz l}Vl/P,P < C lulw1Ip,p · More generally, one may replace (0, 1) by (0, l) n , but then U has to be in the clo,g7J,re .
'
,
of §l-valued smooth maps.
Their result is stated only for p = 2, but the proof works also when 1 < p < 2.
P ROOF . The construction in [BB] is explicit, but not elementary; it relies on a Littlewood-Paley decomposition of u . In one dimension, it is easy to establish a weaker form of (3.2), namely U = e''P, with I cp l w1,1+W1 /P,p S C ( l ul w / p p + l ul�r1;p,p ) · For simplicity, we work only with p = 2 and u E Coo, though this is not relevant. Let cP be any smooth lifting of u . It suffices to prove that I cp'IL '+ H -l / " < C ( lulHl / 2 + lul�'/ 2 )' By duality, this amounts to proving
(3.3)
(3. 4)
1
1 1 cp'(
Using the identity
11
cp'(
S
' cP
,
C (luIHl/2 + lul�'/2) ( lI ( I I L'>O + 1(IHl/2 ),
=
=
'
-lUU ,
E CO' (O, 1 ) .
integration by parts shows that,
1 u' (u() 1
' 2 dimensions, it is unclear whether D Dcp E Ll + H-1 /2 implies cP E W 1 , 1 + H l/2! OPEN PROBLEM 4. Is it true that Theorem 3. 2 is still valid for p > 2 ?
This is not known even in one dimension. An estimate weaker then (3,2) was established in [BBM3] , Theorem 0 . 1 .
424
PETRU MlRONESCU 3.3. Lifting of VMO maps. This was settled in [BNI] .
THEOREM 3.3. Each u EVMO(C; §l) has a lifting
constants.
'P EVMO, unique modulo
PROOF. Uniqueness comes from the fact that integer-valued VMO maps are constant, cf proof of Proposition 3 . 1 . Concerning existence, the idea is to regularize u. l:
0 and y E (O, c:t , The cubes C = Cm,o,y = Y + c:m + (0, c:)n , with m E zn , cover ]Rn ; let F = Fe,y be the collection of these cubes. The (n - 1)-dimensional skeleton Cn-1 of F is the lll1ion of t he faces of the cubes. The (n 2)-skeleton Cn2 is the union of the boundaries of these faces, and so on. The O-skeleton Co is formed simply by the vertexes of the cubes. -
PETRU MIRONESCU
428
By Fubini, one may find y = y (e) such that lul�ll , p(C�_ l) < C/e. For such y, u is in W l ,p (thus Holder continuous) on en-I . The idea is to approximate u on each cube C without modifying u on ac, and to glue these approximations, The way the approximation is performed on C depends on how much u oscillates on C. With 0 > 0 small, a cube C is " bad" (=u oscillates a lot on C) if lul�!1'p ( c > ) oen-p+ 1 or lul �!l 'P(8C) > 8en- p- l ; the choice of y implies that the union A = A,; of bad cubes is small (its measure tends to 0 with e). The remaining cubes are "good" , On a bad cube, the main care is to construct a map with few singularities and not too large norm; since there are few bad cubes, this will suffice. E. g., one may
consider a solution
v
r IDw l P ;
of min
le
w
;
C
-;
N, w
less energy than u. By a deep regularity result [HLJ ,
v
= u
on 8C
;
thus
is continuous in
finitely many points in C. It is easy to see that l l DU - Dv l P -; 0,
C
v
has
except
If C is good and 0 sufficiently smail, then the image of 8C is contained in some small ball B, by the Sobolev embeddings; the center of B will depend on C, but not its radius r . If B is the ball concentric to B and twice larger, it is possible to project N on B n N through a map , Lipschitz uniformly in B (flatten locally N, next take the neare:;t point projection; this is where smallness of balls is needed) . On C, we approximate u with VI = 0 u, which agrees with u on 8C. The lIlap VI is B-valued, Morally, this means that VI is IRk -valued; a standard technique allows then to approximate VI with a continuous map v agreeing with u on 8C. (A similar argument appeared in the proof of Theorem 2.1.) One has to check next that VI is close to u. These two maps differ only on the set D where (u(x» # u(x). If X E D and if C is the good cube containing x, then -
-
(4 .3 )
I U I Wl,p(E) _ ue >
THEOREM 4.2. Let 1 < P < n =dim M and I = n - [P]
-
1 . Then
R = {u E COO (M \ A; N) ; A = finite union of I-dimensional submanifolds of M}
is dense in WI,P(M; N ).
Gluing is a key ingredient in the proof. In WI ,P, this works since two W1,p maps defined on neighbor cubes which have the same trace on their common face are in W l ,p of the union of cuhes. This leaves the hope of adapting this argument in ws,p for s < 1 + lip; when s > 1 + lip, higher order traces appear, and the method needs to be supplemented with entirely new ideas.
SOBOLEV MAPS ON MANIFOLDS: DEGREE , APPROXIMATION, LIFTING
429
4.1.3. Homogeneous extensions method. This works only for s < 1 and is taken from [BBM4] . We explain it when M = �n and sp is not an integer. Let m = [sp] . For a. e. y, the restriction of U to the m-skeletoll Cm of re,y is in W >,P , thus uy,£ as follows: on Cm , uy = u. continuous. For any such y, we define uy Assuming uy defined on Cj , j < n, we proceed to define uy on an arbitrary face F of Cj+ l . If z is the center of F and t E of, we take uy uy(t) on the whole segment from z to t. (This is the " homogeneous extension" technique.) Vie extend in the same way uy from Cn- l to �n, and we end with a map defined in �n . If, for example, m = n - I , then uy is continuous except at the centers of the cubes C. In general, ?Ly is continuous except a countable union of (n - m - I)-planes. Next, the key ingredient is the estimate =
_
f(O,e)n
Iuy,£ - u l � dy -> 0 • .p
as c
->
0,
valid when sp < n and s < 1. It implies that we may find y = y(c) such that Uy("),,, -+ n in ws,p Further mollification allows to replace the uy s by maps that are CDC outside a countable union of (n m - 1 ) planes, and thus prove the following '
-
THEOREM 4.3. Let sp < n =dim M, s < 1 and l = n - [sp] - 1 . Then
n {u E COO (M \ A; N) is dense in W>,P(M; N) . =
;
A
=
finite union of I-dimensional submanifolds of M}
Since gluing is part of the method, one may hope to use it for s < 1 + 1/p. This will not work, even for s = 1: for u E CIf \ {O}, we have ID(uy - U)ILP > C > O.
4.2. Density of smooth maps. We step forward Direction I: density of COO ( M ; N) in X WS,P (M; N). The main known result concerns Wl ,p . It Y was obtained in [Be2] , but both the statement and the proof were incomplete. The corrected result is due to Hang and Lin [HaL2] . It is simple to state only when n - 1 < P < n. =
X
=
THEOREM 4.4. Let n - 1 < P < n =dim M. Then Y = COO (M; N) is dense in
=
WI,P(M; N) if and only if the homotopy group 7rn-1 (N) is trivial.
PROOF. Assume that 7rn- l (N) I- {O}; we will construct a map u which cannot be approximated with smooth maps. There is a map v E CX?(Bn - l ; N) , identically equal to a constant C near §n-2, and such that v is not homotopic to a constant. After locally flattening M, we may assume that M contains Bn Let D (respectively E) be the cone with vertex the North Pole (respectively South Pole) of §n - l and base Bn-l , and set F = D U E. On each horizontal slice S of F, we may transport v by translation and dilation. If u is the map obtained in this way, then, for - 1 < t < 1, the restriction of u to F n {xn t} is smooth and not homotopic to a constant. We extend u to M with the value C outside F; then u E l-V1,p. Assume now by contradiction that Y is dense in X. Possibly after passing to a subsequence, we find {Uk} C COO (M; N ) such that, on a. e. S, Uk u in W1,p , thus in VMO. The homotopy class being stable with re�pect to VMO convergence (cf Section 2.7) , we find that, for large k, Uk IS is not homotopic to a constant, which is absurd. Conversely, assume that 7rn -l (N) = {O}. It suffices to prove that maps in R can be approximated by smooth maps. We explain the method when u has only one singularity; in the general case, we apply this procedure near each singular point. We may assume that M contains Bn, and that the singular point is the =
-+
430
PETRU MIRONESCU
origin. For a < l' < 1 , we extend U from S(O, r) to B(O, r) by homogeneous extension (thus the extension Vr is constant on rays). Let Ur be the map that equals U outside B(O, 1') and Vr inside B(O, 1'). By a Fubini type argument, there is U in lV I ,p . It suffices thus to consider maps U a sequence rn � a such that Urn which are smooth in M \ B (O, '1') and homogeneous in B(O, 'r ) ; assume, e. g., that r = 1 . The map v = 'U lsn -l is smooth and, as a continuous map, homotopic to a constant C. By regularization, there is a homotopy H E COO ( [0, 1] x §n -\ N) such that H(l, , ) = v and H(t, ') = C for t < 1 /2 . For 0 < £ < 1 , we define outside B(O, £) u(x) . It is easy to see that Ue -> 1L in W l ,p. uE (x) = H( lxl /£ , x/lxl ) in B(O, E ) Since the ue ' s are continuous, they can be approximated with smooth maps. D .......
When I < P < n - 1 , the condition 7l"[pl ( N) = {O} is necessary, but not sufficient for density; we send to [HaL2J , Section 6 for details.
OPEN PROBLEM 8. Find, for arbilmry s, p, M and N such that sp < n =dim
M, a necessary and sufficient condition for the density of Y in X .
The answer is known when s < 1 [BBM4J, and the density condition depends on the value of sp. For example, the analog of Theorem 4.4 is
THEOREM 4.5. Assume that n - 1 if 1rn- 1 (N) = O.
::;
The answer is also known when N
=
sp < n . Then Y is dense in X if and only
§ l [BBM4] :
THEOREM 4.6. Assume that N = §l . Then Y is dense in X except when n > 2 and I < sp < 2.
We emphasize the fact that even for the space W2,p the answcr is not known. It is quite likely that understanding this case will unblock the general situation.
4.3. The singular set of a map. Assume that Y is not dense in X , but that we are able to approximate a map U E X with maps 7J.k in the class R. Question:
can one " pass to the limits" the singular sets of the uk 's7 If so, one has a natural notion of singular set of u, and can even dream of proving that u is in the closure of Y if and only its singular set is cmpty. (Thus this question is related to Direction 3 . ) Most of the work in this direction has been done when N = §k. Except at the very end of this section, we let N = §k. To start with, we take n = k + 1 and M = §k+ l ; however, M could be any (k + I)-dimensional manifold. The maps we consider are W l ,P, with k ::; p < k + 1. (When p is not in this range, Y is dense in X.) Actually, it suffices to know how to pass to the limits the singular set when k u E Ufl, ; we take thus p = k. In this case, R consists of maps u smooth outside Home finite set A = A(u) . To each a E A, we may associate a degree, defined as the degree of u on a small geodesic sphere around a on §n , positively oriented with respect to the outward normal at a; this integer is independent of the small sphere and will be denoted deg (u, a ) . Brezis, Coron and Lieb [BeL] discovered the fact that the singular set A can be obtained from u via all analyl.ic formula. deg (u, a) 6a and Ck = l /l §k l , then the action of More specifically, if T = Tu = the distribut.ion
(4.4)
T(()
T is given by
=
-Ck
L
aEA
( OJ ( det(olu, . . L J5k+1
.
, OJ- l U, u, OJ+lU, . . . , 8k+ I 1L) ·
SOBOLEV MAPS ON MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
431
Here, the derivatives are computed in an orthogonal positively oriented frame. Note that the right-hand side of (4.4) makes sense for u E Wl,k If we endow WI,oo (§k+I ; lR.) with the semi-norm ( I--> IID(lI v'o , then T given by (4.4) lies in (Wl ,=) * , and depends continuously on u. Since each u may be approximated by maps in n, one may intuitively think of T as an infinite sum of Dirac masses. This is indeed correct.
PROPOSITION 4 . 1 . For u E WI,P(§k+\ §k ) , one may write Tu for two sequences {P;}, {Nd
PROOF. Assume u E
n.
c
§k+ 1 such that I Pi - Ni I < 00.
L
Then
aEA
deg (u, a)
=
=
L(OP, - ON. ) ,
O. This may be seen either
from topological considerations, or by noting that T(l) = O. Thus the points in A counted with the multiplicity of their degree (a point with degree 2 appears twice as a " positive point" , a point wit,h degree - 1 appears once as a " negative point" , a point of degree 0 does not appear at all) form a list PI , . . . Pm , NI , . . . , Nm of positive and negative points, the positive points being as many as the negative ones. With the points in A listed in this way, we have T«()
m
=
L «((Pi) - ( N; » . i=1
The key ingredient is the following sup-inf inequality devised in [BCL];
( 4.5)
sup
m
L «((P ) - ( Ni» i=1
i
;
ID(I < 1
here, d is the geodesic distance on §k+1 and Sm is the mth symmetric group. Formula (4.4) gives ITu«()1 < CIiDu ll tk IID(II Loo . In view of (4.5), T may be written as T«() = «((Pi) - ( Ni» with 'L d(Pi, Ni ) < CIiDulltk ' A Cauchy sequences argument, combined with the fact that the geodesic distance is equivalent to the 0 Euclidean one, allows to conclude.
L
is tempting t.o consider the set { Pi} U {Ni } as the singular set of u. This is not realistic, since there is a high degree of non uniqueness in the choice of these points; see Ponce [PI] for a thorough discussion on the infinite sums of Dirac masses. In a somehow non intuitive way, one has to identify the singular set of u with the distribution Tu ; when u E n, Tu can further be identified with a set of points. Proposition 4 . 1 has a converse [ABO]: given sequences { Pi } , {lVi} C §k+1 such that I Pi - Ni l < 00, there is a map u E W1,k(§k+\ §k ) such that Tu = L(Op, -ON.). The map u is explicitly constructed using the " dipole construction" in [BCL] . All the above results can be summed in the following IL
L
PROPOSITION 4.2. The map u (WI,OO)', and its mnge is
Tu is continuous from Wl,k (§k+1 ; §k) into _,.------, ( W',oo ) ' I-->
L (Op, - ON.)
•
finite
It turns out that one may define Tu for u E ws,p when k < sp < k + l [BBM3] . By Sobolev and Gagliardo-Nirenberg, it suffices to consider the case sp = k, s < 1 .
432
PETRU MIRONESCU
4.7. Assume that sp k and s < 1 . Then the map u Tu , initially E 'Il, extends by density as a continuous map from W8,p(§k+l ; §k )
THEOREM
defined for
U
=
into ( WI,DO Y , and its range is
-,----____ ( w"
L (OP. - ONJ
finite
f-t
=)-
Formula (4. 4) does not make sense for u E W·,p. The idea is to find another formula for Ttl; this is very much in the spirit of Section 2.2. We take v / l ul, if lui > 1/2 u , with u the harmonic extension of u. Let as there, i. e., v = 2u, if l ui < 1/2 also � be any smooth extension of ( to Bk+2. For u in W I - 1/ (k+1),k+1 n Wl,k, u smooth outside a finite set (call such a 'U a good map), we have, with dk (k + 1 )Ck ' PROOF.
=
this can be easily checked for smooth u, next by approximation. Since good maps are dense in WS'P(§k+l; §k ) , it suffices to prove that the right-hand side of (4.6) depends continuously on u and �; this is done as in the proof of Theorem 2.3. Finally, the range of u f-t Tv- is determined by adapting the dipole construction. 0 While the above result allows to define Tu in ws,p if k < sp < k + 1, it says nothing about the set of all the T" 'so
T k
Assume that k < sp < k + 1 . Charar:te1"ize the distributions which are of the fo rm T = T" for' some u E W ',P (§k + \ §k) . OPEN P ROBLEM 9 .
The answer is not known even when s 1 . Parti al results were obtained, for 1 , by Bousquet [Bo] . E. g., we may find u E Wl,p (§2 ; §1 ) such that T Tu if =
=
�,(W',p/(p- l ) r
-,-
L (op, - ON, )
=
_ _ _ _ _
and only if T E
finite
range of 1L ...... Tu is
. Presumably, in W1 ,p(§k+l; §k) the
L (Op, - ON. )
finite
The following result, due to Bethuel [Bel]' suggests that Tu really describes the singular set of u. THEOREM
4 .8. Let u E X
=
Wl.k (§k+l ; §k ) . Then u E Y if and only ifT"
=
O.
When u E C''''', we have Ttl 0; by continuity, T" 0 if u E Y. The key ingredient in the proof of the converse is the following result, whose proof relies on an explicit construction similar to the dipole one. PROOF.
=
=
LEMMA 4.1 . If A is the singular set of u E 'Il, then there is a map v E W1,k (§k+\ §k ) , locally Lipschitz outside A, such that deg (v, a) 0, a E A, and IID(u - v) 1I1. < C I i Tu ll(Wl.OO)" =
Restricted to a small geodesic sphere S around some a E A, the above v has degree 0; thus VIS is homotopic to a constant. The proof of Theorem 4.4 shows that v E Y . Lemma 4.1 implies that dist (u, Y) < CIiTu ll ��,,= ) , (here, the distance is computed with respect to the W1,k-semi-norm) . By continuity, this inequality holds for each u E W1,k. This completes the proof. 0
SOROLEV MAPS ON MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
433
The proof of Lemma 4.1 works only for p = k. It can be adapted in ws,P, but only if sp = k (see, e. g., [BBM2]) . This leaves us with the following . OPEN PROBLEM 10. Assume that k < sp < k + I and let 1L E ws,p (§k+ l ; §k ) Is it true that u E Y if and only if T" = 0 7 The answer is yes when s < I Ponce [P2] or when k = 1 [Bo] . We next consider maps in w l ,k (§nj §k ) , with n > k + 1 . In this case, the singular set of a map u E R is an (n - k - I )-dimensional manifold A . We may still associate to u an object Tu, but this time it acts on (n k - I ) forms ( =sections of An- k -1 (T* (§n» , not on functions. For simplicity, we explain how this is done when n = k + 2. (In higher dimensions, the ideas are the same [JS] , [HaLl], [ABO] .) The singular set A of 1L is a finite union of compact flimple curves. To start with, assume that A consists of only one curve, say r. We choose an orientation on r and let T be the tangent unit vector positively oriented on r. \Ve may define the degree of 1L around r as follows: we take an ( n - 1 )-submallifold P of §n , transversal to r at some point x, and oriented positively with respect to r (i. e., the orientation on Tx (r) x Tx (P) is the positive one on §n) . This orientation induces a positive orientation on small geodesic spheres S on P around x . We define degen, r) =deg u l s ; this integer does not depend on x or S. Then the object associated to u is T = Ttl =deg(u, r)Tor, with Or the Dirac mass on f (=the I-dimensional Hausdorff measure restricted to r). When A = U f;, we let T = deg(u, f;)n)r, . This object acts on I-forms, i . e. on smooth sections W of the -
L
cotangent bundle to §n through the formula T(w) = (In case of p-dimensional manifolds, p >
formula T(w) =
L deg(u, I';)
r
Jri
Ldeg(u, fi) l. < w, T > ds .
2, one defines similarly
(4.7)
T (w ) = -Ck L t ; here, T is a unit p-vector positively
oriented, and W is a p-form.) If wc write, in an orthogonal positively oriented frame, Wij = OiWj OjW;, then the analog of (4.4) is -
-
v
as in (4.6), !1 a
(Bn+l !1ij det (ol v , . . , 8i-1V, Oi+1 V , . . , Gj - 1 V, Oj+1V, . . , On+1V) . J
Recall that, in case of point singularities, positive and negative points in A are in equal number. This may be translated as A = oe, where C is a union of curves in §n, each one with starting point a negative point and endpoint a positive point. Here, the boundary has to be understood in the distributions sense: if we orientate each curve from the negative to the positive point, then it defines a current (still denoted C) as above, and the equality A = oC means Tu (() = C(d(), 'if ( . The count.erpart of these properties in the case of curves is that that the fi'S are closed, which in turn implies that we may write Tv. = as, where this time S iD the current associated to a finite union of surfaces. The analog of (4.4) is
(4.9)
sup{Tu (w) ;
I dwl
Tv. is continuous from WI ,k(§k+2; §k ) into the dual of Lipschitz I -forms, and its range is {8S ; S is a rectifiable surface} .
The delicate part of the proof is the construction of u when S is given; this is done by adapting carefully the dipole construction, A similar statement holds in higher dimensions. A straightforward adaptation of the proof of Theorem 4.7 gives the existence of Tv. when u E yVS,P and sp k. Thus one may consider Open Problems 9 and 10 when §k+l is replaced by §", with n > k + 2, We end this section by considering the case where §k is replaced by a general k-dimensional manifold N. If n = k + 1 and M is n-dimensional, then maps u with point singularitiPB are dense in Wl.k (M; N), If A is the singular set of u, it is natural to associate to a E A a homotopy class [u] (a), namely the class of u restricted on a small geodesic sphere around a . Thc proof of Theorem 4,4 shows that u E Y if and only if [u] (a) is trivial for each a. It is not known whether one can associate to u a distribution that " hears" the singularities of u. =
OPEN PROBLEM 1 1 . Is there a way to associate to a map u with point singu larities a distribution Tv. supported in the sing1Jlar set of u, depending continuously on the W1,k -norm and such that Tu = 0 if and only if u E Y ?
4.4. Relaxed energy. Though the questions raised in this section make sense for general manifolds M and N, we shall consider only M = §k+ 1 and N = §k;
even this special case is not well-understood, If k < sp < k + 1 , then smooth maps are not dense in X = �vs,p(§k+l; §k), However, one may hope weak density of smooth maps, i. e., that given U E X, there is a sequence {uieJ of smooth maps, bounded in X, and such that Uk U a. e, There is no weak density if sp > k. Indeed, let U ( X' , Xk+2 ) x'/ lx' l , which is singular at the poles of §k+l Argue by contradiction and assume that there is a sequence {ud as above. Then, up to some subsequence, on a. e. geodesic sphere S around the North Pole of §k+1, we have Uk U in ws,k/S (S) , and thus in VMO. This leads to a contradiction, since the degree of u on S is 1 , while the one of the Uk 'S is O. Tn view of this example, from now on sp = k. The relaxed energy introduced by Bethuel, Brezis and Coron [BBe] is " the least energy required to approximate u" : ......
=
......
(4.10)
Ere1 (u) = inf{lim inf l u k l ev
"
p
; {uieJ
C
COC (§k+ 1 ; §k), Uk
�
u
a,e,}.
Clearly, Ere1 (u) > lulfvs,p , If R is dense, then the relaxed energy is always finite; this relies on a dipole construction. It is very likely that R is always dense (cf the discussion aft.er Theorem 4,1), The exact formula of the relaxed energy is known only when 8 = 1 , p = k, This formula is related to the singular set of u and establishes a bridge between Directions 2 and 3. If u E R, let L(u) be the right-hand side of (4.5), In a suggestive way, L (u) is called the minimal connection between the negative points Ni and the positive points Pi [BeL] . For a general u, L(u) is defined as II Tu l l (Wl,�) • .
THEOREM 4,10. In
(4. 1 1 )
W 1 , k (§k + 1 ; §k ) , we have Ere1( u )
=
/ k §k Du k II lltp + l l 2 L ( u ) .
sonOLEV MAPS 01\ MANIFOLDS: DEGREE, APPROXIMATION, LIFTING
435
For k > 2 , this result is from [BBC] . We take k = 2 ; when k > 2, the argument is similar. Inequality < in (4. 1 1) is established, for U E 'R, via the explicit construction of a sequence {ud such that Uk U a. e. and I I Duk 117,2 IIDulli2 + 87r2 L(u). By density, such a sequence exists also for a general u. For >, the key argument is that, for fixed ( with ID(I < 1 , the map u >-+ IIDulliO +87r2Tu «() is lower semi-continuous on the convex set of Hl-maps of modulus < 1 . By taking the supremum over (, this implies thaL u >-+ I I Dulli2 + 87r2L(u) is lower semi continuous. Thus, for any sequence {Uk} such that UTe --'> u, we have (4. 12) II Dulli2 + 87r 2 L(11) < lim inf( llDuk IIi2 + 87r2 L (Uk» = lim inf IIDuk ll i2 . PROOF.
-+
-+
This argument does not apply when k = 1 ; in this case, this result was proved using a different method in [GMS2j; for an elementary proof, we refer to [BMPj. 0 Nothing is known when
s # 1.
.-
We end with the following challenging
OPEN PROBLEM 1 2 . Assume that
u E w',P (sk+1; §k ) , s and k.
we have
Ere1(u)
=
0
0 by (H3)' Therefore b is symmetric and positive definite and by [8, 2, 131, see Theorem 2. 1 of [2], the (symmetric) product ab is positive definite, provided A �-1
that is
A ,-1 A
( ";1 + e - 1 )
0 ,
since we already proved that ab is positive definite. When ° E [�, TJ] ' we can exploit the same arguments in ['T) , O) and [o, �] , using the fact that A is continuous. 0 If a(x) = 1I we have
LEMMA 2.2. (Lemma 2.3.2 of [11)) . A ssume (H3 ) . Then for all � and ,,/ in ]RN , with � =1= TJ,
PARTIAL AND FULL SYMMETRY
441
PROOF. If either of the vectors is 0 the assertion is trivial since A(O) - 0 . Otherwise, since A(t) > 0 for t > 0 and (� , 71) < I� I . 17]1 , we have (A( IW� - A( lrJl)7] , � 7]) -
A( I W I�12 + A(I 7]1 ) 17]12 - A(I W (� , 7]) - A( lrJl ) (� , 7]) > ( IW I � I + ( 17]1 ) 1 171 - ( IW lrll - ( I7] I ) I � 1
=
= { ( I W - ( irJl ) } ( I� I - I7] I ) and the conclusion now comes from the strict monotonicity of .,
o
Exploiting now Lemma 2.1 and Lemma 2.2, we prove the following Comparison Principle, see Theorem 3.3 . 3 of [11] .
PROPOSITION 2.l. Let 0 C l�N be a bounded domain and let 1.1 , v E Wl�';' (0) n C(O) be such that o
(2.2)
( a(x)A(IDuI)Du , D'P)dx
1. Finally, in the case a(x) = p(lxl)lI with p positive and locally bo'unded, the TeSU/t follows 1Jrith the assumptions (Hd , ( H2 ) , (H4) replaced by the weakeT condition (H3) . PROOF. For c > 0, define
f
=
f"
'P = 'Pc
=
=
(1.1 - V - c)+ . and
{x E O : u(x)
-
v ex) > c } .
Then since 1.1 < v on ao we have supp 'P C f and r c c 0 so rp E W5'OO(0). Therefore, recalling that the matrix a is locally bounded, by density arguments we can use 'P as test-function in (2.2) and get r
(a(x) A ( I Du I Du - a (x)A( IDv l ) Dv , Du - Dv)dx
v so that u = v. Now let us define
ii(x)
for any x
u(i)
E O. By the change of variables x -> x it follows that for all
0 u=0
n
1 B (x, U)'P dx n
an,
with B(x, z) E L�c(l1 x R) non-increasing in z . A ssume that conditions (HI ) ' (H2 ) , (H4) are fulfilled and suppose B (x, z ) B(x' , z ) . a(x) a(x' ) , (3 .3 ) =
=
Then u is non-dec1'eas'ing in the C l -direction in n- = {x E n J Xl < a}. The same result holds for solution u only of class WI�''; (l1) n C(n) if (t) const. tp- l , p > 1 . PROOF.
For A < 0 and
x
E
x the reflection of the point By convexity if
By (3.3)
x E l1>.
we see that
x
l1 - , define
(X l + 2( A - X l ) ' x' ),
=
across the plane Xl
then X E
n,
specifically
=
z
be two points in
and so
define
l1>.
=
{x E n :
u(y) < v(y)
=
=
n>..
with y
YI + Zl
u(y)
in nA
in
n-
2 =
Xl < A} .
u(:r,) .
2 . 1 applied
A Then Y
=
can
A . Let
nA , as also of course is u. u > 0 in n , one has u < v on an>. n an and u 0. The papers [5]-[8] use minimization arguments to get these solutions. Bangert [4] used rather different nonvariational methods to get his single transition hete roclinic. More recently it has been shown in [9]-[11] that (PDE) under (Ft}-(F2) together with certain gap conditions possesses a remarkable number of single and multi-transition solutions. These gap conditions are both necessary and sufficient. A mixture of ideas from the calculus of variations, partial differential equations, dynamical systems, and geometry is used to obtain these solutions. Thus the class of equations treated here can be viewed as a proving ground for tools, methods, and results which one might hope for in other phase transition problems. In the remainder of this paper, the various kinds of results that have been ob tained for (PDE) will be surveyed. In §2, Moser ' s results will be discussed more fully and Bangert's work on single transition solutions will be described. A vari ational approach using minimization arguments to find single transition solutions will be presented in §3 to treat the simplest case of Bangert. It requires the use of a renormalized functional. \Ne also discuss solutions of mountain pass type. In §4, "doubly" heteroclinic solutions which require a second renormalization are treated. The final two sections handle multi-transition solutions. There are two basic types of such solutions. Those that lie in a gap are di�cllssed in §5 while those that are monotone and cross gaps are studied in §6.
SINGLE AND MULTI·TRANSITION SOLUTIONS
OF
A FAMILY
OF
447
PDES
2. The work of Moser and Bangert As was mentioned in §l, Moser studied (2) which arises formally as the Euler equat.ion of _
Rn
F(x, u, Du)dx.
The function F(x, z, p) E C2 (Rn x R x Rn , R) and is assumed to be convex and coercive in p, and periodic in XI . ' . . , Xn and z. For simplicit,y take these periods to be one. Furthermore F and itH derivative satisfy conditions implying that weak solutions of (2) are classical solutions. In the spirit of Aubry-Mather Theory, Moser was concerned with solutions, u, of (2) that are (i) minimal and (ii) wit.hout self intersections. Here (i) means for all r.p E WI�'; (Rn , R) with compact support,
(3 ) i.e. for any
f (F(x , u + r.p, D(u + r.p» JR"
- F(x, u, Du» dx > O,
n c Rn with a smooth boundary, u minimizes
l F(x, u, Du)dx
over the class of functions which equal u on 80.. Such minimizers were studied by Giaquinta and Giusti [12J and, as we have learned from L. Nirenberg, even earlier by Morrey. Expressed analytically, property (ii) means for any j E zn and jn + 1 E Z, either u(x + j) - j,,+1 > u(x), == u(x) , or < u(x) for all x E Rn. Assuming the structural conditions for F, among other things, Moser proved THEOREM 1 . If u is a solution of (2) of type (i)- (ii) , there is an M > 0 and unique a E Rn such that l u (x) Q x l :::; M for all x E Rn.
The unique a = a(u) E Rn given by Theorem 1 is called the rotation vector of the solution. Now one can ask whether there are solutions of the above type. As a first step in this direction, Moser proved this is the case for a E Qn. Since it is relevant for the material that follows, the proof will be sketched. However for simplicity assume F(x, z , p) = � lpl2 + F(x, z) , i.e. we are in the setting of (POE) . Suppose Q = (p d ql , ' " , P /q ) , n n where Pi E N, qi E N\ {O}, and Pi and qi are relatively prime, 1 < i < n . In particular if Pi = 0, qi = 1 . Let e l , . . . , en denote t.he usual orthonormal basis in Rn. Set Ao = {u E WI�; (Rn, R)lu(x + q,Ci) = u (x) + Pi, 1 < i = n } . -
•
Let Q = rr� [0, qiJ . For u E An , set
1 ( - I'\7uI2 + F(x, n» dx Q 2
and define (4 ) Then we have:
1° NOt == {u E A"IIOt (u) = co} i 4> , THEOREM 2. 2° Any u E No is a solution of (POE) of type (i) (ii), 3° NO/. is an ordered set.. -
PAUL
448
ll.
!tABINOWITZ
Aside from the fact that u is of type (i) , the remaining assertions of 1°-2° follow from standard minimization and regularity arguments. That u is of type (i) is based in part on 3° which follows from (4) and the maximum principle. H aving Theorem 2 for a E Qn , Moser used an approximation argument to prove: THEOREM 3. For each with rotation vector Ct.
a
E Rn, there is a solution of (PDE) of type (i)-(ii)
It is an opcn question as to whether there is a direct minimization argument giving existence for the case of Ct E Rn\Qn There has been some work in this direction by Bessi [13] in the spirit of the minimal measures of Mather [14] . Henceforth we will only study the case of rotation vector a = O. All the relmlt.s for this case have analogues for a E Qn . To fix the notation, let L( u) = � lV'ul2 + F(x, u) {U E W;�'; (Rn , R) l u is I-periodic in Xl , . . . , Xn } , fo
(5)
Jo (u)
Tn
L(u)dx,
inf JO (U) ,
Co
"Ero
and Mo -
{u
E
fo I Jo (u)
=
co} .
Note that since F is I-periodic in Xi , it is also 1!i periodic in X, for any 1!i E N. In fact there are infinitely many choices for an orthogonal family of directions of periodicity for F. For each such choice,
J L(u)dx
(6)
can be minimized over the corresponding class of "Vl�'; functions periodic in these directions. This produces a solution of (PDE) as in Theorem 2. Any member of Mo is a candidate for the variational problem for the functional in (6). In fact Moser showed the set of minimizers in this broader class of functions is simply Mo. Retum.ing to fo , by Theorem 2, Mo i= ¢ and if u E Mo, by (F2) so is u + j for any j E Z. Since Mo is ordcred, either Mo foliates Rn+l or there are gaps in Mo, i.e. there exist adjacent Vo < Wo in Mo .
(*) 0
If (*)0 holds, we refer to Vo and Wo as a gap pair. In [11], it is shown that (*)0 is generic in the sen�e that if it fails, for any c > 0 and Vn E Mo, F can be replaced by F with IIF - Fllcl(T"+l) < c such that if M is the set for F corresponding to Mo for Fo, Nt = { va + j Ij E Z } . Assuming (*)0 holds, (PDE) was studied by Bangert [4] who fmmd a. solution of type (i)-(ii) which is heteroclinic in X I from Vo to 1JJo a.nd periodic in xz, . , Xn. Thus for all xz, . . . , xn , lim U(x) - wo (x ) . lim U(x) - vo(x) = 0 XJ Similarly there is a heteroclinic from Wo to 'Uo . The proof of these statements involves an approximation argument together with estimates derived by Moser. "
....
"'
...
. .
-. - 00
=
X l -OO
SINGLE
A N Ll M U LTI-TRANSITION
SOLUTIONS OF A FA:vIILY O F PDES
449
RBMAItK 1 . More generally, Bangert showed if �i = �aijei (with a'i.1 E Z ,and having no common faclor for fixed 'i) and the vectors Wi are orthogonal < i < n , then there is a solution of type ( i) - (ii) of (PDE) which is periodic in the W2 " " , Wn directions and heteroclinic from Vo to Wo in t he WI direction. Moreover for each
1
fixed set of wi's, as in 3° of Theorem 2, the set of corresponding solutions is ordered. The effect of varying Wi will be discussed after Theorem 6.
(el " ' " en ) , lct Nl denote Returning to the silIlple�t case of (WI , ' " , wn ) the ordered set of heteroclinics in XI from Vo to Wo which are of type (i)-(ii) and periodic in X 2 , " " Xn· If UI E NJ , so is r:j U1 (x) UI ( x + jel l and as for A1 0 , either Nl foliates the region in Rn+l between Vo and Wo or there are gap pairs in /Vl. If VI < W I is such a pair, B anger t showed there iH it Holution U2 of (PDE) of type (i) -(ii ) which is heteroclinic in X2 from VI to WI and periodic in X3, , , . , X n . SiJl(�e each of VI and W I are heteroclinic in Xl from VI) to W(), so is U2 . Furthermore, as above, the set, N2 , of such doubly heteroclinic U2 's is ordered. Thus if there arc gaps in .11/2 , there are triply heteroclinic solutions, et c . Whenever one has a HOlution , U , of (PDE) that is heteroclinic in X l from Vo to WI and likewise, U* , heteroclinic from Wo to vo , t he theory of dynamical systems suggests that there should be infin it ely many multi-transition solutions that shadow formal chains obt ained by gluing together phase shifts Df U and U·. The simplest kind of such solutions are therefore 2-transition homoclinics from Vo to 1)0 in X l which are periodic in X2 , . . . : Xn . Likewise there should be 2-transition homo clinics from 1)1 to VI which are periodic in X3" . . , X n . Such solutions w ill be disc ussed in §5. While not necessarily corresponding to a formal chain, there are aJso 2 (or possibly more) - transition heteroclinics from '1)0 to 1 + 'll!o which wi ll be treated in §6. Variational methods for finding such shadowing solutions have been developed in various settings, See e,g. Mather [15] and references [7]-[11] . These methods require a minimization formulation of the basic solutions which are used as build ing blocks to construct multi-transition solutions. Thus such a characterization is needed for the members of Nl , N2 , etc. However Bangert ' s existence results are not variational i n n ature Therefore the first step in finding multi-transition solutions is to develop a variational approach to obtain the basic heteruclinics. This will be the object of the next two sections. =
-
.
3. Renormalization and minimization A natural way
to find
heteroclinics from Vo to Wo is to minimize
rJRx
(7)
Tn-l
L ( ) d:r -/1,
over a class of functions with the desired asymptotics in X l . Unfortunately in generaJ the functionaJ in (7) will not be finite for any admissible function. E g. This will be the case for F(x, z ) > b where b is a p ositive constant. Thus (7) must be modified or as we prefer to say, renormalized in order to make the resulting functionaJ finite valued on a reasonable class of functions. Towards that end, let i E Z and Ti = [i , i + 1] x T,,-I , The cla.-;s of admissihle functions wi l l be taken to .
,
450
PAUL H. RABINOWITZ
be { u
lI u and Il u
+ Wl�'; (R x
-
VO Il£2(Ttl
Tn+ l R) l vo < u < WO,
--->
'Wo il£2 (T,J
,
0,
i
0,
i
--->
-
00
---> 00
,
}.
implies u is periodic in X 2 , , X n and the asymptotic behavior is achieved in a mild way. For E Z and U E r 1 Vo , wo) set
N ate
that u E
r 1 (vo , 'Wo)
-
•
i
For p
(5).
< q E Z,
J1 ,p,q(u)
[11J
that
r 1 ( Vo , wo) .
JI ;p , q (U)
•
-
Co
set q
L Jl ,i (u) , p
is bounded from below independently of p, q E
With the aid of Theorem in
2
=
.
(
r L (u)dx JTi
Jl ,i(u) =
where CO is defined by
--->
and some of the remarks following (6), it was shown
Z
and U
This suggests defining
J1 (u)
lim
=
p - - oo
J1;p,q (u).
q - OQ
This definition of .11 �eems awkward but as shown in
(I.{)
J1 (u)
lim
p - - oo
(9)
The properties (9) possessed by
.II
[11],
if J1 (u)
< 00,
then
J1 :p ,q(u)
i . e. the lim inf is a limit and moreover
lI u - Vo I I W,,2(T,J
E
0, i ->
0,
---> -00
2 -> 00. .
suggest that some compactness is built into it .
A� further evidence, define
ct (vo, wo)
(10) Then it was shown in
[11]
=
uEr, (vo .wo) inf
J1 (u) .
that any minimizing sequence for ( 1 0) is bounded in a
WI�'; (R x Tn-l , R). This implies it. has subsequence (Uk) converging weakly to some U E WI�'; (R x Tn-I, R) as k -> 00. But actually (Uk) eonverges strongly in WI�': (R x Tn-I , R) . Thus .11 satisfies a Palais-Smale type property for minirni7,ing
sequences . These facts lead to
4. ( [11] ) : (F2 ) If F satisfies (F1) THEOREM 1°
2°
(*)0 holds, M J (vo, 'WI)) {u E r 1 (vo , wo ) I JI (u) = c(vo , 'Wo ) } i= ¢ Any 1L E M 1 (vo , 'Wo) is a classical solution of (PDE) with 00 (a) I I U vo l l c2 (Td -> 0 , i II U wo llc2(Til -> 0, i -> - 00 (b) Vo < U < T�I U < 'Wo -
and
-
-
->
-
and 3° M1 (vo , wo)
is an ordered set.
-
,
SI' 0
(; E 6(,5, t) such that
and
'rrL2 - 'lnl possibly still larger, there is a Ua E Go (s, t),
(28)
and
(24) To prove Theorem 10, as earlier it can be assumed there is a minimizing se quence for (20) which converges in Wl�'; (R x Tn- I , R) to U E Wl�'� (R x Tn-I , R) with U < 7:'IU and J1 (U) < 00. Using (18) and ( 1 9) leads to (2 1 ) . Therefore U E Ym and earlier arguments show J1 (U) = bTn• Next comparison arguments like those of Theorem 8 show for m2 > > m l , the constraints hold with strict inequality. This fact with a local minimization argument from [7] shows U is a solution of (PDE). Use of the maximum principle yields (22) as in earlier results. Lastly to prove the shadowing estimates (2a)-(24) requires a result of indepen dent interest. A
PROPOSITION 1 . For any E > 0, there is a 0 = 0 (E) such that if u E r 1 (va , wo) with J1 (u) < Cl (vo , wo) + 0, then there is a W E M l (Vo, wo) such that for all i E Z, Ilu - w ll wl .2(Ti_1LJTiLJT,+1 ) < f.
(25)
Theorem 10.
Proposition 1 and comparison arguments then yield the rest of t he proof of
RRMARK 5. (i) Using th at U as given by Theorem 10 sati sfies U < T:'1 U, it is shown in [18] that there is a solution of mountain pass type between U and 7:' 1 U . (ii) One can construct formal chains of solutions of (P DE) from those given by Theorem 8 and 10. The arguments of Theorem 8 can then be employed to get actual such solutions that shadow the chains. Nothing is yet known about corresponding solutions of mountain pass type. (iii) As was mentioned in §2, all of the results of this paper for rotation vector (Y = 0 have analogues for lY E Qn and B anger t has done some work when a E Rn\ Qn. However essentially nothing is known about solutions of the type constructed in §5-6 when a E Rn \ Qn aside from a paper of Kessi
[13] .
(iv) Another completely open question concerns analogues of the results of t his paper for systems. •
PACL H. RABINOWITZ
458
References [IJ
Moser, J . : Minimal solutions
of a variational pr o blem on a torus.
AIHP, Analyse
Nonlineaire ,
3, 229-272 (1986) [2J Aubry, S . , LcDacron, P.Y.: The discrete Frenkel-Kantorova model and its extensions, 1. Exact results for the ground states . Physica D, 8 (1983), 381-422 [3J Mather, J . N . : Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology, 2 1 , 457-467 (1 982) [4J B angert , V.: On minimal laminations of the torus. AIHP, Analyse Nonlincaire, 6 , 95-138
(1989) [5J
Alessio,
F.,
Jeanjean,
L.,
Montecchiari,
Stationary layered solutions for
Var .
,
11,
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[7J
[8J [9J [10J [11 J [12J [13J [14J [15J [16J [17J [18J
a class of nonau
277-302 (2000) Alessio, F., Jeanjean, L . , M ontecchiari, P.: Existence of infinitely many stationary layered solutions for a class of periodic Allen-Cahn equations. Comm. P D . E , 27, 1537-1574 (2002) Rahinowit.z, P.H . , St.redulinsky E.: M ixed states for an Allen-Cahn type equation. Comm. Pure Appl. Math., 56, 1078-11 34 (2003) Rabinowitz, P.H., Stredulinsky, E.: M ixed states for an Allen-Cahn type equation , II. Calc. Var. , 2 1 , 157-207 (2004) Rabinowitz, P.H . , St redulinsky, E.: On some results of Moser and B angert. AIHP, Analyse Nonlineair e, 2 1 , 673-688 (2004). Rabinowitz, P. I I . , Stredulinsky, E.: On some results of Moser and Bangert, II. Adv. Non linear St.ud. , 4, 377-396 (2004) . Rabinowitz, P.H . , Stredulinsky E.: In progress Giaquinta, M . , Giusti, E . : On the regularity of the minima of variational integrals. Acta. Math., 148, 31-46 ( 1 9 82 ) Bessi, U.: Many solutions of elliptic problems on Rn of irrational slope. Comm. P. D.E., 30, 1773-1804 (2005) M ather J.: M ore Denjoy minimal sets for area preserving cliffeomorphisms. Comm. Math. Helv., 60, 508 557 ( 1 985) M at her J . : Variational construction of connecting orbits. Ann. Inst. Fourier (grenoble ) , 43, 1349-1385 (1993) Bolotin, S . , Rabinowitz, P.H . : A note on heteroc1inic solutions of mountain pass type for a class of nonlinear elliptic PDE's. C ontributions to nonlinear analysis, 105-114, Progr . Nonlinear Differential Equations Appl. , 66, Birkhauser, Basel, 2006 Rabinowitz, P.H . , Solutions of a Lagrangian system on T2 . Proc. Nat!. Acad. Sci. USA, 96, 6037-6041 (1999) Bolotin S., Rabinowitz , P.H . : In progress tonomous Allen-Cahn equations. Calc.
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DEPARTMENT OF MATHEMATICS, l' NIVERSITY OF WISCONSIN, MADISON, WISCONSIN 5370 6 RABINowr@MATH . WI SC . EDU
Contempora.ry Mathematics Volume 446, �OU7 •
Some Methods and Issues in the Dynamics of Vortices in the Parabolic Ginzburg-Landau Equations Sylvia Serfaty
Dedicated to Haim Brezis with best wishes on the occasion of his 60th birthday. Thank you Haim, for constantly sharing your enthusiasm for mathematics and pushing them forwU1'd, and for the encouragement and a.ttention you prOVide to many and in particular to young mathematicians.
1. Introduction We are interested in the parabolic flow for the Ginzburg-Landau energy ( Ll )
E,, (u)
=
1 2
n
l \7u l 2
+
( 1 - l u I 2 )2 2c:2 '
.
where 0 is a smooth bounded (simply-connected) domain of � 2 and u maps 0 to 1 is in fact higher (like dt l log c l ) so for a given degree d > 0 on the boundary, in order to minimize the energy, one needs to choose d vortices of degree + 1, and then to minimi7.e the remaining lower order interaction term Wd , which is independent of c: and governs the locations of the limiting vortices. From now on, we will reduce to t.he case di = ±l and will use the following re�ult
(1) Assume E. (u,,) THEOREM 1 .4 (Gamma-convergence of Ginzburg-Landau). $ Cllog c:1 and u. = 9 on an o r �� = 0 on an, then, up to e:r.tmction, curl (iue 'Vue )
and if V i, di
=
±l
lim (E,, (ue)
0-0
-
->.
27f
n
L diOn; i=1
7fnllog EI) > Wd (a 1 , ' " , an ) ,
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
463
(2) For all (ai, d;) , di = ±1, there exists ue s'l.u;h that lim (E. (ue ) - 1rnllog cD
IIV' F (u(t» II�.
(2.5) 2 ')
(construction) If u, � u, for any V E Y, any v defined in a neighborhood of ° satisfying = v (O) U Dtv (O) = V there exists v,, (t) such that v,(O) = u" ,
lim l I atv, ( O) II� < l I at v (O ) IIi-E-O �
lim
,_0
-�
dt I t=o
E, (v,) >
-
�
dt It=o
=
F(v)
11V 1li-- , =
- (V'F(u) , V)y.
Then if D ( O) = 0 (i. e., it is well prepared) we have D (t) inequalities above are equalities and Vt E [0 T), uo (t) ..:i u(t)
=
0 Vt E [0, T), all
,
- V' yF(u) 11.(0) = UJ) at'll
i. e.,
11.
=
is a solution of the gradient flow for F for the structure Y .
SYLVIA SERFATY
466
2.3. Interpretation and further remarks. This theorem means that under conditions 1 ) and 2), or 1 ) and 2') (since 2') implies 2)), solutions of the gradient flow of E" for the structure X, converge to solutions of limiting gradient-flow (for the structure Y) if well-prepared. Let us make a few additional comments: (1) The difficulty is not in proving this theorem but in proving that in specific cases the conditions hold. (2) The limiting structure Y is somehow embedded in the conditions 1) and 2). The time rescalings are embedded in X, . (3) In general we expect 1 ) and 2) to be satisfied for any u, 3. u or u,,(t) 3. u(t) not necessarily solutions (here we required it only for solutions) (4) 1) and 2) do provide the extra Cl-order couditions on f-convergence. 2) in particular implies that critical points converge to critical points. (5) We can weaken conditions 1) and 2) to
lim
t 1l8tu,, 11 2 > t 1 18t u ll 2 Jo
.-+o Jo
-
O (D (t ))
lim IlVx, E. ( u,J llt > IlVF(u) ll� - O(D(t) ) I
$--+0
where D(t) is the energy-excess, and handle the terms in D(t) in the proof via a Gronwall's lemma (finally obtaining that D(t) == 0 if D(O) = 0). (6) The method should and can be extended to infinite-dimensional limiting spaces and to the case where the Hilbert structures X. and Y (in partic ular Y) depend on the point: such as Y" = L� forming a sort of Hilbert manifold structure. It would thus be interesting to see how, through r-convergence, the structures underlying the gradient-flows can become "curved" at the limit, even though they are not curved originally at the [ level, and also become possibly nonsmooth and nondifferentiable. In fact we can write down an analogue abstract result using the theory of "min imizing movements" of De Giorgi formalized by Ambrosio-Gigli-Savare [AGS] , a notion of gradient flows on structures which are not differen tiable but simply metric structures.
2.4. Idea of the proof. Since the proof is elementary, let us see how 1) and 2) imply the result. We assume
8
8
467
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
Then, for all t
F (u(t)) . Therefore we must have equal ity everywhere and in particular equality in the Cauchy-Schwarz type relation (2.6), that is or
1
2
0
t
II\1y F(u) lI� + lI atull} ds
=
1t0 (-\1 yF( u(s) ), atu( s))y
lot 1 1 \1F(u) + atull� ds
=
O.
Hence, we conclude that at u = -\1y F(u), Vi E [0, T). The idea is thus to show that the energy decreases at least of the amount expected, on the other hand it cannot decrease more because of the f-convergence hence it decrease exactly by the amount expected, all along the trajectory.
PROOF OF 2')
;-
2). (2') is a constructive proof of 2)). Observe that here
u. does not depend on time. For every V E Y we may pick vet ) such that
v(O)
at v (O)
i.e., pick a tangent curve to V at u . \Ne v" (t) such that
v. (O)
u
=
V
=
ume there exists (we can construct)
a..�s
u. lim,,-->o l I at v (0) l it < IIVII� =
.
lim,,-->o - ft 1t=o Ee (ve ) > - :ft 1t=o F(v)
=
- (\7yF (u), V)) y
that is a curve v,,(t) along which the energy decreases by at least desired amount. Then, choosing V = - \1y F (u) , we have ( - \7 E. (u,,) , atv,,)x,
=
d . -d E,, (ve) = t 11 0
�
- (\7y F (u , V )y
=
II\7F(u) lI�
468
SYLVTA SERFATY
thus
lIV'y F (u) lI} < (-V'x. Ee(ue ) , atVe (O)))x, < IIV' x. Ee(Ue ) II x. ll 8t ve (0) IIx. < II V'x, Ee (ue) IIx. (I Wlly + 0(1)) . Recalling that V = - V' yF (u), we conclude that IIV' Ee(ue ) lIx. > II V'F(u) lIy + 0(1).
o
The idea was to rely on the fact that steepest descent is characterized as the evolution which maximizes the energy-decrease for a given II Bt ue ll �. We compare it to a test-evolution obtained by "pushing" Ue in the direction V (and in fact choose the steepest descent direction V = V' F ( u)), i.e., find a curve v(t) and "lift it" to a curve Ve that pushes Ue in direction V' y F(u) with a decrease of energy of at least the expected one, and a cost I I Bt v. 1I 2 which is at most the expected one. We can in fact achieve this in such a way that OtU,(O) depends linearly on V . In "pedantic" terms, we show that there exists a linear embedding -
-
Ie : TuN V
) 1-+
Tu, M
atve (O)
which is an "almost-isometry" in the sense : lim I IIe (V) llx£ e---+ O
=
lim I; V' x,Ee(u.) = V'y F (u) . IWlly and e---+ O
2.5. Application to Ginzburg-Landau. In order to retrieve the dynamical
law for vortices, we need to prove that conditions 1) and 2') of Theorem 2.4 can be proved for Ginzburg-Landau. As seen in Theorem 1 .4, we need to consider the energIes 1 = (2 .7) Fe (n) = Ee(n) 7rnl log .01 2 •
-
and F = W (the renormalized energy) so that Fe r ) to define are 1 2 = 11 . 11 £2(0) II . II �, I log .01 N = nn\diagonals (2.8) 2
II . l I y
(2.9)
=
1
7r
F. The structures we need
2 I I . 1 I(&2)n
where n is a prescribed number of vortices of a priori fixed degrees ± l . Applying Theorem 2.4, we retrieve the dynamical law that the vortices flow according to some rescaled gradient-flow of the renormalized energy: THEOREM 2.5 (Ginzburg-Landau vortex dynamics - [Lil, JSl, SS4]) . Let UE
be a family of solutions of
BtU
_.
I log .01
with either
=
Uo
u €
ll.1L + -2 ( 1 =
9
�; = 0
an an
2
- lu i )
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
such that curl (i�LE ' V'uE) (O) -" 27r with a? distinct points in n, di = ±1, and
'"
L d, 8a� i=1
as c
-->
469
0
(2. 10) Then there exists T* > 0 such that Vt E [O, T' ) , curl (iu, V'u) (t) -" 211" as
c -->
(2 . 1 1 )
0, with
dai dt
n
L di 8a,(t) i=1
-
ai (O)
where T* is the minimum of the r;olli.�ion time and exit time of the vortices under this law. Moreover D(t) = 0 for every t < T* . Thus, as expected, vortice� move along the gradient flow for their interaction TV, and this reduces the PDE to a finite dimensional evolution (a system of ODE's) . This result was obtained in [Lil, JS2] , but with PDE methods, it is reproven in [SS4] with the r-convergence energetic method exposed here. By the same method, we obtained the dynamics of a bounded number of vortices for the full Ginzburg-Landau equations with magnetic field, i.e., the gradient-flow of ( 1 .3) , for large applied fields (the result for bounded applied fields had been obtained by Spirn [Spi]) . We will assume that
hex = ..\Ilog EI 0 < ..\ < 00. In this regime we have obtained various results about the minimizers and critical points of .I, see [885, Sl, SSl, SS2, SS3] . The (heat flow) Ginzburg-Landau equations as proposed by Gorkov-Eliashberg are (2. 12)
(2. 1 3)
atu + iu = V'�u + ; (1 - luI2) in n E
atA + V' = V'l.h + (iu, V' AU) (iu, V'AU) . n = 0 h = hex
in n on 8n on 8n.
These are the gradient-flow for essentially the same £2 structure as in the case without magnetic field. Observe that here there is no need to rescale in time to see motion of vortices. The quantity makes the equations invariant under the gauge-transformations : (2.14)
'U
ue1W A I--> A + V'w if> H
\ '"' .c . 2 1TA L.- d, curl
with
,
,
a such that, for all t E [0, T* ) , n
« iue, 'V A,ud + A,,) (t) .....l. 21T L diOai(t)
i= l
Vi
and T* is the minimum of the collision time and of the exit time from 0 for this law of motion. 2.6. Remarks. ( 1 ) The result holds as long as the number of vortices remains the initial one (so that the limiting configuration u = (al , ' " , an) belongs to the same space N) . It ceases to apply when there are vortex-collisions or some vortex exits the domain under the law (2.11), even though these can happen. Then a further analysis is required, see Section 5 below. (2) Under the same hypotheses, if u� is a solution of the time-rescaled gradient flow Otu. = -Ae 'Vx,Ee (u,) with D(O) = a then if Ae « 1, 'ue (t) 3. Uo , 'It, i.e., there is no motion, while if At » 1 , u. (t) 3. u, 'It, with 'Vy F(u) = 0 i.e., there is instantaneous motion to a critical point. Thus, we see that the structure Xe and the relation 1) in Theorem 2 .4 contain the right time rescaling to see finite time motion in the limit. For Ginzburg-Landau without magnetic field, it is necessary to accelerate the time by a Ilog cl factor in order to see motion of the vortices (this is due to the fact that the renormalized energy W which drives the motion is a lower order term in the energy). (3) The method works for Ginzburg-Landau with or without magnetic field as long as the number of vortices remains bounded. It is more difficult to apply to other models such as Allen-Cahn, or 3D Ginzburg-Landau, because what is missing is a more precise result and understanding on the profile of the defect during the dynamics. For example, for Allen Cahn, we need to know that the energy-density remains proportional to
471
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
the length of the underlying limiting curve during the dynamics (which is true a posteriori) . It is also an open problem to apply it when the number of vortices is unbounded as I:: --; O.
3. How to prove 1) and 2') for Ginzburg-Landau 3.1. A product-estimate for Ginzburg-Landau. The relation 1) which
relates the velocity of underlying vortices to OtU, can be read (3. 1 )
1 lim IOtu, 1 2 ds > ,-0 IIog I:: I 10, tl x n
71'
L
0
t Idt ai l2 ds
assuming curl (iu, V'u,) (t ) ---' 2rr L.,i d;ba, ( t) , as I:: --; 0, 'lit. This turns out to hold as a general relation, without asking the configurations to solve any particular equation. It is related to the topological nature of the vortices. It can be embedded into the more general class of results of lower-bounds for Ginzburg-Landau functionals. The setting is now 0, a bounded domain of ]Rn (n � 2) (we will need n = 3), and Ee (u)
=
1 2
i.
1 2 ( 1 - 11.1 1 2 ) 2 . l V'ul + fl 21::2
We define the "current" ju associated to 1.1 as the I-form ju = (iu, du) Then the Jacobian Ju is the 2-form Ju =
� d(ju)
=
=
L., (iu, Ok U}dXk.
�d(iu, dlt) .
It can be identified to a (n - 2)-dimensional current through J'u((/J)
=
� J Ju
1\
l.l l (n ) , lim , �o Ilog 10 1 in 2 an estimate that was previously proved in [JS2] . Our result extends to higher energies EE < N, l log EI with N, un bounded. In that case we just need to rescale by N, and replace .1 by the limit of REMARK.
JUE . N�
3.2. Idea of the proof. The method consists in reducing to two dimensions.
By using partitions of unity, we can assume that X and Y are locally constant. We may then work in an open set U where X and Y are constant. If they are not parallel, they define a planar direction (if I.hey are then .l(X, Y) = 0 and there is nothing 1.0 prove) . \Ne then slice U into planes parallel to that plane. Assume X = el and Y = e2 orthonormal vectors. In each plane we have the known 2D lower bounds of the type
�1
planenu
I V u, 1 2 >
7r
L i d; I llog 101
where di is the degree of the boundary of the balls, constructed with the ball construction method (see [Sa J, 885] ) . This is possible as long as there is a good bound on the energy on that planar slice, and the number of balls can be unbounded. The main trick is to observe that this is true for any metric in the plane, and use the metric >"dx + � dy, leading to ,
�
2
I D1Ue l 2 +
planenU
�1
planenU
I 02U 1 2 2
7r
..
L I di l l log E I ·
Integrating with respect to the slices yields
fu 2�
I Vue X I 2 + .
�
U
2> YI Vu I E •
Optimizing with respect to >.., we conclude that lim 1 1
.. �O
1
og
10
I
U
I Vu, . X I 2
J
U
I V u, . Y I 2
J(X, Y) Ilog 10 1 .
>
J .l(X, Y)
and we may finish by adding these estimat,es thanks to the partitions of unity. u
3.3. Application to the dynamics. In order to deduce a result for dynamics
in 2D, the idea is to use this theorem in dimension n = 3 with 2 coordinates corresponding to space coordinates and 1 coordinate corresponding to the time coordinate (this can be done in any dimension, but we restrict to 2D here for the sake of simplicity). The vortex-lines in 3D are then the trajectories in time of the
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
473
vortex-points in 2D, and clearly the length of these lines is somehow related to the velocity of these points. Doing the coordinate splitting, we write .
) 11,
THEOREM
3.2 ([883]) . Let ue(t, x) be defined over [O, T]
n = 2) and such that
x
n (n
c
]Rn, here
Vt E,,(u,, (t)) < C l log EI [O,TJ x n
IOtu,, 1 2
< Cllog E I
then v"
with
z
--'
V,
Moreover, VX E C� ( [O, T]
x
dt J.L + div V = O. n, ]Rn) , and f E C� ( [O, T]
x
n) ,
1 . hm .,-----, ,,�O I log E I In 2D, the vector V
=
(Vl , V2) really is 1r L, di (Ot ai )Oai(t) , such that
Ot COROLLARY
1r
L d; O",(t) ,
3.2. 1 . If in addition di
=
+
div V
=
O.
± l andVt , � In l \7u,, 12 < 1r ( L I di D I log EI ( 1 +
0(1)), then for all intervals [tl , h) on which the ai 's remain distinct, we have 1 lim IOtu,, 12 > ,,�O I log E I n x !t" t,J
1r
L i
t2
t,
IOtai l 2 dt.
This is the desired estimate 1 ) in Theorem 2.4. To prove this corollary, recall that from (3.3) if E,,(u,,) � lrn l log EI then Ilo� "I In IX · \7u,, 12 � lr Li I X(ai ) 1 2 and optimizing over X and f gives the L2 bound on V. Theorem 3.2 allows one to bound from above the motion of the vortices for solutions of (1.4) . Using the fact that for them 10; " I IoT IOt u l2 is the variation of 1 energy between time 0 and T, it implies the crucial relation (3.4)
-
where l::.p denotes the variation in position of the vortex, l::.T the increment of time, and l::. E the variation in energy.
SYLVIA SERFATY
474
3.4. Proof of the construction 2') . We wish to prove that 2') holds for
Ginzburg-Landau so that we deduce 2) i.e., if UE .!!" U then limE--->o IIVx, EE ( UE ) II X, > II V W ( u) Ily· Observe that this is a static result. We thus assume that curl (iuE, VUE ) ----'27r Li difJai and may consider disjoint balls B(ai , p) of fixed radius p. If IIVE(UE ) I l x, +00 there is nothing to prove. If II Vx, E., (uE) l l x, = 0(1) then we can prove that DE = 0(1) where DE = EE (uE) - 7rn l log 0: 1 - W(u) is the " energy-excess" . For a proof of thifl nlflUlt, see [S3] , it relies on the fact that IIVEE (u) II x, < C means 2 < IO� . 1 = 0( 1) and one can take advantage of the fact u u n + I l .6. :, ( 1 - l I2) I that UE is thus an "almost-solution" . Once this is proved, we may deduce
(3.5) (3.6) where ,
with the appropriate boundary conditions. The rough idea is that V'PE ::::: V � o outside of the vortex balls. Through these relations, everything is well-controlled outside the balls and inside the balls we shall only perform a pure translation. Given V = (VI , . . , Vn) , we want to push each ai in the direction V; . For that purpose, define Xt (x) = x + tV; in each Bi, and extend it in a smooth way outside of the B; into a family of smooth diffeomorphisms that keep an fixed and are independent of 0: . Choosing the deformation .
v. (x, t)
=
uE (X;l (x »
does the job of pushing the vortices ai along the direction V; . However it is not enough, and we need to add a phase correction 'ljit : i (3.7) v. ( Xt Cu), t) = uE (x) e ,pt(x)
so that for every t , the phase of harmonic conjugate of
.6.t
=
VE
is approximately the optimal one, that is the
2 7r � d; clad t)
ai(t)
=
ai + tV;.
,
It is possible to construct 'ljit single valued, independent of 0:, so that V�o + V'ljit
�
V � (t 0 Xt) .
We will now check that the VE constructed this way works. First, 1 2 at v 1 (0) l I log 0: 1 in
r
,,
,
--->
.
DYNAMICS OF VORTICES IN GINZBURG-LANDAU
475
'
because Xt achie\'es a translation of vector V; in the Bi'S while the contribution outside of the Bi 's is negligible; then, we use the relation (3. 3). The first requirement for 2') is thus fulfilled. Let us check the second requirement, i.e. , the energy-decrease rate, by evaluating : t =o Ee (ve (t)) . With a change of variables,
q
Ee (ve(t))
� 10 l \7ve l 2 + 2�2 (1 _ 1ve 12) 2 2 2 2 � 10 I DX;-l \7(ve Xt l l + 2�2 (1 _ lu I ) 0
Now, recall that Xt is a translation in Ui Hi hence I Jac Xt l of U,B, there is almost no energy, hence
=
l Jac Xt l ·
cst there, while outside
d d i e Ee (ve(t» = 1/> t ) I 2 I Jac Xt l + o (l). D Xtl \7 (u e dt I t=o dt It=O Expand \7 (ue ei1/>, ) as \7 ueei1/>t + iUe \7'1Pt, expand the squares, and apply 1ft t=0 ' 1
The crucial fact is that the terms which get differentiated do not depend on c . For the other terms, we use (3.6) , so that there remains
d 1 .L .L DXt \7 0 ' \7 0 fl\UiBi dt I t=o d d .L 1 2 IJac xd + 0(1) \7 t + + I \7 1\7u . 0 '1j J 2 dt I t=o fl\UiBi dt It=o 1 d o + \7'l,0t W I Jac Xt l + 0 ( 1 ) IDXt (\7.L 1 dt l t=o 2 fl\u,Bi But observing that 'Ii't was conHtructed in such a way that \7 .Lo + \7 'l,0t = \7 .L ( t 0 Xt), and doing a change of variables again, we find d d 1 2 + 0 (1) Ec(v,, (t)) t 1\7 1 dt It=O dt It=o 2 O\Ui Bi d t) , . . · , an (t)) + 0(1) Wct (al ( dt I t=o
1
-
i.e., the desired result.
4. Second order approach - stability issues We extended the method to second order in order to treat stability questions for this 2D Ginzburg-Landau equation. Here is the abstract result, pushing the method of condition 2') to second order. The setting is as in Section 2 . 1 . We say a critical point is stable if the Hessian is nonnegative, unstable otherwise.
THEOREM 4 . 1 (Abstract result - [S4]) . Let u" be a family of critical points of Ee with Ue ....,;"S U E N, such that the following holds: for any V E /3', we can find ve (t ) E M defined in a neighborhood of t = 0, such that Bt ve (O) depends on V in a linear and one-to-one manner, and
( 4. 1 )
•
limc _o ft 1t=o E,, (v,, (t))
(4.2) (4. 3)
'U"
lim _o � lt=oE. (v,,(t» ) ..
=
(0) = 1l"
=
ft1t=o F(u + tV )
Jt'2 I t= o F(u + tV)
=
=
dF(u). V
( D2F ( u) V, V) .
SYLVIA SERFATY
476
Then - if (4 . 1}-(4 · 2) are satisfied, then u is a critical point of F - if {4. 1}-{4.2}-(4 · 8} are satisfied, then if UE are stable (resp. purely unstable) critical points of Eo u is a stable (resp. purely unstable) critical point of F. More generally, denoting by nt the dimension (possibly infinite) of the space spanned by eigenvector's of D2 Eo (uo) associated to positive eigenvalues, and n + the dimension of the space spanned by eigenvectors of D2 F(u) associated to positive eigenvalues (resp. n; and n- for negative eigenvalues); then for E' sm.all mough we have n; > n - . Thus, we reobtain that critical points converge to critical points of the limiting energy F (proved in [BBH] for Ginzburg-Landau), but in addition we obtain that under certain conditions, stability/instability of the critical point also passes to the limit. The previous l'esull. (Theorem 1.2), was an analysis of the C1 structure of the energy landscape, thus suited to give convergence of gradient-flow and critical points; while this is the C2 analysis of the energy landscape around a critical point. For ( 1 . 1 ) , the construction done in Section 3.4 can be fe-used, and the calcu lation pushed to second order, to obtain conditions (4. 1 )-(4.3). Thus, from the theorem above, we deduce in [S4] the corresponding theorem for solutions of (1.2) (which was not proved before) : stable/unstable critical points of Ginzburg-Landau converge to stable/unstable critical points of the renormalized energy. An interesting application is for Neumann boundary condition, for which it is known that the corresponding renormalized energy W has no stable critical point. Hence from Theorem 4. 1 there are no stable critical points of Eo with vortices. THEOREM
4.2 ( [S4) ) . Let
UE
be a family of nonconstant solutions of
au = 0 an
-�u = eno'ugh,
(on 12
C
IR2
1Le
�(l lul'l) -
simply connected) such that is unstable.
in 12 on an
£e(u,J
0, with possible (but not necessarily) limiting degree O. More precisely, it is known that if u is a static solution of Ginzburg-Landau in the plane, with vortices (ai , d;) then we mw;t have 2 2 d = "' 0 i ,
equivalent to the fact that L:i#j di dj = 0, or to the fact that the forces exerted by the vortices balance each other. This follows from suitable applications of the Pohozaev identity, as in [BMR). Similarly, as seen in [eM! J, if u is a static solution of ( 1 .2) in a bounded domain and has some vortices Ui of degree di accumulating (as € -> 0) around a single point p, then the same rule (L:i d;)2 = L:i d; holds. Now, if u is a configuration with say, t,wo vortices, one of degree 1 , one of degree - 1 , at a distance 0(1) as c -> 0 (which is what happens during a vortex-collision of a +1 with a - 1) then this rule is obviously violated (and it's the same for any situation with (L:i d;) 2 1= L:i df ) , so we can trace how much it is violated in the Pohozaev identity for (5.2), and get a lower bound for Ilf" IIL2 . This is the method we applied to get the result below. In a first stage, we may forget that u solves (1.4), forget time, and jus\' focus on studying the static equation (5.2) which is ( 1 .2) with an L2 perturbation. Before stating the result, let us make a few assumptions. It is natural to assume
( 5 .4) since the energy decreases during the flow ( 1.4) , and (5 . 5)
luEI < 1
M , l '\lu" I < €
which are satisfied at all times for solut.ions of ( 1 .4). We assume in addition that 1 2 (5.6 ) 11 10 1I L2 (0 ) < -".fJ for some (3 < 2 . If this assumption is not true, then clearly we have a large lower bound on II i" I I £2 . If it is true, then after blow-up at the scale c, solutions of (5.2) converge to solutions of Ginzburg-Landau in the plane
-tl.U = U(1 - I U I 2 ) which enables us to define what we shall call a "good collection of vortices" ai with degrees di (depending on 6) for U" . Without going into full details of what
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it means and how they are found, these are points (depending on c) such that UiBi ;= UiB(ai, Ree) (with some Re < I log el) are disjoint and cover all the zeroes of u" , and d; = deg (u", aB(ai, Ree» .
THBOREM 5.1 (Analysis of solutions of (5 . 2) - [83]) . There exist constants 1o > 0 and Ko > 0 such that, assuming that Ue is as above and that there exists a nonempty subcollection {B;}7 1 of the balls {Bi} which are included in B(xo , 1/2), e Vl log el « I < In as e --> 0, and such that for some K > Ko, either (1 ) B (xo, Kl) c n and B(xo, Kl ) intersects no other ball in the collection {Bi } , and we have 2
(5.7)
•
(2) Xn Then
i= l i=l E an and B(xo, Kl) inter'sects no other ball in the collection {Bi } . C c l 2 110g 10 1 ' 12 10g2 l
( 5.8)
•
All the constants above depend only on /3, M, n and g.
As a byproduct, we retrieve the fact that for 10 small enough, solutions of ( 1 .2) have no cluster of vortices with (l:i d; ) 2 of. l:i d; at mutual distances I < lu. The analysis for Theorem 5. 1 , which is partly inspired by the one of Comte Mironescu [CMI, CM2] for solutions of (1.2), combined with Pohozaev identities, also allows us to obtain the following result. THEOREM 5.2 ( [83] ) . Let u" satisfy (5.2) -(5. 6), then we have,
(5.9)
Be (ue) < 7r
i >7 10g � + Wd (a l , ' " ,=1 .
c
, a n ) + Cllf" 111,2(0) + 0(1),
wheTe C depends only on /3, M, 12 and g.
Observe that under certain conditions, this gives relations of the form De < - I lo�el Jt Ee (t) for solutions of (1 .4) , where De again denotes the energy-excess.
5.2. First application to the dynamics: well-prepared implies very well-prepared. Coming back to the dynamics (1.4), in Theorem 2 . 5, we made the
assumption (2.10) initially, as required by Theorem 2.4, which we can call a "very well prepared" assumption, as opposed to the weaker requirement in [Lil, J82] which was only Ee (uE ) < 1I"n l log e l + C, which we now call "well-prepared" . Our requirement turns out not to be really stronger because within a time 0(1), solutions with well-prepared initial data (or even a weaker requirement, see below) become very-well prepared, as shown in the following result, which is proved through Theorems 5.1 and 5.2. Ug
THEOREM 5.3 (Instantaneous "very-well preparedness" - [83]) . Assume that is a solution of (1.4) such that
(5.10)
curl (iu�, Vu�)
->.
211"
n
L Di6p? i=l
as
e
-.
0,
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479
with D; = ±1, and such that (5.11)
Ee( u�)
27r log l et)
1t to
C dt F (t)
It: 12�t) dt and solving an ODE, we find eCM(t)
C(t - to ) _ -1 t2 (to ) •
1 . Then, there exist Hl ( (O, T*) ) trajectories bj,k (t) such that for evenJ t [0, T* ) ,
(Jo�2 1!�:g
curl (iuj , Y'ui) (t)
---I.
E 211" L Dj,k6bj , . (t)
as
k
c ->
0
(5.25) and T' is the first collision-time under this law. Observe that this result allows us to treat a varietv of cases such as collisions of vortices of opposite degrep.s, separations of vortices of same degree. . . "
5.5. Post-scriptum. Since these notes were first written up, t.he study of the dynamics of vortices in the whole plane ]R2 under the general hypothesis E" (u�)
