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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris
Subseries: Fondazione C. I. M. E., Firenze Adviser: Roberto Conti
1713
Springer Berlin Heidelberg New York Barcelona Ho ng Ko ng London Milan Paris Singapore
Tokyo
E Bethuel G. Huisken S. Mfiller K. Steffen
Calculus of Variations and Geometric Evolution Problems Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy, June 15-22, 1996 Editor: S. Hildebrandt, M. Struwe
~:~ w ,~ Fonda
one
C.I.M.E.
Springer
Authors
Editors
Fabrice B e t h u e l Universitd P a r i s - S u d L a b o r a t o i r e d ' A n a l y s e N u m e r i q u e et E D P U R A C N R S 760, Bfitiment 425 9 1 4 0 5 Orsay, F r a n c e
Stefan Hildebrandt M a t h e m a t i s c h e s I n s t i t u t d c r Univcrsit~.t Beringstral3e 6, 53115 Bonn, Germany
Gerhard Huisken Alexander Poldcn M a t h e m a t i s c h e s insti~ut U n i v e r s i t a t Ttibingen A u f der M o r g e n s t e l l e 7 2 0 7 6 Ttibingen, G e r m a n y
Michael Struwe E T H - Z e n t r u m , R ~ i m i s t r a s s e 10 8 0 9 2 Ziirich, S w i t z e r l a n d
Stefan M011er M a x - P l a n c k Institute for M a t h e m a t i c s in the S c i e n c e s [nselstrage 22-26 0 4 1 0 3 Leipzig, G e r m a n y K l a u s Steffen M a t h e m a t i s c h e s Institut Universitat Dtisseldorf Universit~itsstraf3e 1 4 0 2 2 5 D/isseldorf, G e r m a n y
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Die Deutsche Bibliolhek - CIP-Einheitsaufnahme C a l c u l u s o f v a r i a t i o n s a n d g e o m e t r i c e v o l u t i o n p r o b l e m s : held in Cetraro, Italy, June 15 - 22, 1 9 9 6 / F o n d a z i o n e C I M E . F. B ethuel ... Ed.: S. Hildebrandt ; M. Struwe. - Berlin ; H e i d e l b e r g ; N e w York ; B a r c e l o n a ; H o n g K o n g ; L o n d o n ; M i l a n ; Paris ; S i n g a p o r e ; T o k y o : Springer, 1 9 9 9 (Lectures given at the ... session of the Centro Internazionale Matematico Estivo (CIME) ... ; 1996,2) (Lecture notes in mathematics ; Vol. 1713 : Subseries: Fondazione CIME) ISBN 3-540-65977-3
M a t h e m a t i c s S u b j e c t Classification ( 1991 ): 35-06, 49-06, 58-06, 73-06 ISSN 0 0 7 5 - 8 4 3 4 I S B N 3 - 5 4 0 - 6 5 9 7 7 - 3 Springer-Verlag Berlin Heidelberg N e w York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re use of illustrations, recitation, broadcasting, reproduction on microfihns or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law 9 Springer-Verlag Berlin Heidelberg 1999 Printed in Germany Typesetting: Camera-ready TEX output by the authors SPIN: I0650190 46/3143-543210 - Printed on acid-free paper
PREFACE
The international summer school on
Calculus of Variations and Geometric Evolution Problems
was held at Cetraro, Italy, June 15-23, 1996.
Tile lecturers, F. Bethuel, G. Huisken, S. MiJller, K. Steffen had complete freedom in choosing the topics of their courses within the themes of the conference. The contributions to this volume reflect quite closely tile lectures given at Cctraro which have provided an image of a fairly broad field in analysis where in recent years we have seen many important contributions. Among the topics treated in the courses were variational methods for Ginzburg-Landau equations, variational models for microstructure and phase transitions, a variational treatment of the Plateau problem for surfaces of prescribed mean curvature in Riemannian manifolds, - both from the classical point of view and in the setting of geometric measure theory. The second theme of the conference was presented in lectures on geometric evolution equations for hypersurfaces in a Riemannian manifold. G. Huisken has included his student A. Polden as coauthor, because the notes presented in this volume present a hitherto unpublished part of Poldcn's thesis providing, for example, a shorttime existence proof for the gradient flow of the Willmore functional.
The organizers would like to express their gratitude to the speakers for their excellent lectures and to all participants for contributing to the success of the summer school.
S. Hildcbrandt, M. Struwe
TABLE OF CONTENTS
F. Bethuel
G. Huisken, A. Polden
S. Mtiller
K. Steffen
Variational Methods for Ginzburg-Landau Equations ....................................
1
Geometric Evolution Equations for Hypersurfaces ........................................................
45
Variational Models for Microstructure and Phase Transitions ..................................................
85
Parametric Surfaces of Prescribed Mean Curvature ...................................
211
Variational methods for Ginzburg-Landau equations F. Bethuel
I. I N T R O D U C T I O N Ginzburg-Landau flmctionals were first introduced by V. Ginzburg and L. Landau in 1950 [GL] in tile context of superconductivity. They were aimed to model (on a macroscopic scale) the energy state of a superconducting sample, in presence of an exterior magnetic field. Similar energy functionals appeared thereafter in various contexts, and under different forms. In particle physics one may mention the Abelian Higgs model, the Poliakov-t'Hooft monopole, and more generally various models of chromodynamics. A common feature of the above models is that they involve a nonconvex potential. A typical example for such a potential is the function V(u) = (1 - [u[2) 2 (u E ]R or u E IR2 for instance). The vacuum manifold is the set of point where V achieves its minimum. If u E JR, in our example, then the vacuum manifold is {+1, - 1 } , whereas if u E IR2, then the vacuum manifold is S 1, the unit circle. The topology of the vacuum manifold turns out to be crucial in the study of the model : it will induce various topological defects, called in our context vortices. To make things more precise, we will start with a very simple model situation, which was studied in particular in a joint book with H. Brezis and F. H61ein [BBH]. An important part of these notes will be devoted to the study of this model. Nevertheless, in the last sections, we will show how the technics introduced can be useful for attacking more realistic physical situations : as we will see, although progresses have been obtained, outstanding mathematical problems remain open in that direction. II. A SIMPLE
MODEL
Let f / b e a smooth bounded domain in ]R2 (throughout this paper, we will restrict ourselves to two-dimensional problems). We will consider complex-valued maps on f~, that is maps from ~2 to ]R~. The simplest Ginzburg-Landau functional for such maps
takes tile form
g'(") = ~1 ~
1 f~ ( 1 - [ v i a ) :2 Iv"l= + 4--7
Here r is a positive parameter, homogeneous to a length. In the sequel, we will mainly be interested in the case e is small : the asymptotic limit e tends to zero will be central in our analysis. The nonconvex potential V is here 1 (1 -1.1=) v ( . ) = 4-7
=
and the vacuum manifold is the unit circle S 1. For critical maps v of the energy, the potential V forces (for small r [v[ to be close to 1 : hence there are almost SX-wlued. However, at some points v may have to vanish : this introduces defects of topological nature (which will be called vortices). In order to have a well-posed mathematical problem, we have to prescribe boundary conditions. The simplest idea will be to impose Dirichlet boundary datas (although this might not correspond to any realistic physical situation...). For that purpose, let g be a smooth map from 0 ~ to the circle S 1. We prescribe v to be equal to g on c3fL It is then natural to introduce the Sobolev space
H~(~; ~ 2) = { . ~ HI(n; z : ) , . = ~ on 0 ~ } . Tile functional E~ is indeed well-defined, smooth on H i , and satisfies moreover the Palais-Smale condition. Critical points of E~ then verify the Ginzburg-Landau equation
(1)
-Av
= ~1-
v (1 - M ~) in
, = g
on On.
We will often refer to (1) as (GL),. Since the nonlinearity on the right-hand side is subcritical, solutions to (1) are smooth on ~. Moreover one has P r o p o s i t i o n 1. A n y solution v to (1) verities
(2)
M -< z oi1 ~,
and (3)
C [Vvi _< -g
where the constant C depends only on f~ and g.
Proof : Inequality (2) is a consequence of the maximum principle. Indeed we have
~ A t , I ~ = ~ Av + IVvl 2
= ~lvl = (I,,I 5 1
-
1) + IVvl =
2
>__~-Ivl (I,,P - 1). tlcnce the function w = I.I ~ - 1 satisfies
- A w + a(z)w 0. By the maximum principle, we conclude that w _< 0. For (3), we note that, by (2) 1
IAul _< 2/
and the conclusion follows by elliptic cstimates (see [BBH2]). In the next section, we will be interested in minimizing solutions. The existence of such solutions is easy to establish. Since E~ is positive, one has
ge =Inf {E~(v), v E H 1} >0, and any minimizing sequence for he is bounded in H i , hence converges weakly up to a subsequence to some map ue. By lower-semicontinuity of Ee (for the weak topology), ue is a minimizer. In the sequel, we will always denote minimizers by u,. Remark : Minimizers might not be unique (for sufficient small ~, at least). We will give later examples of nonuniqueness, when symmetries are present. Next, we will study the asymptotic limit as e tends to zero.
III. ASYMPTOTIC
ANALYSIS OF MINIMIZERS
The winding number d of g (from 0~ to S 1) plays a crucial role in the asymptotic analysis, inducing, in the case d ~ 0, the appearance of vortices and the divergence of the minimal energy tr as ~ -~ 0. This is deeply related to the following P r o p o s i t i o n 2. Assume ~ is simply connected. Set HI(~'/;S 1) = {v E H~(~;]R2), [ v [ - 1}.
4
Then ~ (~; S') is .o,,cmpty
if ~,a o , a y if e = o.
With similar notations, the fact that C0(f/; ocl) is non-empty if and only if d is zero reduces to standard degree theory (and is of course well-known). For H 1 maps the proof is slightly more involved, and relies on the following. L e m m a 1. A s s u m e ~ is s i m p l y c o n n e c t e d . L e t v be a map in Hl(f'/; Sx), where gl(~"~;,..q'l) ~- {11 9 HI(~'~;IR2), I1}[~- 1}. T h e n there e x i s t s a r e M - v a l u e d map ~o 9 HI(f~;IR) s u c h t h a t
v = exp i v. Moreover
Ivvl = Iv~l a.e.
(4)
Proof of Lemma 1. Since v is Sl-valued, if (Zl, x2) are cartesian coordinates on f~, then v,, is parallel to v,=. This writes (5)
v,, x v,= = 0 .
[Here, we have embedded IR2 into IRa, and x denotes the cross-product in IRa]. We may rewrite (4) in divergence form
o (vx
(6)
Ozl
v~=
)+---o (-vxv~,)=0. Oz~
Since f / i s simply connected, by Poincar6's Lemma, there is some map ~b in H1(12; IR) such that v x vxl = %b~t V X /}z2 ~ ~dz~t,
[Here v x v~ is orthogonal to IR2 in IRa, and considered as a scalar]. Next consider the S 1- v a l u e d map w - - e x p -i%b-v where the multiplication stands for complex multiplication. Then, for i = 1,2, w~, =-i~b~,(exp ~0.
-i~b.v)+(v
xv~, exp - i # ) . v )
Hence Vw = 0, thus w is a constant. The conclusion follows. Remark : The conclusion of Lemma 1 would be false if, instead of Hl(fl; $1), we had considered W 1m(~; $1), for p < 2. Take for instance gt = D 2 the unit disc and =
Then v belongs to W 1'I' for any p < 2, but of course cannot be written as the exponential of a W l,v function. The proof fails because, although (5) still holds almost everywhere, (6) is no longer true (for proving (6) in Lcmma 1, one may approximate v in H I ( ~ ; ]R2) by smooth functions and pass to the limit : this uses H 1 bounds). Proof of Proposition 2 : Assume HI(F/; S 1) is not empty and let v be in HI(Q; El). Then, by Lemma 1 there exists some function q0 in H~(~'/; 0~ v=expi~o
such that
on~.
In particular g = exp i~ o11 0~, which implies that the degree of g is zero. The proof is complete. Next we are going to turn to the asymptotic analysis when d = 0. I I I . 1 . T h e case d = 0. In this case, there is no topological need for vortices, and indeed, they do not appear (for minimizers at least). First, we notice that n, remains bounded independently of ~. To see this, let v0 be any map in H I ( ~ ; S 1) (this is possible by Proposition 2), and take v0 as a comparison map. We have
1 fn IVv~
(7)
a, 0 such that the Morse Index of v, is larger than ~0ldl ~ provided d >_ 2, and r is suttlciently small Sketch of the proof ([AB]) : In tile neighborhood of v, we may write
E~(v, + ~) = E~(v,) + Q d w ) + O(llwl13),
v w E H01(a).
Here Q~ is the quadratic form given by 1
1
1
that is Q,(w) = (L~w, w>,
where Le is the linear operator given by
L,(w) =
- A w - F1 ( 1 -
2
Ifdl 2) w + fi(w.v~)v~.
The Morse Index of v~ is given by the number of negative eigenvalues of L~ : this number is finite by standard Riesz-Fredholm theory. Moreover, if V is a subspace of H01(~) such that Q(z) < O, V z e V then
(3s)
dim V < d i m / / _ ,
where H _ is the space spanned by the eigenvectors with negative eigenvalues. We are going therefore to construct a space V~ with the previous property.
31 First, consider the unique p~ E [0, 1] such that 1 fd(Pe) = ~. we
h a v e for a n y ~ e
(39)
HI(D(p~)) C H~(D~) Q(w) _ 2.If ~ is sumciently small, then (GL~) has at least three distinct solutions, among which one at least is non-minimizing. Remark. Other non-minimizing solutions have been produced by F.H. Lin [Lil]. For special b o u n d a r y conditions g, he was able to produce solutions with vortices of opposite sign, which are local-minimizers, using heat flow methods.
In contrast the solution
p r o d u c e d in Theorem 5 has probably a non-zero Morse index. The proof of Theorem 5 is based on Morse theory. We consider the level sets E~ = {v 9 H~(fl;IR~), E~(v) < a}. If E a and E b have different topologies, for some a and b in ]It, then s t a n d a r d arguments of Morse theory assert that there is a critical value in (a, b), hence a solution to (GLe) (recall t h a t the functional E~ satisfies the Palais-Smale condition). Since E ~176 = Hi is a contractible space (it is an affine space), we will apply the previous argument for b = +c~ and show that for some a > ~ , E a has a non trivial topology. More precisely, we will prove :
33 Proposition
18.
There e x i s t s a c o n s t a n t Xo > 0 s u c h ~hat, for a = RE + XO, and r
s u t ~ c i e n t l y s m a l l t h e r e e x i s t s a l o o p in E a w h i c h is n o t contractible, i.e. a c o n t i n u o u s map 7
: S 1 --* E a w h i c h cannot be e x t e n d e d to D 2 in a c o n t i n u o u s way.
Proposition 18 is of course tile main ingredient of the proof of Theorem 5. As in many other variational problems in P D E ' s (see for instance J.M. Coron [C], Bahri and Coron [BC], C. Taubcs IT], ...) the topology of level sets can partially be reduced to a finite dimensional problem. In our case, we already saw (at least for minimizers) that the energy functional (which is defined on an infinite dimensional space) is deeply related to the renormalized energy which is defined on a finite dimensional space : for minimizers oil E = ~ d \ A , where A is the diagonal. The bottom idea in the proof of Proposition 17 is that the topological properties of level sets E" as stated above, that is for a close (but yet not too close !) to the infimum of the energy, are related to the topological properties of E. In particular, we will use the fact that 7ri(E) # 0. However two new difficulties appear in the procedure above, which are mainly of analytical nature. 1) First, we have to define the notion of vorticcs for maps in E% Indeed, this notion was only defined at this stage for critical points, and the equation (in particular Pohozaev's identity) played a very important role in the analysis. Moreover some continuity in H 1 for the singularities has to be derived. 2) Second, we have to relate the energy of a map u to the renormalized energy of its vortices, as for instance in Theorem 1. V I . 2 . 1 . V o r t i c e s for m a p s in E a In order to define vortices for maps u in E a, we will proceed indirectly. Let 0 < 7 < 1 be given, and set h = e~. Consider, for a given u in E a, the minimization problem
(41)
Inf F h ( v ) uEH~
where Fh is given by I
Fh(v) = E,(v) +
-
Clearly Fh is achieved by some map Uh (we do not claim uniqueness) which verifies the "perturbed" Ginzburg-Landau equation 1
Uh--U h2
Aua =
-fiUh
(1
luhl 2) --
9
In view of our choice h = e ~, the perturbation is small (in some appropriate sense). Adapting the method of local estimates (cf. Section IV.5, proof of Theorem 2 bis), we may prove
34 Proposition
19. Let K be an arbitrary constant, and assume that a verifies the bound
(42)
a < /(([log c[ + 1).
Let u be in E ~, and ua be a minimizer/'or (41). There exist constants N E ~q*, A > O, r > O, CI > O, depending only on g, K, and 7 such that if r < r points al, ...,at in ~, such that
then there exist f
g-
1
t
on
f/\ U B(ai,Ar i-1
B(ai, 2Ar (7 B ( a j , 2Ac) = 0 if i r j and t
i=1 Note first that assumption (,12) is much weaker than the assumption of Proposition 18, leading to the hope that more solutions (of higher energy and Morse index) can be found. For a r b i t r a r y maps u in E a, one might have many (that is a number diverging with r regions where u vanishes : for instance, for a given map, one may insert a very large number of dipoles, i.e. a pair of vortices of opposite charge separated by a distance, say of order r for some 0 < 3' < 1. At the end, this leads to a very blurred image, a m a p with many "details" on a smale scale (of order r Nevertheless, these details are basically unrelevant ]br Morse Theory. The idea behind Proposition 18 is that, if we omit the details h occuring on a length scale less t h a n h = ex, then things look more or less as in Theorems 1, 2, or 2bis. In other words, our approximation u h (which is of parabolic type) smooths out details of small scale (of order < h). However, there is a price to pay : the vortices themselves are only defined up to a small error, which corresponds to the scale of resolution we (arbitrarily) introduced. Therefore the m a p which assigns to an element in E a its vortices (or more precisely the vortices of u h) can certainly not be continuous. Nevertheless, it is rbalmost continuous in the following sense. D e f i n i t i o n 1. Let F and G be two metric spaces. Let r1 >_ 0 and f be a function from F to G. We say that f is u-Mmost continous at a point uo in F if, given any 6 > O, there exists O > 0 such that if d(uo,v) _O, V x E O a
~0(x)
=
o n [0, + o o [ •
v
9
e a,
where u : [0, +cx~[• f't ~ ]R2, and the initial data u0 is smooth and in H i. By standard arguments a solution exists for all time, and is unique. Moreover, we have the equality (45)
=
/0'/ok
-Ou ~
+ Ee(u(t)) = E~(uo),
hence the energy decreases along the flow (44). When we are able to define vortices for u0, an important question is to derive the motion law for these vortices. In the case the energy of u0 is close to ~c (i.e. is less than ~ + C, for some constant C independent of r this question was settled by F.H. Lin. He proved that, if the time t is scaled by Ilog r then the vortices move (in the limit r --~ 0) according to the opposite of the gradient of the renormalized energy (see also Jerrard and Soner for related results [GS]).
VIII. THE SCHRODINGER
EQUATION
Here we assume that the domain is IR2. The Schr6dinger equation related to the Ginzburg-Landau functional
i,,, = Au + u (~ - {u{ ~) u(x, 0)
=
on [0, + o o [ •
u0(x).
It appears in various models in physics, for instance superfluidity, nonlinear optics, or fluid dynamics. It is often termed Gross-Pitaevskii equation. Many problems remain open, as existence, motion low for vortices... In a joint paper with J.C. Saut [BS], we have studied the existence problem for travelling wave solutions of the form. These solutions have the form
U ( ~ l , X , , t ) = ~(~, - a , x ~ ) where (xl,x2) are cartesian coordinates on IR2, v is a function on IR2, and c > 0 is the speed of the wave. The equation for v reads av
icg;7~, = A~ + v (1 - I~1~).
39 We establish the existence of a solution for small speeds. These solutions have been studied on a more formal level in a serie of papers (see for instance for references, Jones, P u t t e r m a n [OPrq, or Pismen and Nepomnyashchy [eN D. The existence proof is based on the Mountain-Pass theorem for the functional 1 F(u)=~/r
1 , V u , = + ~ j~It2 ( 1 - ] u ' 2 ) 2 - c / ~ ,
(i~--~l,U) .
The small parameter c plays here the role of the small parameter c in our previous analysis.
IX. SUPERCONDUCTIVITY As mentionned in the introduction, Ginzburg-Landau functionals have been first introduced to model superconductivity. The functional is however slightly more involved than thc simplc modcl we have considcrcd so far. In order to account for electromagnetic effects one has to introduce a vector potential A, which can be considered as a 1-form
A = Aldxl + A2dx2 where the functions A1 and A2 defined on ~2 axe real-valued. The Ginzburg-Landau functionals involve u and A and write (45)
F~(u, A) = ~1 ~ IVAul = + IdA - Ho 12 + 1 (1 - l u l = ) =
where H0 is a given function on ~2 (the exterior applied field), u is complex-valued and
•AU = h=dA=
0u
Ou _ i A l u , ~ - i A 2 u OA1 Ox2
),
OA2 Oxl '
and r > 0 is a parameter (which depends on the material). Some words on physics are in order. At low temperature, some material exhibits very special properties : they lose electric resistivity, and repel magnetic fluxes. This phenomenon is termcd superconductivity. It turns out (according to the theory developped by Bardeen, Schaeffer and Cooper) that the electric current is not mediated by isolated electrons (as in usual conductors), but by pairs of electrons (with opposite sign), which behave llke bosons. On a macroscopic level, these pairs of electrons
40 (called Cooper pairs) are modelled by a complex-valued wave function u. The norm of u squared, lut 2 represents the density of superconducting pairs of electrons : after some renormalizations, one may assert that if [u(x)[ ~ 1, the sample is superconducting at the point x e fl, if [u(x)[ _~ 0, the sample is in the normal state (i.e. not superconducting). Hence a sample may have regions where it is superconducting, and others where it superconductivity is lost. For (45) we have restricted to the situation the sample is two-dimensional, and all magnetic fields are perpendicular to the sample. H0 represents the exterior applied magnetic field. Stable configurations are supposed to be local minimizers for Fe(u, A), on all possible configurations in H 1(~, ]R2) x (H 1(it, ]R2), e is a parameter depending on the material. The function h = dA = oA__4_x _ ~Ox2 represents the induced magnetic Oxt flux, and the electric current is given by J = (iu, VAU) = ((iu, u~, -- i A , u ) , ((iu, u~, - i A 2 u ) ) .
An important feature of the functional F, is that it is gange-invariant. More precisely, for every function ~ E H2(~), we have F~(u, A) = F~(v, B),
where v =expi~o.u
B=A+d
.
All physically relevant quantities like [u h J, h are gauge-invariant. In order to remove the invariance one may impose a condition on A, like the Coulomb gauge (46)
( d i v A = 0 on
t
A.v = 0
on 0f~.
Then, (45) and (46) define an elliptic problem. When H0 is small, the minimizing solution to (45) verifies (in the Coulomb gauge) u ( x ) ~- 1 and h = dA satisfies (approximatively) the London equation -Ah+h=O
infl
h = H0
on 0f~.
Hence all the material is superconducting. When H0 is large, and e is small, vortices appear : they trap regions where u ( x ) ~_ O, i.e. where the material is in the normal state.
4] An interesting problem is to determine the critical value Hc of H0 for which vortices appear. A computation by the physicist Abrikosov shows that H~ "~rllog c I. However this estimate has not been completely rigorously proved on a mathematical level (see [BR2] for a discussion). Another interesting question is to describe the location (and the number) of vortices, when H0 > He, and to prove the (observed) fact that they have all winding number +1 (as in Theorem 1). The asymptotic analysis of [BBH] (for r tending to zero) has been extended to F,, in the case H0 = 0, and a Dirichlet type of boundary condition is imposed, that is [u] = 1 on 0 n
deg(u, 0~) = d is prescribed and V.V au = g on0~/
where T is the unit tangent vector to 0n, and g : 0~ --4 lit is a smooth real valued function. In an other direction, an important physical experiment has attracted much work from mathematicians. Consider a superconducting sample that has the shape of annulus, or a ring. The experiment is the following : put the sample at ambiant temperature in a magnetic field H0. This magnetic field induces (by the standard rules of electromagnetism) a current, that circles around the annulus. Next cool down the sample, and later remove the magnetic field : a current persists. In view of the previous discussion, this phenomenon is related to the existence of local minimizers of F~, which are not constants (for H0 = 0). This was investigated in work by Jimbo, Morita, and Zha~ [JMZ], Rubinstein and Sternberg [RS], and Almeida [A]. The fact that the topology of ~ (and more precisely, not trivial 7rl) enters into the discussion is related to the following 20. Let ~ be a smooth bounded domain in ]Rn (for n > 3). Lf ~rl(~) {0}, then H I ( ~ ; S ~) has m a n y connected components.
Proposition
The proof is essentially similar to the proof of Proposition 2. For instance if ~/ is the annulus D 2 \ D (89 then the different connected components are labelled by the degree on OD 2. A consequence is that the Dirichlet energy has infinitely many local minimizers. For small e, some of these minimizers yield local minimizers for F~, and
42 L. Ahneida proved that the levels set F~ of F~ have many components for small ~ (see [A]).
REFERENCES [A] L. Almeida, Thesis. [AB1] L. Ahneida and F. Bethuel, Multiplicity results for the Ginzburg-Landau equation in presence of symmetries, to appear in Houston J. of Math. [AB2] L. Almeida and F. Bethuel, Topological methods for the Ginzburg-Landau equation, preprint. [BBH] F. Bethuel, H. Brezis and F. Hdlein, Ginzburg-Landau vortices, Birkha/iser, (1994). [BBH2] F. Bethuel, H. Brezis and F. H61ein, Asymptotics for the minimization of a Ginzburg-Landau functional, CMc. Var. and PDE, 1, (1993) 123-148. [BCP] P. Bauman, N. Carlson and D. Philipps, On the zeroes of solutions to GinzburgLandau type systems, to appear. [BHe] F. Bethuel and B. Helffer, preprint. [BR] F. Bethuel and T. Rivi~re, A minimization problem related to superconductivity, AnnMes IHP, AnMyse Non LindaJre, (1995), 243-303. [BR2] F. Bethuel and T. Rivi~re, Vorticit~ dans les modules de Ginzburg-Landau pour la supraconductivit6, S6minaire Ecole Polytechnique 1993-1994, expos6 n ~ XV. [BS] F. Bethuel and J.C. Saut, Travelling waves for the Gross-Putaevskii equation, preprint. [DF] M. Del Pino and P. Felmer, preprint. [GL] V. Ginzburg and L. Landau, On the theory of superconductivity, Zh Eksper. Teoret. Fiz, 20 (1950) 1064-1082. [JMZ] S. Jimbo, Y. Morita and J. Zhai, Ginzburg-Landau equation and stable steady state solutions in a non-trivial domain, preprint. [JS] LR.L. Jerrard and H.M. Soner, Asymptotic heat-flow dynamics for GinzburgLandau vortices, preprint, (1995). [Lil] F.H. Lin, Solutions of Ginzburg-Landau equations and critical points of the renorrealized energy, AnnMes IHP, Analyse Non Lindaire, 12 (1995) 599-622. [Li2] F.H. Lin, Some dynamical properties of Ginzburg-Landau vortices, to appear in CPAM. [MCd] D. Mac Duff, Configuration spaces of positive and negative particles, Topology, 14 (1974) 91-107.
43 [Mill P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. AnM., 130 (1995) 334-344. [Mi2] P. Mironescu, Les minimiseurs locaux pour l'fiquation de Ginzburg-Landau sont 5. symdtrie radiale, C. R. Acad. Sci. Paris, 6, (323), 593-598. [PN] L. Pismen and A. Nepomnyashechy, Stability of vortex rings in a model of superflow, Physica D, (1993) 163-171. [RS] J. Rubinstein and P. Sternberg, Homotopy classification of minimizers of the Ginzburg-Landau energy and the existence of permanent currents, to appear. [Sta] G. Stampacchia, Equations elliptiques du second ordre 5. coefficients discontinus, Presses Universit~ de Montreal (1966). [Str] M. Struwe, On the asymptotic behavior of the Ginzburg-Landau model in 2 dimensions, J. Diff. Int. Equ., 7 (1994) 1613-1324 ; Erratum 8, (1995) 224.
Geometric evolution equations for hypersurfaces GERHARD HUISKEN AND ALEXANDER POLDEN
1
Introduction
Let Fo : A4 ~ -* (N =+1, ~) be a smooth immersion of a hypersurface r = F0(A4 ~) in a smooth Riemannian manifold (N"+l,~). We study one-parameter families F : .h4 = x [0, T] ~ (N "+l, ~) of hypersurfaces J~r = F(., t)(A4") satisfying an initial value problem OF - ~ (p,t) -- - f u ( p , t ) ,
F(p,0) = F0,
p 9 A4", t 9 [0, T],
p 9 .h4 n,
(1.1) (1.2)
where u(p, t) is a choice of unit normal at F(p, t) and f(p, t) is some smooth homogeneous symmetric function of the principal curvatures of the hypersurface at F(p, t). We will consider examples where f = f ( A l , " - , ~ ) is monotone with respect to the principal curvatures Al,' "-, Am such that (1.1) is a nonlinear parabolic system of second order. Although there are some similarities to the harmonic map heatflow, this deformation law is more nonlinear in nature since the leading second order operator depends on the geometry of the solution at each time rather than the initial geometry. There is a very direct interplay between geometric properties of the underlying manifold (N"+l,~) and the geometry of tile evolving hypersurface which leads to applications both in differential geometry and mathematical physics. Here we investigate some of the general properties of (1.1) and then concentrate on the mean curvature flow f = - H = --(Al+..-+A,,), the inverse mean curvature flow f = H -1 and fully nonlinear flows such as the the Gauss curvature flow f = - K = - ( A I ' " An) or the harmonic mean curvature flow, f = -(A] "1 + ... + )~1)-1. We discuss some new developments in the mathematical understanding of these evolution equations and include applications such as the use of the inverse mean curvature flow for the study of asymptotically flat manifolds in General Relativity.
46 In section 2 we introduce notation for the geometry of hypersurfaces in Pdemannian manifolds and derive the crucial commutator relations for the second derivatives of the second flmdamental form. In section 3 we study the general evolution equation (1.1) and obtain evolution equations for metric, normal, second fundamental form and related geometric quantities. We discuss the parabolic nature of the evolution equations, a shorttime existence result and introduce the main examples. We study the mean curvature flow in scction 4. In this case the evolution law is quasilinear and the knowledge of the flow is more advanced than for all other cases. We give some examples of known results concerning regularity, long-time existence and asymptotic behaviour. In particular we discuss the formation of singularities and give an update of recent new results (joint with C.Sinestrari) concerning the classification of singularities in the mean convex case. The section concludes with an isoperimetric estimate for the one-dimensional case, ie tile curve shortening flow. Section 5 deals with hilly nonlinear flows such as the Gauss curvature flow and the harmonic mean curvature flow. Without proof wc review in particular results of Ben Andrews concerning an elegant proof of the 1/4-pinching theorem, the affine mean curvature flow, and a conjecture of Firey on the asymptotics of the Gauss curvature flow. The inverse mean curvature flow is discussed in section 6. We explain the basic properties of this flow in its classical form relating it to the Willmore energy and Hawking mass of a twodimensional surface. In view of these properties the inverse mean curvature flow is particularly interesting in asymptotically fiat 3-manifolds which appear as models for isolated gravitating systems in General Relativity. It is briefly explained how in recent joint work with T.Ihnanen an extended notion of the inverse mean curvature flow was used to prove a Riemannian version of the so called Penrose inequality for the total energy of an isolated gravitating system represented by an asymptotically flat 3-manifold. While the first part of this article just described stems from lectures given by the first author at the CIME meeting at Cetraro 1996, the last section of the article is a previously unpublished part of the doctoral dissertation of Alexander Polden. It provides a selfcontained proof of shorttime existence for a variety of geometric evolution equations including hypersurface evolutions as above, conformal deformations of metrics and higher order flows such as the L2-gradient flow for the Willmore functional. 'Phe author wishes to thank the orgaafisers of the Cetraro meeting for the opportunity to participate in this stimulating conference triggering joint work with Tom Ilmanen on inverse mean curvature flow, as well as for their patience in waiting for this manuscript.
2
H y p e r s u r f a c e s in R i e m a n n i a n m a n i f o l d s
Let (N '~+l, g) be a smooth complete Riemannian manifold without boundary. We denote by a bar all quantities on N, for example by .~ = {~,,~}, 0 < a, fl < n, the metric, by ~ -{z9~ } coordinates, by f' = {r'~} the Levi-Civita connection, by V the covariant derivative and by 15dem -- (ISdeln~6} the Riemann curvature tensor. Components are sometimes
47
taken with respect to tim tangent vectorfields (O/Oy~'), 0 < a < n associated with a local coordinate chart y -- {ya} and sometimes with respect to a moving orthonormal frame {ca}, 0 < a < n, where O(e~,,ea) = 3~. We write 3 -1 = {9aa} for the inverse of the metric and use the Einstein summation convention for the sum of repeated indices. The Ricci curvature 15dc = {/~oa} and scalar curvature ft of (N"+I,O) are then given by
and ttle sectional curvatures (in an orthonormal frame) are given by #a~ = / ~ . Now let F : 3d" --~ N "+1 be a smooth hypersurface immersion. For simplicity we restrict attention to closed surfaces, ie compact without boundary. The induced metric on Ad n will be denoted by g, in local coordinates we have OF
OF 9
=
OF"
OF a
.hi".
./)/,,a(l,(p))-~xi(p)-~xj(p) ,
p e
l,~lrthermore, {l'}k}, V and llJem = {lt.ukl} with latin indices i , j , k, l ranging from I to n describe the intrinsic geometry of the induced metric g on the hypersurface. If u is a local choice of unit normal for F ( . M " ) , we often work in an adapted othonormal frame u, e l , . . . , e , ~ in a neighbourhood of F(3/I ") such that e l ( p ) , ' " ,e,,(p) e Tp.M '~ C T p g "+1 and 9(p)(ei(p), e~(p)) = (f,i for p e .A4'~, 1 < i , j < n. The second fundamental form A = {ho} as a bilinear form m(p): T p M " x T,.hd '~ ~ IR and the Weingarten map W = {h}} = {9a'hki} as an operator w : T . M " --, T , 3 4 "
are then given by hij
= ,Q ~ e t l / , e j
> -~ _ < tJ, V e l e j > .
In local coordinates {xi}, 1 < i < n, near p e 3d" and {y~}, 0 < a < n, near F(p) 9 N these relations are equivalent to the Weingarten equations 02 F~" Ox ~Ox~
k OF'~ a,~ OFa OIN rq~ + I a6~ ~
Ou" Ox--7 +
~
-
OF~u6 p Ox i
h Ou',
. =
~OF'~
hijg J Ox 1.
Recall that A(p) is symmetric, ie W is sclfadjoint, and the eigenvalues ) q ( p ) , . . . , ,~(p) arc called the principal curvatures of F(34") at F(p). Also note that at a given point p 9 34n by choosing normal coordinates and then possibly rotating them we can always arrange that at this point gij = ~ij,
7"e / = O, Ve, -
hi~ = h ii = diag(Al,...
A,~).
48
The classical scalar invariants of tile second flmdamental form are then symmetric homogeneous polynomials in the principal curvatures: Thc mcan curvature is givcn by H := t r ( W )
= h~ = giJhlj = )~l + " "
+ )~.,
the Gauss-Kroncckcr curvaturc by K := d e t ( W )
= det{h}} - det(h,~} det{gi~}
- ,~l . . . . .
)~,,,
the total curvature by ]A[2 : = t r ( W t W )
, ~ = h i i h q = g ik g i, hilh~a = ,~ + . . . + ~ , = ttjl~
and the scalar curvature (ill Euclidean space 1R"+l) by R =
II 2 -
]A[ ~ = 2(~1A2 + )~Aa + " "
+ )~,-~)~,,).
More general, the mixcd mean curw~turcs S , , , 1 _< m __< n, arc given by thc clcmcntary symmetric flmctions of the A~,
il("'(im
such that $1 = H, 5'2 = (1/2)R, S,, = G. harmonic mean curvature
Other interesting invariants include the
f l := (~i-I + . . . + ~ 1 ) - 1 = S , / S , _ I
as well as other symmetric functions of the principal radii )~-1. All the invariants mentioned or powers thereof are candidates for the speed f in our evolution problem (1.1). For the purposes of analysis it is crucial to know the rules of computation involving the covariant derivatives, the second fundamental form of the hypersurface and the curvature of the ambient space. We assume the reader to have some background in differential gcometry, but rcstate the formulas used in this article for convenience (in an adapted orthonormal frame). The commutator of second derivatives of a vectorfield X on Ad '~ is given by V~VjX k -
VjV~X
k =
P~zmg~aX"~,
and for a one-form w on .h.4'~ by V~Vjwk - V j V i w k
= l~jcagZmwrn.
More generally, the commutator of sccond derivatives for an arbitrary tensor involves one curvature term as above for each of the indices of the tensor. The corresponding laws of course also hold for the metric ~.
49
The curvature of the hypersurface and ambient manifold are related by the equations of Gauss
l~t = R4~ = 1r =
[ ~ t -k hikltjt - h a h j k , 1 < i , j , k , l < n, fla~ - [~o~o~+ Hhi~ - hahn, 1 < i, k < n, /-~ - 2/-~oo+ H ~ - IAI~,
and the equations of Cod~zi-Mainardi
Vilt,~ - V~H
=
/~o~.
The following commutator identities for the second derivatives of the second fundamental form were first found by Simons [48] and provide the crucial link between analytical methods and geometric properties of A4" and N n+~. See also [47] for a derivation of thc following facts from the structure equations. T h e o r e m 2.1 Thc sccond dcrivativcs of A satisfy the idcntitics V~V~h,~
V~V~h~ + hkjh~..h,,,~ - ht~,hah.,j + hk~h~,,,h.a -h~,~h~ih.a + [~a,,.h,,,1 + [l~i,,,h,,a +/~,~nh~,,, + P ~ o j h ~ - [ ~ h ~ + Tl,,a~h~m
The trace of these identities plays an important role in mimimal surface theory and is of particular importance for mean curvature flow and inverse mean curvature flow: C o r o l l a r y 2.2 The Laplacian A = ~ i V,Vl of the second fundamental f o r m satisfies Ah~
=
V i V j H + Hh~,,,h,,,i - h~jlAI 2 + Hft?.~.I - Roohij -t- [~kit-mhmj § [:~kjkmhirn
AIAI
=
h i j V i V j H + IVAI 2 + H t r ( A 3) - IAI4
+ H h i i [ t . ~ 1 -/~oolAI z + 2[tkik,,,h,,~thi j - 2[-lki,,,~hkmhij
P r o o f . By the Codazzi equations we first get VkVlh~j = Vk(V~hlj +/~,jn). Then compute from the definition of h O Vk(/2o~l)
=
VkRoj, + hkm[~i~
50 and commute V~ and Vk to derive VkVll~,j
=
V~Vfl~ 0 -I- Rku,,,h,,, 1 + Rko.,h.a
Then use the Codazzi equations again to get V~Vkh o
Vi(V~/~k~ + [la3k)
=
Employing the Gauss equations we finally conclude
VkVlhij
= ViVjhkl + [~a.J~i + Rkijrnhma -t-f~,,ajkhi,n- [~o~okt~ij- Roijoh.or -t-Vk ~ojil -t- Vi[~oUk
and the conclusion follows from the symmetries o f / ~ o ~ .
3
The evolution equations
Let Fo : .Ad'~ --+ IR~'+1 be a smooth closed hypersurfaceas as in the introduction in a smooth Riemannian manifold (N"+l,~), n > 2. Assume for simplicity that N, M are orientable and choose a unit normal field v on M . If .Ad" C llV'+~, we choose the exterior unit normal such that the mean curvature of a sphere is positive. We then consider the initial value problem (1.1), where f is a smooth, homogeneous function of the principal curvatures A~. Shorttime existence for (1.1) can in general only be expected when the system is parabolic. to investigate the linearisation of (1.1), notice that due to the symmetry of f in an equivalent setting wc may consider f as a function ] of the Weingartcn map W or a s a fimction ] of the second fimdamental form A:
](W) : / ( { h ~ } ) = ](m) = ]({h,~}) = f ( A , . . . A,). In view of the Weingarten equations the linearisation of (1.1) is then an equation of the form ^
O G- -Of Thus the "symbol"
g'kg ~z " 02G
v) v+lowerorder.
51 of the RtlS is always degenerate in tangential directions, reflecting the invariance of the original equation undcr tangential diffeomorphisms. It is strictly positive definite in normal direction if
o/
o-~,j(p) r162 > 0 or equivalently
ol
0,~ (p) > 0
v0 r
e n~-,
Vl 0,
1 < i < n,
(3.2)
holds everywhere on Fo(.h4"), then (1.1) has a smooth solution at least on some short time interval [0, T), T > 0. E x a m p l e s . i) In the case of mean curvature flow f = - H we have -(Of/OAi) = 1 and the flow admits a shorttime solution for any smooth initial data. it) For Gauss curvature flow f = - G we get -(Of/O)~,) = )~'(IG and we have shorttime cxistcnce if the initial data are convex. More generally, the elementary symmetric functions - f = S,~, 1 < m _< n, satisfy -(Of/O)~i) > 0 on the convex cone F,,, = {)~ E lR'*lSl(A ) > 0, 1 < l < m}, yielding shorttime existence for corresponding initial data. iii) The quotients Qk., = Sk/Sl for 1 _< l < k _< n again satisfy (3.1) on Fk. In particular, this yields a shorttime existence result for the harmonic mean curvature flow on convex initial data, since - f = [t = S,,/S,,-t. iv) The inverse mean curvature flow with f = H -~ satisfies -(0f/0,k~) = H -2, yielding shorttime existence of a classical smooth solution for any initial data of positive mean curvature.
52
Working in the class of surfaces where shorttime existence is guaraalteed, tile interesting task is now to understand the longterm change in the shape of solutions, and to charactcrise their asymptotic behaviour both for large times and near singularities. For this purpose evolution equations have to be established for all relevant geometric quantities, in particular for the second fundamental form. T h e o r e m 3.2 On any solution ~4"~ = F(., t)(.M") of (I.I) the following equations hold: (i) -~g~i~ = 2fh~,
(ii) a(d#) = fU(d#), (iii) a v =-V (iv) ~
f,
= - V , V j f -t- f(h,kh~ - Rowj),
(v) ~ I f = - A f
- f(tml 2 + ~ c ( . , L.)).
llcrc dlz is the induced mcasure on the hypcrsurface and A is the Laplace-Beltrami operator with respect to the time-dependent induced metric on the hypersurface. Notice that - A f - f(lAI 2 + l~ic(~, ~)) = J f is the Jacobi operator acting on f , as is wellknown from the second variation formula for the area. P r o o f . The computations are best done in local coordinates {x ~} near p E .A4'* and {y~} near F(p) in N. Arranging coordinates at a fixed point p such that g~j(p) = ~,j, (O/Ox')gjk(p) = O, ~,a(F(p)) = ~,,a, (O/Oy~')ga6(F(P)) = 0 all identities are straightforward consequences of the definitions and the Gauss-Weingarten relations. The computations have been carried out in detail for f = - U in [34], [35] and [4]. A short derivation by the second author is also contained in the last section of this article. We will now use the commutator identities in Theorem 2.1 to convert the evolution equations for the curvature into parabolic systems on the hypersurface. For this pupose we introduce for each speed function f the nonlinear operator L / b y setting
LIU = L~iV~Viu : -
Of V~V~ u , 0--~j
where ] as before is the symmetric function f considered as a function of the h~i. Notc that for mean curvature flow LH = A is the Laplace-Beltrami operator, for inverse mean curvature flow with f = H -1 we have L/ = (1/H2)A and in general L1 is an elliptic operator exactly when f is elliptic, ie satisfies (3.1). The following form of the evolution equations exhibits their parabolic nature. C o r o l l a r y 3.3 On any solution .h4'~ = F(., t)(2k4 '~) of (1.1) the second fundamental form
hi! and the speed f satisfy 0 L~tVkVthij -~ h~i =
02f VihtaVjh~ Oh,klOh~
53
0f
0
~f
"" "0
P r o o f . From V~f = (Of/Ohkl)Vd~kl we see that V~Vjf
02/
0f
=
zy--ViV~hkt
+
u l l , k!
-
-
Oh~Oh~
VihktVih A.
This yields the first identity ill view of Theorem 2.1 and Theorem 3.2(iv). Similarly we get
o__: _ at =
o]o
o]
o
gl,} = g,,g (g'kl,kj) 0/ + 01 0, -2:h,
= L)~V, V i f _ f Of (h,kh~ + ['Qo'j), Oh~j as required. The curvature terms in this nonlinear reaction-diffusion system provide the key for understanding the interaction between geometric properties of the hypersurface and the ambient manifold. They are the tool to study these geometric phenomena with analytical means. For some choices of f we will now describe recent developments.
4
M e a n c u r v a t u r e flow
In the case of mean curvature flow f = - H it is well known [34] that for closed initial surfaces the solution of (1.1)-(1.2) exists on a maximal time interval [0,T[, 0 < T < co. In some cases the behaviour of the flow has been completely understood: For closed convex surfaces in IR"+1, n > 2, it was shown in [34] that the solution contracts smoothly to a point, becoming more and more spherical at the end of the evolution. In [35] this was extended to general Riemannian manifolds under the assumption that the initial hypersurface is sufficiently convex: Each principal curvature )% of the initial surface has to be bounded below by a constant depending on the curvature and the derivative of the curvature in the ambient manifold. While the constants are optimal in locally symmetric spaces, the dependence on the derivatives of curvature in the general case is not desirable form a geometric point of view. Some of the fully nonlinear flows discussed in the next section have a better behaviour from this point of view.
54
Regularity and longtime existence was also obtained for surfaces that can be written as graphs, compare the joint work of the first author with Ecker in [16] and [17]. In the one--dimensional case Grayson proved that any embedded closed curve on a 2surface of bounded geometry will either smoothly contract to a point in finite time or converge to a geodesic in infinite time, compare [25], [26] and earlier work of Gage and IIamilton in [22]. In higher dimensions it is well known that singularities will in usually occur before the area of the evolving surface tends to zero. If T < (x), as is always the case in Euclidean space, the curvature of the surfaces becomes unbounded for t --* T. One would like to understand the singular behaviour for t --* T in detail, having in mind a possible controlled extension of the flow beyond such a singularity. See [37] for a review of earlier results concerning local and global properties of mean curvature flow azid its singularities. We will not discuss singularities in weak formulations of the flow, a good reference in this direction is [54] and [43]. Since the shape of possible singularities is a purely local question, wc may restrict attention to the case where the target manifold is Euclidean space. Nevertheless, in the light of an abundance cvcn of homothctically shrinking examples with symmetries, the possible limiting behaviour near singularities seems ill general beyond classification at this stage. In recent joint work of C. Sinestrari and the author [41], [42] the additional assumption of nonnegative mean curvature is used to restrict the range of possible phenomena, while still retaining an interestingly large class of surfaces. We derive new a priori estimates from below for all elementary symmetric functions of the principal curvatures, exploiting the one-sided bound oll the mean curvature. The estimates turn out to be strong enough to conclude that any rescaled limit of a singularity is (weakly) convex. Recall that l __[E21,
whcre [E21 is lhc area of any conncclcd component of ON 3. Equalily holds if and only if N 3 is one-half of the spatial Schwarzschild manifold. The spatial Schwarzschild manifold is the manifold IRa \ {0} equipped with the metric := (1 + m/2]x])45, representing the spatial exterior region of a single static black hole of mass m.
7
Short-Time
Existence
Theory
Classically, the existence theory for nonlinear parabolic equations is treated in two stages: first, the use of linearisation techniques to prove that a solution may be found for a short interval of time; and second, derivation of the all-important a priori estimates which enable us to extend the short-time solution to a maximal time interval. In this chapter, wc carry out the first half of the process. 7.1 E v o l u t i o n E q u a t i o n s for M a n i f o l d s a n d H y p e r s u r f a c e s This section introduces the primary concern of this work: evolution equations for geometric structures. Typically, we considcr motions of manifolds and submanifolds drivcn by forces which stem from their curvature. Specifically, we address two problems:
Conformal Deformation of a Manifold: Let (M'*,g) be a smooth Riemannian manifold, and consider the deformation process 0
~ g = ~(x,t) 9g,
(7.1)
for some function ,k. This defines a continuous, conformal change in the metric - - conformal because the metric changes only by a scaling factor; angles are not affected. The
61
best-known examples of this are tile Kicci flow on a compact 2-surface (described and solved completely in [27]) and the Yamabe flow on a manifold of dimension at least three [33]; in both thcsc cases, the defining equation is _Og = - R 9 g, where R is the scalar Ot curvature of g. Normal Deformation of a ltypersurface: Let F : M'* "--* (N "*+l, g) be a smooth immersion of a hypersurface in a Riemannian manifold. M'* is assumed orientable, so that there is a smoothly varying, globally defined unit normal vector. In this case, we consider deformation of F according to the equation 0 0t
--F
= ;~(z,t).
n.
(7.2)
The best-known example of this is the mean-curvature flow of hypersuffaces, where the speed is (up to a sign) the mean curvature of F. This was first introduccd by Mullins in [45] (an unjustly little-known work); it was later found independently by Brakke, who cxpresscd the equation in the languagc of gcomctric measure theory in [9]. These are the standard examples of such problems, and they share a common structure. In both cases, the dcformation process can be shown to be equivalent to a quasilinear scalar partial differential equation on M '~. When the impetus comes from the curvature, as in these examples, the scalar equations are strictly parabolic and of second order. Such are known to possess solutions under very general conditions, at least for some short period of time. The total curvature problems we wish to study in this work also lead to quasilinear parabolic scalar equations, but of fourth or higher order. It will surprise nobody that such equations still admit short-time solutions. Nevertheless, when the setting is a manifold rather than a euclidean domain, this does not belong to the standard theory and requires proof. The question of existence will be taken up in later sections; the question of how solutions actually behave will be taken up in later chapters. For the remainder of this section, we assume that we already have a solution to (7.1) or (7.2), and derive a handful of basic properties. (7.1) and (7.2) imply evolution equations for the curvature and other geometric attributes of g and F. Consider first the conformal deformation. L e m m a 7.1 Let M'* be a smooth manifold; let g~ be a one parameter family o/metrics on M '~ varying according to (7.1). Then, gt can be written as exp2u(x, t)'go, where the function u evolves by the equation ~uat = ~A.l P r o o f . It is obvious that gt may be so represented. It follows that 0 0 (exp 2u. go) = 20u cOu ~ g =- O-t Ot " e x p 2 u , go = 2 - ~ . g and this provides the equation for u. Any other metric g~ in the same conformal class could take the place of go in this lemma. The equation for the conformal factor is unaffected.
62 Notation: In what follows, we drop the subscript t from the time-dependent metric. The calculations will relate g to a fixed background metric; for convenience, we take this to be go. The covariant derivative and laplacian operators of g and go will be represented as V, A and V ~ A~ the curvatures will be represented in the same way. The zero may appear as a subscript or a superscript, whichever happens to be more convenient for typesetting; the mcaning remains clear. This accords with the usage in subsequent chapters. L e m m a 7.2 The Christoffel symbols and curvature of g may be expressed in terms of those of go and the conformal factor u:
r,~
=
~,
=
R
=
(ro),~ + (~,~v~ + 5jV~u k o - g~go o k,-o, v~u) _ o o o o _ ( n~ (n - 2) ( v , v ? - v , u v ? ) e-'" (R ~
VOu ~).g,~o
~ou+(n_2)
2 ( n - I)A~ - ( n - l ) ( n - 2) V ~
Similarly, the laplacian operator corresponding to g can be related to that of go: for any smooth function r : M" --* R, A r = e-2"A~162 + (n - 2)g~176
V~162
P r o o f . The first three equations may be found in the discussion of the Yamabe problem in [8]; the fourth follows easily. Let r be a fixed smooth function on M"; then, in local co-ordinates,
Am =
g'~ \ O z ' O z J
r,j~/ f
=
t' k Ou
k Ou
=
e-"~~
~'~x~ + 6 ~
=
k, Ou 0r e-2"A~162 + (n - 2) go ~xSx ~0x ~
o ~ Ou ~ 0r
-g'jg~ ~ )
0~'
and this establishes the final equation. Combining the previous two lemmas gives the variation of the curvature under (7.1): L e m m a 7.3 The change (7.1) produces in the curvature o f t is given by: 0-t0/~i = - ~1 (AA 9gij 4- (n - 2)V,V3A)
and
O Ot_ R = - ( n - 1)AA - RA.
P r o o f . Differentiating the equation above for the Ricci curvature and substituting A = a
1
(_
o
+ (._ )vovo ) +
V~
63
However, for any Ca function f, Af.g,j+(n-2)V,Vjf=A~176
+(n-2)V,
o +LI
in which the error term is given by LI : ((Fo - , ,,~k , o,,,, j,,.~,j j g,jo + ( n - 2 ) ( r ~ - r ) , ~ ) 9 V ~
and using the transformation rule for the Christoffel symbols of Lcmma 7.2, this simplifies to L ! : 2(n - 2)(g~176 V~ 9gO _ V~AVOu). But this matches precisely the final term in the evolution of R~j, and so, cancelling, ~0P ~ J : - 51 (AA. g'J + (n - 2 ) V , V ~ ) which proves the first claim of the lemma. The evolution of the scalar curvature is simpler:
~
n
=
g
=
-(n-
I ) A A - RA.
and this proves the second claim. This is as much as we need say for now about the con formal problem. Now let Ft be a one-parameter family of immersions M ,) ~ N ,`+I which vary in accordance with (7.2). Let gt denote the induced metric Ft*~. As above) we shall drop the t subscripts wherever this would not lead to confusion. From (7.2), we compute evolution equations for the geometric features of F. This is simplified immensely by the use of well-chosen co-ordinates. These calculations are purely local in nature; so we focus on some point (x*, t*) in spacetime. Let y* be the image of x* under Ft.. We may assume that the co-ordinates on N n+l are normal at y*, and that those on M n are normal at x* relative to the metric induced at this one instant of time. In particular, the Christoffel symbols F-~~ ( y *) and F~(xk*,t*) all vanish, and the Gaul] and Weingarten equations reduce to 02F~
ox, o~j(~',t') =
One" * *"
-h,A~',t*). n~
L e m m a 7.4 Under (7.2), the induced metric on M '~ evolves according to 0 It follows directly that the inverse of the metric and the measure evolve by
~0g ~i = - 2 A h ~
OF~ x* t*
-5-ir~Cz ,t ) = h~(z*,t*).-5~-~ ( , ).
and
0 d/t = AHd#. 0--t
64
P r o o f . Let (x*, t*) be a given point of spacetime, and assume the co-ordinate systems on M" and N '~+1 are normal at (x*, t*) and Ft. (x*), as above. We compute the evolution of g~i at the point (x*,t*). The induced metric is by nature given by go = r
,
,
and hcnce, noting that .~ is symmetric and has no covariant derivative of its own, ~~go=2~
N
~
'~
=2~
~xTx ~
'~xJ
'
Now expand the product derivative. The derivative term in A clearly vanishes because of orthogonality, and all that remains is 0
(0n
0F)
and it follows directly from the rcduccd Wcingarten equation at (x*,t*) that the final factor is simply ho; with that, the first claim of the lemma is proved. The rest is easy. To compute the evolution of the inverse of the metric, we differentiate the equation gik. g~ =
~:
0 - b(gik" gkt) Og~k ,kOg~ Ogik Ot - cot gk~ + g ~[ - cOt g~ + 2Ah~" 'lYacing with gO gives OgO - 2 M r O, which establishes the the second claim. The final Ot ~part follows from the rule for differentiating a determinant:
-0~ d t t = ~ (
d~dx)
= ~1 d ~ . g
00
~ g . ,j d x = ) ~ H d # ,
and this completes the proof. Next we derive the variation of the normal vector: L e m m a 7.5 The change in the normal is given by
On Ot
F, (gradM"A).
P r o o f . n is a unit normal vector; thus, ~(n,n) = 1 everywhere. Differentiating this equation, we see that the derivative (any derivative) of n must be normal to n itself, and hence tangential to F(M'~). It may therefore be represented in the form
COn .. (On, OF) COF co--[ g'9 \ ot ~ " coxJ"
(7.3)
=
Now differentiating the equation 9(n, ~ . F ) = 0, we have
O = ~ ~,Ot , Ox, ) + ~ n, ot Ox, j = ~ \ Ot Ox, ) + [? n, Ox' ) = g ~,ot ~
+COx -'-~ '
65 noting here that another term vanishes because n is orthogonal to all its derivatives. This allows us to substitute for g(bTn,-s - o o in (7.3), giving On ~j OA OF 0--[ = - g ~ OxJ'
and this is cxactly -F.(gradM"A). L e m m a 7.6 The variation in the second fundamental form is given by the equations 0
0 i -~h~
=
- V ~ V ~ A - A(h~kh k1 + R i e m j , )
OH Ot
=
-a~
=
- 2 h i i V i V i , ~ - 2,~(tr A s + hUll,iemi,~,)
O]A]
- (IAI ~ + k-~(n,
~))
Proof. The second fundamental form is, by definition, hij = - 3
Ox-----], n
,
and so, differentiating, 0
[-
OF On'~
Now we reinstate the assumptions of Lemma 7.4. In the normal co-ordinate system, the rightmost term vanishes altogether because On is tangential and the spatial derivative normal. So, expressing the remaining covariant derivative in co-ordinates, ~h~j=-~
-~ L Ox.Oz j +
~ . O x i OzJ Oy~' ,n
.
Expanding the time derivative, and noting that the terms containing I'~.y all vanish, this becomes ~h,i
=
-g k~
+ Ot t~' Ox i Ox~ Oy~
=
- 3 kOxiOxJ + ~ v . r ~
[O2()~n)
OFeOF~ O o~i o ~ , ou ~
] "~
J
At the point (x*, t*), the Weingarten equation for the derivative of the normal gives On" ~OF '~ Ox i = h i Ox i
and
02n " Oh} OF ~ + h} 02F'~ OxiOxi - Ox i Ox k Ox~O ~x-
OF'~ O f ~ ~ ~ n ~.
66
It follows that
This enables us to expand the product derivative in (7.4), which gives
a
o51
lh,~h; + 19 ( ( v
r-o o r ~ 0 F -
-
_ OF~ ,~ O , n
~
IIowever, in our normal co-ordinates, the second partial derivative of I coincides with the covariant derivative, and the final term is simply the Riemann tensor of N '~+l. Permutation of the indices in one or other of the summands allows this term to be rewritten as
and now the first factor matches the definition of the Riemann tensor; the product simplifies therefore to Ih" '~OF~OF'r ~ = ~ . j . en~n~ ~ n ~
~
n
In view of all this, the evolution equation for hit may be rewritten as 0
-~hij = - V i V i l - 1(-hikh~ + Pdem,,,j.)
(7.4)
which settles the first claim of the lemma. The equation for h~ follows quickly: 0 i = ~-~(g 0 .ikhik) = ~-~hi
=
- 2 1 . h~khik - 9jk .( V i V k l
-v,Wt
A(-hah~ + Pderni,,~))
- ~(h,~h{ + men~&),
and tracing over i and j gives the equation for H at once. Lastly, _
0 im12= Otfl-~(h~h}) i k ik + hiJRiemi,,j) ' = -2hiJVivi)~ - 2,k(hih~h
Ot
which is the final claim. The curvature equations hint at the structure concealed in (7.2) and (7.1). In the examples mentioned earlier, where the speed ,k was the curvature itself, they are clearly parabolic. The speeds which interest us in this work feature the laplacian of curvature as their leading term. These too will lead to parabolic equations. The assumption of orientability demanded in order to make sense of (7.2) is heavy-handed. In real-life examples, it can typically be avoided. The mean curvature flow, for instance, may be rewritten using the Weingarten equation simply as
OF -- AF, Ot
-
-
67
where the laplacian is computed ill the induced metric; and this is now perfectly meaningful even when the manifold is not orientable. The hypersurface flows we shall consider later can be redefined ill tile same way. The first step towards understanding these problems is to prove that solutions can be found at all. The strategy here is inherited from the second-order theory. The geometric equation is shown to be equivalent to a quasilinear scalar equation, which may be solved for some short interval of time using linearisation techniques; this short-time solution may then be continued as long as it does not beconm singular. In the remainder of this chapter, we construct the short-time existence theory. In the problems of higher order we consider, even the linear theory is incomplete, and we have to develop it for ourselves. This is the goal of the following two sections. 2.2 T h e l i n e a r p r o b l e m In this section, we prove tlle existence of solutions to the linear parabolic equation of 2p-th order on a closed manifold. There are numerous proofs of the corresponding result in a euclidean domain - - see, for instance, [18] or [50]. But these typically rely on the construction of a fundamental solution to the equation, a technique whidl is not easily adapted to the manifold setting. Friedman [21] describes an abstract approach based on a variant of the Lax-Milgram lemma, developing ideas originally due to J. L. Lions [44] and F. q'~hves [51]. However, Friedman's account of tlle result is inaccurate. He proves the existence of a t i m e - W k'2 solution to the linear equation Dtu + Atu = g, Uo - 0 under the unacceptably strong assumption that g vanishes at time zero along with all its derivatives of order up to k - 1. Considering the equation as a physical process, this is tantamount to assuming there are no external forces at time t = 0. This is clearly an undesirable condition, and in no way a natural one. This is in itself not a mistake, though it limits the usefulness of the theorem, bS-iedman goes on to claim in a remark, however, that one may prescribe the initial values Uo freely by considering the equation for u - Uo. But this gives an equation whose forcing term no longer satisfies the vanishing condition; the theorem as Friedman states it does not apply. In this section, we use techniques related to Friedman's to prove a minimal existence result. This will be strengthened later when we prove the natural a priori estimates for the linear problem. Let ( M ", g) be a smooth, compact Riemannian manifold. Let A be a linear differential operator of order 2p on M"; that is, for a 2p-times differentiable function u : M n --* IR, we set A u ( x ) = ~ A "'='"'~ k (z)V,,i~...~u(x), k Q and handling the forcing term with Young's inequality, the lemma is proved. To establish that u has derivatives of higher order, we prove estimates for its difference quotients. Defining the difference quotients requires a continuous co-ordinate system, so we have to focus on a single co-ordinate patch. This means multiplying u with a cut-off function in space. L e m m a 7.11 l f u E W W , is a solulion to (7.7) with initial values uo E WP'2(M"), then u E L W ~ , with the estimate
+ IIglILL~ ' I1 T)'P if' + l v p (A,u)
This controls the terms in At in (7.10); now consider those which remain. The first term is of the form handled in (7.9):
(Dt(r
r
2
: a zr~+'Akhu12LL~-- 21 rf+,A~uo L~(M.)
(7.12)
while the forcing term is easily estimated if we now assume k _< p: g, A k- h k/.2p+2Aku~\ q h ]ILL~
~-
IM[LL." A[a('I~P+2Akhu)
LL~,
(7.13)
Tc
crystal structure
cubic
g
so(n)
T < Tc
phase transition >
orthorhombic S0(3)U1
UI=
microstructure
none (see Section 2.3)
m
a 0 0
.
.
.
0 3 7
S0(3)U6 0) 7 r
large variety observed (see Section 2.2)
Figure 2: A cubic to orthorhombic transition In view of (1.4) the matrices Ui are related by conjugation under the cubic group.
93
1.4
T h e basic p r o b l e m s
We slightly generalize the setting of the previous section and consider maps u : Q C R n -+ R m on a bounded domain ~ (with Lipschitz boundary if needed). In particular the Sobolev space W 1,~ agrees with the class of Lipschitz maps. Let K C M m• be a compact set in the space M mxn o f m • n matrices. Problem 1 (exact solutions): Characterize all Lipschitz maps u that satisfy
a.e. i n ~ .
Duck
Problem 2 (approximate solutions): Characterize all sequences uj of Lipschitz functions with uniformly bounded Lipschitz constant such that
dist (Duj, K ) -+ 0
a.e. in ~.
Problem 3 (relaxation of K): Determine the sets K e~ and K app C M mxn of all a n n e maps x ~ F x such that Problem 1 and 2 have a solution that satisfies u(x)
=
F x on
Of~,
uj(x)
=
F x on
Of~,
respectively. Problems 1-3 also arise in many other contexts, e.g. in the theory of isometric immersions. An important technical difference is that in geometric problems one is often interested in connected sets K (and hence C 1 solutions u) while we will usually consider sets with more than one component. For further information we refer to Gromov's treatise [Gr 86] and to Svers ICM lecture [Sv 95]. In the context of crystal microstructure discussed in the previous section the sets K e~ and K app in Problem 3 have an important interpretation. They consist of the a n n e macroscopic deformations of the crystal with (almost) zero energy. They trivially contain the set K of microscopic zero energy deformations but can be much larger. For the set K = S O ( 2 ) A U S O ( 2 ) B one obtains (see Section 4.5) that under suitable conditions on A and B the sets K app and K e~ contain an open set (relative to the constraint det F = 1), leading to fluid-like behaviour.
94
Problem 4: Find an efficient description of approximating sequences that eliminates nonuniqueness due to trivial modifications while keeping the relevant "macroscopic" features. We saw in Section 1.2 how failure of minimization can lead to "infinitely fine" microstructure. In practice crystal microstructures always arise on some finite scale (albeit on a wide range from a few atomic distances to 10 - 100 pro). Minimization of elastic energy alone may not be enough to explain this since there is no natural scale in the theory.
Problem 5: Explain the length scale and the fine geometry of the microstructure, possibly by including other contributions to the energy, such as interfacial energy. Another possible explanation for limited fineness is that infinitely fine mixtures are (generalized) energy minimizers but not accessible by the natural dynamics of the system. This is a very important issue, but we can only touch briefly on it in these notes and refer to Section 7.2 and the references quoted there.
95
2
Examples
It is instructive to look at some examples before studying a general theory related to Problems 1-3. These simple examples already show a rich variety of p h e n o m e n a and interesting connections with (nonlinear) elliptic regularity, functional analytic properties of minors and quasiconformal geometry. In the following K always denotes a subset of the space M mxn of m x n matrices,
re, n > 2 .
2.1
The two-gradient problem
Exact solutions: Let K = {A, B}. T h e simplest solutions of the relation DuEK are so called simple laminates, i.e. maps for which Du is constant in alternating bands t h a t are b o u n d e d by hyperplanes x 9 n = const (see Fig. 3). Tangential continuity of u at these interfaces enforces t h a t AT = B~- for vec-
u:
u=A
Figure 3: A simple laminate
tors ~- perpendicular to n and thus A - B has rank one and can be written as
B-A=a| In this case we say t h a t A and B are rank-1 connected. We recall t h a t the m a t r i x a | n has entries (a | n ) i j = ainj. If one assumes t h a t the interfaces between the regions {Du -- A} and {Du = B } are s m o o t h then
96
a similar a r g u m e n t shows t h a t t h e y m u s t be h y p e r p l a n e s with fixed n o r m a l n. Moreover no such s m o o t h a r r a n g e m e n t is possible if r k ( B - A) > 2. T h e following p r o p o s i t i o n gives a m u c h stronger s t a t e m e n t because it shows t h a t also a m o n g possibly very irregular m a p s there are no other solutions. 2.1 ([BJ 87], P r o p . l ) Let ~ be a domain in R ~ and let u : --+ R m be a Lipschitz map with D u C {A, B } a.e.
Proposition
(i) I f r k ( B - A) >> 2, then D u = A a.e. or D u = B a.e.; (ii) if B - A = a | n then u can locally be written in the f o r m
u(x) = A x + a h ( x . n) + coast where h is Lipsehitz and h' E {0, 1} a.e. I f Q is convex this representation holds globally. In particular, D u is constant if u satisfies an affine boundary condition u ( z ) = F x on 0 ~ .
Proof. T h e key idea is t h a t the curl of a gradient vanishes. By t r a n s l a t i o n we m a y assume A = 0 and thus D u = B)CE, for some m e a s u r a b l e set E C 0. For p a r t (i) we m a y assume in addition, after an affine change of the d e p e n d e n t and i n d e p e n d e n t variables, t h a t the first two rows of the m a t r i x B are given by the s t a n d a r d basis vectors el and e2 and thus Du 1 = elXE,
Du 2 = e2XE.
S y m m e t r y of the second distributional derivaties and the first equation i m p l y t h a t OjXE = 0 for j # 1 while the second equation yields OkXE = 0 for k # 2. Hence 1)XE = 0 in the sense of distributions a n d therefore XE = 1 a.e. or XE = 0 a.e. since Q is connected. To prove p a r t (ii) we m a y assume A = 0, a = n = el and thus 1)U 1 = e l ) ~ B , D u k : O, k : 2 , . . . , m . Hence u 2 , . . . , u m are c o n s t a n t a n d OkU1 = 0, for k = 2 , . . . , m. T h e r e f o r e u 1 is locally only a function of x 1 as claimed. If ~ is convex then u 1 is c o n s t a n t on the h y p e r p l a n e s x 1 = const t h a t intersect ~ a n d thus globally of the desired form. Finally if u = F x on c9~, t h e n F = (1 - A)B, A E [0, 1] since by the Gauss-Green theorem
,E, B =
/ Dudx=
/uQndTt~-l=
/ Fdx,
97
where n is the outer normal of ft. Extending u by F x on R n \ ~ we can argue as in the proof of (ii) to deduce u(x) = d x + a [t(x. n ) + b on a n, where h' 9 {0, 1 - A, 1}. Hence u(x) - F x since each plane x . n = const intersects the set where u = Fx. []
Approximate solutions: Consider again K = {A, B} and suppose B-A:a|
F=AA+(1-A)B,A 9
1].
We show t h a t there exist sequences uj with uniformly bounded Lipschitz constant such t h a t in ft
dist(Duy, {A, B}) -+ 0
in measure,
(2.1)
and
uj(x) = F x
Oft.
(2.2)
Note t h a t (2.1) and the bound on the Lipschitz constant imply t h a t convergence also holds in L p, Vp < oc. After translation we m a y assume
F =O,A=-(1-
A)a|
B= Aa|
Let h be the periodic extension of the function given by
h(t)-- { - ( 1 - A ) t A(t-1)
t 9 [0, A), t9
and consider
~;(x)
1
--
- a h(jx.
n).
3 Then Dvj E { A , B } a.e. and vj ~ O. To achieve the b o u n d a r y conditions consider a cut-off function ~ C C~176 oo)), 0 < ~ < 1 , ~ = 0 on [0,1/2], ~ = 1 on [1, oo) and let
uj(x) = ~(j dist(x, Ol2) )vj (x). Then uj -- 0 on Of~, Duj is uniformly bounded and Duy = Dvj except in a strip of thickness 1/j around 0f~. If follows t h a t uj satisfies (2.1) and (2.2). Various modifications of this construction are possible, and we return
98
in Section 3.2 to the question whether all a p p r o x i m a t i n g sequences are in a certain sense equivalent. Note t h a t due to the assumption B - A = a | n, the problem (2.1), (2.2) essentially reduces to the scalar problem discussed in E x a m p l e 3 of Section 1.2. We now consider the case r k ( B - A) > 2. We have shown t h a t in this case there are no nontrivial exact solutions. T h e a r g u m e n t used strongly the fact t h a t Du only takes two values and t h a t the curl of a gradient vanishes. It does not apply to a p p r o x i m a t i n g sequences. Nonetheless we have
2.2 ([BJ 87], Prop.2) Suppose that r k ( B - A) > 2 and that uj is a sequence with uniformly bounded Lipschitz constant such that
Lemma
dist(Duj, {A,B}) ~ 0
in measure in f~.
Then Duj ~ A in measure or
Duj ~ B in measure.
In particular the problem (2.1), (2.2) has only the trivial solution, F E {A, B} and Duj --~ F in measure. T h e proof uses the following fundamental properties of minors. We recall t h a t the semiarrow ---" denotes weak convergence. Theorem
2.3 [Ba 77, Mo 66, Re 67] Let M be an r x r minor (subdeter-
minant). (i) If p > r and u, v C
WI'p(~),
u -- V
E Wo'P(~) then
f M(Du) = f M(Dv). In particular f M(Du) : / M ( F ) ft
f~
if u = Fx on 0fL
(2.3)
99
(ii) I f p
> r
and if the sequence uj satisfies uj ~ u in WI'P(f~, Rm).
Then M ( D u j ) ~ M ( D u ) in L;/~(f~). Remark. Integrands f for which the integral f f ( D u ) only depends on the b o u n d a r y values of u are called null Lagrangians, since the Euler-Lagrange equations are a u t o m a t i c a l l y satisfied for all functions u. Affine combinations of minors are the only null Lagrangians and the only functions t h a t have the weak continuity p r o p e r t y expressed in (ii) (see also Section 4.3). Proof of Theorem 2.3. T h e main point is t h a t minors can be written as divergences. For n = m = 2 one has det D u = 01(ulc~2 u2) - 02(ulc~2~t2),
(2.4)
for all u E C 2 and hence for all u c W 1'2 if the identity is u n d e r s t o o d in the sense of distributions. More generally for n = m > 2 the cofactor m a t r i x t h a t consists of the (n - 1) x (n - 1) minors of D u satisfies div cof D u = 0,
(2.5)
i.e. Oj(cofDu)ij = 0
and thus det D u = l oj(ui(cofDu)ij), since F ( c o f F ) T = Id det F . Similar formulae hold for general r x r minors, see [Mo 66, Da 89, GMS 96] for the detailed calculations. T h e multilinear algebra involved in these calculations can be expressed very concisely t h r o u g h the use of differential forms. In this setting one has for n = m = 2 det D u d x 1 A dx 2 = du 1 A du 2 = d (u 1 A du2), while for the r x r minor M ( D u ) t h a t involves the rows 1 , . . . , r columns 1 , . . . , r one has M(Du)dx 1A...dx n
T A d x r+l A . . . A d x
and the
=
du 1 A . . . A d u
n
=
d ( u 1 A d u 2 A . . . d u TA d S + l A . . . A d x n ) .
100
In either formulation (i) follows from the Gauss-Green (or Stokes) theorem (and a p p r o x i m a t i o n by s m o o t h functions) while (ii) follows from induction over the order r of minors and the fact t h a t uj converges strongly i n / 2 . [] Proof of L e m m a 2.2. We may assume A = 0 and t h a t there exists a 2 x 2 minor M such t h a t M ( B ) = 1. By assumption there thus exist sets Ej such
that D u j - BXEj --+ 0 in measure ,
(2.6)
and hence in L p for all p < ec. Moreover there exists a subsequence (not relabelled) such t h a t XE~ - - 0 in L ~ ( f t ) ,
uj - - u in W l ' ~ ( f t , Rm).
(2.7)
It follows from T h e o r e m 2.3 and (2.6) BXEj M(B)XE,
*" D u = BO
(2.8)
*" M ( D u ) = M ( B ) O 2
Combining the first convergence in (2.7) and (2.8) we see t h a t 0 = 0 2 a.e. Thus 0 must be a characteristic function XE. Hence (2.7) implies t h a t (use e.g. the fact IIXEjlIL~ ~ IIXEIIL2) XEj --+ 0 = XE
in measure.
Therefore by (2.6) D u j ---+ D u = B X E
in measure.
Finally L e m m a 2.1 (i) implies t h a t D u = B a.e. or D u = A = 0
2.2
a.e.
[]
Applications to crystal microstructures
Before proceeding with the m a t h e m a t i c a l discussion of the problem D u C K let us briefly review what can be learned a b o u t crystal microstructure so far. Which microstructures can form and why are t h e y so fine? First let us consider again the r61e of rank-1 connections. In the continuum t h e o r y discussed in the previous section they were related to continuity of the tangential derivatives or to the fact t h a t the curl of a gradient vanishes
101
(in Section 2.6 we still s t u d y the connections with the Fourier transform). T h e condition can also be u n d e r s t o o d in the discrete setting of crystal lattices. Two homogeneous lattices, obtained by a n n e deformations A and B of the same reference lattice can meet at a c o m m o n plane S only if the deformations differ by a shear t h a t leaves S invariant. Analytically we recover the condition B - A = a | n, where n is the normal of S (see Figure 4). n
o
o
o
o
o
~AO
o
o
o
o
~
@
9
9
9
o
o o
|
B
o o
9
o o
|
O O
Q
o
o
o
o
0
0
0
0 /
9
9
9
9
9
9
9
9 ~.~,f
o
o
o
0
0
0
9
aO
oOe
9
9
9
A 0
0 9
B
4/3
Figure 4: C o m p a t i b l e and incompatible lattice deformations. On the left the condition B - A = a | n is satisfied, on the right B = Id, A = 4 / 3 Id, so the condition is violated. After deformation there is no interface on which the two lattices meet.
Under certain additional conditions the two sublattices are referred to as twins. T h e r e are different definitions what precisely constitutes a twin; a c o m m o n requirement is t h a t B = QAH, where Q E SO(3) \ {Id}, Q2 = Id and where H belongs to the point group of the crystal, see [Ja 81] and [Za 92] for further discussion. C o m p a t i b l e lattice deformations can be arranged in alternating bands of different deformations, see Figure 5 (cf. also Fig.3). If the set K E M m• of minimizing affine deformations contains more rank-1 connections then more complicated p a t t e r n s such as the double laminates (or 'twin crossings') in Figure 6 are possible. In this way one can explain the observation of a n u m b e r of mierostructures t h r o u g h an analysis of rank-1 connections. T h e constructions based on rank1 connections, however, involve no length scale. Why, then, are the observed structures often so fine? For the situation of just two deformations A and B Proposition 2.1 (ii) and the discussion of a p p r o x i m a t e solutions provide an explanation. As soon as one imposes a nontrivial a n n e b o u n d a r y condition F = hA + (1 - )~)B there are no exact solutions, and a p p r o x i m a t e solutions become the b e t t e r the finer A and B are mixed (in a real crystal, additional contribution to the
102
Figure 5: Compatible lattice deformations can be arranged in laminar patterns. Schematic drawing (left), atomic resolution micrograph of fine twinning in Ni-A1 (middle; courtesy of D. Schryvers, RUCA, Antwerp), twinning in Cu-A1-Ni (right; courtesy of C. Chu and R. D. James), grey and black represent two different lattice deformations.
energy may eventually limit the fineness, see Section 6). In practise boundary conditions are often not so much imposed globally but by contact with other parts of the crystal where other deformation gradients prevail (e.g. because the phase transformation has not yet taken place there). A typical example is the frequently observed austenite/finely-twinned martensite interface (see Figure 7). In an idealized situation this corresponds to a homogeneous affine deformation C on one side of the interface and a fine mixture of A and B on the other side. Neither A nor B are rank-1 connected to C but a suitable convex combination AA + (1 - A)B is. There is no deformation that uses all three gradients A, B and C and only these (see the end of the proof of Proposition 2.1). However, the volume fraction of gradients other than A, B and C can be made arbitrarily small by matching C to a fine mixture of layers of A and B in volume fractions A and 1 - A. The analysis of the rank-1 connections determines the volume fraction A as well as the interface normals n and rn, in very good agreement with experiment; see [BJ 87], Theorem 3 and [JK 89], Section 5 for a detailed discussion. More complex patterns like the wedge microstructure in Figure 8 can be
103
Figure 6: Twin crossings on Cu-A1-Ni (courtesy of C. Chu and R. D. James) and schematic drawings of the different deformation gradients and their rank1 connections (indicated by solid lines).
understood in a similar vein. In this particular case so many rank-1 connections are required that the microstructure can only arise if the transformation strain satisfies a special relation; see [Bh 91], [Bh 92] for a comparison of theory and experiment. The considerations in this subsection focused on constructions of microstructures based on rank-1 connections. Do these constructions cover (in a suitable sense) all possible microstructures? We return to this fundamental question in the remainder of this Section and in particular in Sections 4.3, 4.6 and 4.7.
104
(1 ~.h
A
Figure 7: Austenite/finely twinned interface in Cu-A1-Ni (courtesy of C. Chu and R. D. James), schematic distribution of deformation gradients and rank1 connections; a simple model for the refinement (branching) of the A/B twins towards the interface with C is discussed in section 6.2.
Id
(
Figure 8: Wedge microstrueture in Cu-A1-Ni (courtesy of C. Chu and R. D. James). The necessary rank-1 connections between the six orthorhombie wells SO(3)Ui and the untransformed phase only exist for special transformation strains U1.
105
2.3
The one-well p r o b l e m
The simplest set K that is compatible with symmetry requirements (1.3) and (1.4) is K = SO(n). In this case approximating sequences must converge strongly. T h e o r e m 2.4 ([Ki 88], p.231) Suppose that
Du C SO(n) a.e. in ~. Then Du is constant and u(x) = Qx + b, Q E SO(n). If uj is a sequence of functions with uniformly bounded Lipsehitz constant such that dist(Duj, SO(n)) --~ 0
in measure,
(2.9)
then Duj -+ const
in measure.
Proof. To prove the first statement recall from (2.5) that div cofDu = 0 for any Lipschitz map. Now c o f F = F for all F E SO(n) and thus u is harmonic and therefore smooth. Moreover IDul 2 = n, where IFI 2 = t r F T F = Y~,i,j F./}, and therefore
21DSu[2 =/XlDul 2
-
2 Du
. D/Xu
=
O.
Thus Du is constant. To prove the second assertion of the theorem we may assume that uj *" u in Wl,~176 Rm). Consider the function
f(F)=IFI n-c~detF,
c ~ = n n/2.
One easily checks that f > 0 and that f vanishes exactly on matrices of the form AQ, )~ >_ O, Q c SO(n) (use polar decomposition, diagonalize and apply the arithmetic-geometric mean inequality). Hence (2.9), the weak continuity
106
of minors ( T h e o r e m 2.3) and the weak lower semicontinuity of the L '~ n o r m imply that 0
=
liminf
f f(Duj)dz
n---+O3 , 1 Q
= lim~f(/]D~jpd~-c~J'detD~dx) ~2
f~
T h e r e f o r e all the inequalities m u s t be equalities and in p a r t i c u l a r
f(Du) = 0 a.e.,
IID~jlI., ~ IID~IIL~.
It follows t h a t
Duj
Vu(x)
--+ Du :
in
L " ( f ~ , M "~•
~(x)Q(;),
A _> o,
(hence in m e a s u r e )
Q(x) e
so(N)
Moreover IDujl 2 = n a.e., whence [Dg[ 2 = n a.e. T h u s by the first p a r t of the t h e o r e m Du = const.
a.e.
Du C SO(n) a.e. and, []
T h e case n = 2 of the above result shows some interesting connections with the C a u c h y - R i e m a n n equations. Identify C _~ R 2 as usual via z = x+iy a n d let Oz = 1/2(0= - iOv) , O~ = 1/2(05 + JOy). Suppose t h a t 1 < p < oo and
dist(Duj, SO(2)) --~ 0 in LP(ft). T h e n in p a r t i c u l a r
(2.10)
[Ozuj[--+ 1 and O~uj ~ 0 in LP(f~, C),
and regularity for the C a u c h y - R i e m a n n o p e r a t o r implies t h a t there exists a function u s.t. u; --~ u in
wl'~(f~, C),
T h u s u is (weakly) h o l o m o r p h i c and Ozu = const.
O~u = O.
IOz~l =
limj~oo
[Oz~jl =
1.
Hence
107
2.4
The three-gradient problem
Theorem 2.5 ([Sv 91b]). Let K = {A1, A2,
A3} and
suppose that rk(Ai -
Aj) # 1. (i) If D u 9 K a.e. then D u is constant (a.e.). (ii) If uj is a sequence with uniformly bounded Lipschitz constant such that dist(Duj, K ) --+ 0
in measure
then Duj -+ const
in measure.
Proof of part (i). For simplicity we only consider the case n = m = 2. The general case can be reduced to this if one considers separately the cases that the span E of A2 - A1 and A3 - A1 contains two, one or no rank-1 lines and uses Lemma 2.7 below, see also [Sv 91b]. We may assume that A1 = 0 and thus det A2 # 0, det A3 # 0. Multiplying by A~-1 we may further assume A2 = Id. Using the Jordan normal form we see that after a change of variables we have either
A3=(
#) #A - A
'
A2 + # 2 r
or
A 3 = ( A0 # a ) '
A#0,#~{0,1}.
In the first case u satisfies the Cauchy-Riemann equations and is holomorphic and therefore smooth. Thus D u - A~ since K is discrete. In the second case D u E K implies that 01u 2 = O.
Hence u2(x) = h(x 2) (locally) and 02u2(x) = h'(x2). Since # • {0, 1} the value of 02u 2 uniquely determines one of the matrices Ai. Thus D u ( x ) = g(x2). In particular 0101 u = 0,
0201 u :
0102 u ~- 0
108
in the sense of distributions. Thus 01 u = const and D u = const | e2. Therefore r k ( D u ( x ) - Du(~)) < 1 and thus D u =- Ai.
+O(z 2) | []
An alternative proof that features an interesting connection with the theory of quasiconformal (or more precisely quasiregular) maps proceeds as follows. After possible renumbering we may assume that det(A2 - A1) and det(A3 - A1) have the same sign. Taking A1 = 0 and multiplying by diag(1,-1) if needed we have det A2 > 0, det A3 > 0. Thus D u C K implies that
[Dul 2 > k det D u for a suitable constant k. Hence u is quasiregular and a deep result of Reshetnyak says that either u = const or u is a local homeomorphism up to a discrete set Bu of branch points and that the (local) inverse u -1 preserves sets of measure zero (see [Re 89]). Hence either D u = 0 a.e. or D u ~ 0 a.e. In view of the results for the two-gradient problem this implies the assertion. The proof of (ii) requires more subtle arguments (see [Sv 915], [Sv 92b]). Svers first shows that after suitable transformations (and elimination of some simpler special cases) one may assume A i = A i T,
d e t A i = 1.
Now a gradient D u is symmetric if and only if u is itself a gradient Dv. Thus assertion (ii) is essentially reduced to a study of approximate solutions of the Monge-Amp~re equation detD2vj-+l,
vj : f~ c R 2 --+ R .
The difficulty is that, different from the usual literature on the MongeAmpere equation, one cannot assume that D2vj is positive (semi-)definite. Indeed a crucial step in the proof that uses ideas from the theory of quasiregular maps is to show that det Dev > 0 a.e. implies that v is locally convex or concave.
2.5
The four-gradient problem
The following example which was found independently by a number of authors (I am aware of [AH 86], [CT 93] and [Ta 93]; see [BFJK 94] for the
109
adaptation of Tartar's construction for separately convex functions to diagonal matrices) shows that the absence of rank-1 connections does not guarantee absence of microstructures (i.e. strong convergence of approximating sequences). L e m m a 2.6 Consider the 2 • 2 diagonal matrices A1 = -A3 = diag(-1, - 3 ) , A2:-A4:diag(-3,1) and let K : { A 1 , A 2 , A3, A4}. Then r k ( A i - Aj) # 1 but there exists a sequence uj d i s t ( D u j , K ) -+ 0
in measure,
and Duj does not converge in measure. Exercise: Show that there is no nontrivial solution of D u E K for the above choice of K. Hint: consult the previous subsection. It is not known whether there is another choice of four matrices with rk(Ai - Aj) ~ 1 for which nontrivial solutions exist. It is known, but not trivial, that for each e > 0 there exist nontrivial maps such that dist(Du, K) < (see the discussion after Theorem 5.4). Note that for small e the set of admissible gradients still contains no rank-1 connections.
Proof. Since K contains no rank-1 connections the key idea is to 'borrow' four additional matrices J~ (see Fig. 9) and to successively remove the regions where D u assumes Ji- We will construct a sequence Vk that satisfies the arlene boundary condition v (x) = a4x
on 0 Q = 0 ( 0 , 1 ) 2.
As a first approximation we may take v (~ = J4x. To increase the measure of the set where the gradients lie in K we observe that J4 is a rank-1 convex combination of A1 and J1, 1
J4--=~al+~
1j
1.
As in Section 2.1 we can thus construct a map v (1) that agrees with v(~ 0Q and uses only gradients A1 and J1 (in layers of thickness 1/2k) except for a boundary layer of thickness c/k where the gradient remains uniformly bounded. In the next step we replace the stripes where Dv O) = J1 by fine
110
J,
Aa
F22
3--
gl
A2
&
1
&
1
3
J3
A4
Ell
A1 Figure 9: Four incompatible matrices that support a nontrivial minimizing sequence
layers of A2 and J2 and k new boundary layers of thickness c/k 2. This yields v (2) (see Fig. 10). The volume fraction of the Ji phases has been decreased to (89 (up to small corrections due to the boundary layers). If we replace J2 by fine layers of A3 and J3 (with k 2 boundary layers of thickness c/k 3) we obtain v (a) and replacing Ja by A4 and J4 we obtain v (4). Up to the boundary layers Dv (4) only uses the values Ai and J4. Compared to v (~ the volume fraction of the set where J4 is taken has been reduced from one to (slightly less than) (1/2) 4. The volume fraction of the boundary layers is bounded by c
k C
~n t- k2 -t-
k2 c
F-I-
k3__c
c
k4 = 4 ~ .
Hence we have 4c 1 [{Dr(4) r K}I -< k- + 1-6 To further reduce the volume fraction of the set Dv ~ K we can now apply the same procedure to each of the small rectangles where D v (4) ---- J4.
111
c/k
1/k 2 ~
~
A3
~
J3
""~
-c/k2
I. . . .
"
J1 A1 J1 Jl
1/k
"', --a]k
"
.-"
v(Z) A2
J2
Figure 10: T h e first three stages in the construction of vk.
After 1 iterations we obtain
]{Dr(4/)~ K}] ~(~+1) T h e upper b o u n d d ( n , n ) = n 2 - n is sharp exactly in dimension n = 2, 4 and 8. See [ B F J K 94] for further information. --
2
115
3
Efficient description Young measures
3.1
of minimizing
The fundamental theorem
sequences
-
on Young measures
We have seen in the examples in Section 1.2 and 2.1 that there are usually many minimizing sequences for a variational problem. We return now to the question whether all these sequences have some common features and whether one can describe the 'macroscopic' features of a sequence without paying attention to unnecessary details. Closely related is the issue of defining a notion of generalized solution for variational problems that do not admit classical solutions. A reasonable condition for an object that describes the macroscopic behaviour of a sequence zj " E --+ R d is that it should determine the limits of
f f(zj) U
for continuous functions f (such as energy-, stress- or entropy density) and for all measurable subsets U of E. Such an object exists and was first introduced by L.C. Young in connection with generalized solutions of optimal control problems. By C0(R d) we denote the closure of continuous functions on R a with compact support. The dual of C0(R d) can be identified with the space M ( R a) of signed Radon measures with finite mass via the pairing
(1~, f ) = .~ f dp. I~ d
A map # 9 E --+ M ( R d) is called weak, measurable if the functions x ~-+ (#(x), f ) are measurable for all f E C0(Rd). We often write #x instead of
T h e o r e m 3.1 (Fundamental theorem on Young measures) Let E C R n be a measurable set of finite measure and let zj : E ~ R d be a sequence of measurable functions. Then there exists a subsequence zjk and a weak* measurable map u : E --+ M (R d) such that the following holds (i) ux ~ 0,
[lu=IIM(R~): f dux ~ 1, Rd
for a.e. x E E.
116
(ii) For all f C C0(R d)
f(zjk ) - - f where
/(x)
in L ~ ( E ) ,
f = ( ~ , f ) = / fdy~. Rd
(iii) Let K C R d be compact. Then supp~x C K
if dist(z/k, K ) -+ 0 in measure.
(iv) Furthermore one has
(i') II,~ll~= 1
for a.e. x 9 S
if and only if the sequence does not escape to infinity, i.e. if lim sup[{Izjk I > M}I = 0 . M--+c~
(3.1)
k
(v) /f (i') holds, i r a C E is measurable, if f 9 C ( R d) and if
f(zjk ) is relatively weakly compact in n l ( d ) , then f(z3k ) -~ ] in LI(A),
f ( x ) = ( ~ , f).
(vi) If (i') holds, then in (iii) one can replace 'if' by 'if and only if'.
Remarks. 1. The map ~ : E -+ A J ( R d) is called the Young measure generated by (or: associated to) the sequence zj~. Every (weakly* measurable) m a p , : E --+ M (R d) that satisfies (i) is generated by some sequence Zk. 2. The assumption IEI < oc was only introduced for notational convenience, cf. [Ba 89]. In fact R d with Lebesgue measure can be replaced by a more general measure space ($, E, #), e.g. a locally compact space with a Radon measure. The converse statement in Remark 1 requires that # be non-atomic.
117
3. The target R d can be replaced e.g. by a compact metric space K. In this case one always has Iru ll= 1 a.e. The condition (3.1) has a simple interpretation if we replace R d by its one-point compactification K = R d U {oc} _~ S d and consider the corresponding family of measures $~ on K. Then II~l[ = 1 a.e., and (3.1) ensures that ~ does not charge the point oc. 4. If, for some s > 0 (!) and all j C N
f lzjl s 1 the choice f = id yields zjk -~ z, z(z) = (u=,id). (3.2)
Pro@ The point is to pass from the functions zj which take values in R e to maps which take values in the space of 2td(R d) of measures in R d. Thus we allow new limiting objects which do not take a precise function value at every point but a probability distribution of values. Let
Zj(x) = Then IIZj(x)[IM(Rd)= 1 and ( Z j ( x ) , f ) = f(zj(x)). Thus Z j belongs to the space L ~ ( E ; Jtd(Rd)) of weak. measurable maps # " E --+ J ~ ( R d) that are (essentially) bounded. Now it turns out L~(E,.Ad(Rd)) is the dual of the separable space n l ( E , C0(Rd)) (see e.g. [Ed 65, p.588], [IT 69, p.93], [Me 66, p.244]), where the duality pairing is given by
(,, g) = f(,(x),
g(x))dx.
, 2
E
Hence the Banach-Alaoglu theorem yields a subsequence such that
Zjk = 5zjk(. ) *" u
in L,~,(E,.Ad(Rd)).
(3.3)
Lower semicontinuity of the norm implies that II us I1< 1 for a . e . x . For e LI(E) and f E C0(R d) we denote by ~ | the element of LI(E, C0(Rd))
118
given by x ~ ,r
The definition of Zj and (3.3) thus imply
f
:
| f> f
E
f>d.
E
Hence (ii) follows, and considering all functions f > O, p > 0 we also deduce ~_>0. To prove (iii) it suffices to show that {uz, f) = 0
V f G C0(R d \ K).
(3.4)
Let f 9 Co(R d \ K). Then for every ~ > 0 there exist C~ such that If(Y)l c + C~ dist(y, K). Hence the hypothesis dist(zjk, K) --+ 0 in measure implies that ( I f l - e)+(zJk) -+ 0 in measure, and in view of (ii) we conclude that (.x, (Ifl - e) +} -- 0
for a.e.x.
Now (3.4) follows since ~ > 0 was arbitrary. The proof of (iv) and (v) is easily achieved by a careful truncation argument and the characterization of weakly compact sets in L 1 [Me 66], see [Ba 89] for the details. Finally the proof of (vi) follows by an application of (v) to the bounded function f = max(dist(-, K), 1). []
Remark. Since the span of tensor products ~ | f, ~ 6 L 1(gt), f E Co (Rd), is dense in L1(9; C0(Rd)) assertion (ii) of the theorem is equivalent to Zjk *" /2,
The measure Uxo describes the probability of finding a certain value in the sequence zjk(x ) for x in a small neighbourhood Br(xo) in the limits j -+ cx~ and r ~ 0. The following useful fact reflects this probabilistic interpretation. C o r o l l a r y 3.2 Suppose that a sequence zj of measurable functions from E to R d generates the Young measure ~ : E -+ A/I(Rd). Then
zj --+ z in measure
if and only if
Px = 6z(x) a.e.
Proof. If zj -+ z in measure then f ( z J -~ f ( z ) in measure for all f E C0(Rd). Hence by Theorem 3.1 (ii) one has @x, f) = f ( z ( x ) ) for all f E Co(R d) and thus ~ = 6z(~). If conversely ~x = 5z(~) a.e. we claim that limsup I{Izj - w I > ~}l-< I { I z - w l > e/2}l, j-4c~
119
for all piecewise constant measurable functions w 9 E --+ a d. To see this it suffices to consider constant functions w - a and to apply (v) with f ( y ) = P ( l Y - a l ) where ~ is continuous 0 _< ~ < 1, ~ = 1 oil [e, ec), p = 0 on [0, e/2]. Thus
limsup I{Izj j-~
-~l
> d l _< limsup [{[zj j-~oo
-~l
> ~/2}1 + I { ! w - zl > ~/2}1
_< 21{1~ - wl > e/4}[. The last term can be made arbitrarily small since measurable functions can be approximated by piecewise constant functions, and the assertion follows (note t h a t z is measurable since {vx}~eE is weak, measurable). [] An alternative approach to the 'if' part of the corollary is to apply Corollary 3.3 below to the Carath~odory function f ( x , y) = min(ly - z(x)l, 1).
3.2
Examples
a) Let h : R --> R be the periodic extension of the function given by
h(x) =
a if0<x S 2 C R 3 be a sequence of magnetizations that generates a Young measure z~. Then
/ ~(mj)dx ---~/ (t/x, ~9}d:c. The limit of f ]hmj 12, however, is in general not determined by the Young R a
measure (see Fig. 11). Indeed let f be the periodic extension of the sign function on [ - 1 / 2 , 1/2], let f~ = [0, 1] a and let
mj = f(jxt)elxn ; rha
f(jx2)etx~.
1 Both sequences generate the same (homogeneous) Young measure Z~x= ~@ + ~c~ ~1. On the other hand it is not difficult to verify t h a t Ilhm, llU~ 1 while II 0. First replace X~ by a smooth function ~ and show t h a t the resulting fields MJ and fI j satisfy curl Mj --> 0, div2~7la ---> 0 in H - l ; then use libra, -- hv, tl2_f
f
f(x, A, #)dS~(z)(A) | du~(p)d:c
f~ R m x M m X n
=f
f
f(x,u(x),.~)d~,x(A)dx.
Mmxn
The proof of the lower semicontinuity is thus reduced to the verification of the inequality
f
g(A)~'x(A) > g(Du(x)) = g(@x, id})
(3.12)
MmXn
for the function
g(A) = f(x,
A)
with 'frozen' first and second argument. To see when (3.12) holds we need to understand which Young measures are generated by gradients. This is the topic of the next section.
126
4
W h i c h Y o u n g m e a s u r e s arise f r o m g r a d i e n t s ?
To employ Young measures in the study of crystal microstructure we need to understand which Young measures arise from sequences of gradients {Duj}. As before f~ C R ~ denotes a bounded domain with Lipschitz boundary. D e f i n i t i o n 4.1 A (weakly. measurable) m a p ~ : f~ -~ A J ( M "~x~) is a W I'p gradient Young measure if there exists a sequence of m a p s uj : f~ --+ l:t m such that 2zj ~ L/, in W I , p ( ~ ; R m) (~x i f p = oo),
Using this notion we may reformulate Problem 2 (approximate solutions) as follows. Problem 2' Given a set K C M "~• Young measures u such t h a t
supppx C K
for
characterize all W 1,~ gradient
a.e.x.
An abstract characterization of gradient Young measures due to Kinderlehrer and Pedregal will be derived in Section 4.3 below. It involves the notion of quasiconvexity. Quasiconvexity, first introduced by Morrev in 1952, is clearly the natural notion of convexity for vector-valued problems (see Section 4.2) but still remains largely mysterious since it is very hard to determine whether a given function is quasiconvex. Therefore further notions of convexity were introduced to obtain necessary or sufficient conditions for quasiconvexity. We begin by reviewing these notions and their relationship. 4.1
Notions
of convexity
For a matrix F C M m•
let M ( F )
denote the vector t h a t consists of all
min(n,m)
minors of F and let d(n, m) :
E
(',~) (7) denote its length.
r=l
D e f i n i t i o n 4.2 A function f : M m•
-+ R U { + o c } = ( - o c , ec] is
(i) convex i f f ( A A + (1 - A)B) < A f ( A ) + (1 - A ) f ( B ) V A, B c M'~•
E (0, 1);
127
(ii) polyconvex if there exists a convex function g : R d(n'm) --+ R 0 {+oe} such that f(F) = g(M(F)); (iii) quasiconvex if for every open and bounded set U with ]OU] = 0 one has
f(f
+ D~)dz >
u
I(F)& = [glf(f)
V~ ~ W o'
(g; R ), (4.1)
u
whenever the integral on the left hand side exists; (iv) rank-1 convex, if f is convex along rank-1 lines, i.e. if f ( A A + (1 - A)B) < Af(A) + (1 - A)f(B) V A, B C
M mxn
with rk(B - A) = 1,
V k E (0, 1).
Remarks. 1. If f C C 2 then rank-1 convexity is equivalent to the LegendreH a d a m a r d condition 02 f OF 2 ( F ) ( a | b, a | b) -
02 f ~
(F)aib~aJbo > O.
2. Quasiconvexity is independent of the set U, i.e. if (4.1) holds for one open and bounded set with tcOUI = 0 then it holds for all such sets. If f takes values in R it suffices to extend ~2 by zero outside U and to translate and scale U. For general f one can use the Vitali covering theorem. 3. If f takes values in R and is quasiconvex then it is rank-1 convex (see L e m m a 4.3 below) and thus locally Lipschitz continuous (use t h a t f is convex and thus locally Lipschitz ill each coordinate direction in Mmxn; see [Da 89], Chapter 2, Thm. 2.3, or [MP 98], Observation 2.3 for the details). In this case the integral on the left hand side of (4.1) always exists. It is sometimes convenient to consider quasiconvex functions t h a t take values in [-cx~, oo). The argument below shows t h a t such functions are rank1 convex and thus either take values in R or are identically - o c . If n = 1 or m = 1 then convexity, polyconvexity and rank-1 convexity are equivalent and they are equivalent to quasiconvexity if, in addition, f takes values in R.
128
Lemma
4.3 If n >_ 2, m >_ 2 then the following implications hold
f
convex
f
polyconvex
f
quasiconvex ~ f < oc ~( if rn > 3 rank-1 convex
f
T h e most difficult question is whether rank-1 convexity implies quasiconvexity. Svers [Sv 92a] ingenious counterexample solved this long standing problem in the negative if m > 3; the case m = 2, n _> 2 is completely open.
Proof. T h e first implication is obvious, the second follows from the fact t h a t minors are null Lagrangians (see T h e o r e m 2.3) and Jensen's inequality. To prove the last implication let f be quasiconvex, consider A, B E M re• with r k ( B - A) = 1, and a convex combination F = AA + (1 - A)B. After translation and rotation we may assume t h a t F = 0, A = (1 - A)a @ e~, B = - A a | el. Let h be a 1-periodic sawtooth function which satisfies h(0) = 0, h' = (1 - A) on (0, A) and h' = -)~ on (A, 1). Define for x C Q = (0, 1) '~ uk
= ak-lh(kzl),
vk
= a m i n { k lh(kxl),distoo(x,Q)},
where d i s t ~ ( x , Q) Ilxll~
= inf{llx - YlI~: Y e Q}, = sup{Ixil,/= 1,...,n}.
T h e n Dvk E { A , B } U {+a| = 0 on OQ, and I{Dvk r k --+ 0 (see Fig. 12). It follows from the definition of quasiconvexity t h a t
0 as
Af(A) + (1 - )~)f(B) = k~lim / f(Duk)dx = k.o~limf f(Dvk)dx > f(O), Q Q as desired. Note t h a t the inequality Af(A) + (1 - A)f(B) >_ f(O) still holds if f takes values in [ - o c , oc). As for the reverse implications, the minors (subdeterminants) of order greater t h a n one are trivially polyconvex but not convex. An example of a
129
-a
|
e2
I I
A
B
A
B
A
B
A
B
~(~ - x ) / k
a| Figure 12: The gradients of vk, for n = 2. quasiconvex but not polyconvex function is given below. Sver~k's counterexample of a rank-1 convex function that is not quasiconvex will be discussed in Section 4.7. []
Remark. The proof that quasiconvexity implies rank-1 convexity is similar to Fonseca's ([Fo 88], Theorem 2.4). In fact her method yields a slightly stronger result: if f : M m• ~ [ - e c , cc] is finite in a neighbourhood of F and quasiconvex then f does not take the value - o c on any rank-1 line through F a n d f isrank-1 Convex at F, i.e. Z(F) < A I ( F - ( 1 - . ~ ) a | +(1 - A)f(F + Aa | b), V a C R n, b C R "~, A E (0, 1). To obtain this refinement it suffices to replace disto~(x, Q) in the definition of vk by E distoo(x, Q) for small enough e > 0. The following example, due to Dacorogna and Marcellini [AD 92], [DM 88], IDa 89], may serve as a simple illustration of the different notions of convex-
130
ity. Let n = rn = 2 and consider
f(F)
= Ill
- vlfl
a c t F.
(4.2)
Then f f f f
convex polyconvex quasiconvex rank-1 convex
~=~ ~ ~ ~
17l _< 5 I?l < 2, 171 < 2 + e, bl-
0; whether or not 2 + e = ~a is open. Alberti raised the following interesting question which shows how little we know about quasiconvexity. Let 2 _< n _< m and let 9 " M'~Xn ~ R, .~: M n• ~ R , ~ ( F ) = g ( F ) . 9
Question (Alberti): g quasiconvex -', )- ,0 quasiconvex. Obviously equivalence holds for the other three notions of convexity. Kru~ik recently answered Alberti's question in the negative if 9 is allowed to take the value +oc and m _> 3. Refining his argument one can show that SverSk's quartic polynomial provides a finite-valued counterexample (see the end of section 4.7). Ball, Kirchheim and Kristensen [BKK 98] recently solved a long-standing problem by proving that the quasiconvex hull of a C 1 function f (i.e. the largest quasiconvex function below f) is again C 1, provided that f satisfies polynomial growth conditions. The representation of the quasiconvex hull through gradient Young measures (see Section 4.3) plays a crucial r61e in their argument.
4.2
Properties of quasiconvexity
Quasiconvexity is the fundamental notion of convexity for vector-valued variational problems. It is closely related to lower semicontinuity of integral functionals, existence and regularity of minimizers and the passage from microscopic and macroscopic energies. Quasiconvex functions are the natural dual objects to gradient Young measures (see Section 4.3).
131
In the following Q always denotes a bounded (Lipschitz) domain in R n and we consider maps u 9 f~ --+ R m and the functional
I(u) = . ~ f ( D u ) d x
In this section we merely summarize the results. p = oc are given in Sections 4.8 and 4.9 below. references can be found at the end of these notes. Theorem
4.4 Suppose that f 9 M m•
Some of the proofs for Further comments and
-+ R is continuous.
(i) The functional I is weak, sequentially lower semicontinuous (w*slse) on W I ' ~ ( Q ; R m) if and only if f is quasiconvez. (ii) Suppose, in addition, that
0 Jt4(M mx~) is a W l'p gradient Young measure if and only ifvx > 0 a.e. and the following three conditions hold (i) f
f
IF]Pdvx(F)dx < oc;
ft M - ~ x -
(ii) (vx, id)
-- Du,
u 9 Wl'P(f~;Rm);
(iii) (v~,f} _> f(@~,id}) for a.e. x and all quasiconvez f with [ f l ( r ) < c ( I F I p + 1). Young measures arise naturally as generalized solutions of variational problems that have no classical solution. To this end extend the functional
I(u) = f f ( D a ) d x f~
135
on functions to a functional ,
J(@ =
@~,f}dx
on Young measures. For v E WI'p(~; R ~") consider the admissible classes ,A
{~t E W l ' p ( ~ ; a r r ~ ) : u - v E ~'~'P(~'~;Rrn), ~- {11" ~ -"+ ./~(1~ rn) "12 I'V l'p gradient Young measure, @x, id) = D u ( x ) , u C A } .
=
T h e o r e m 4.9 Suppose that f is continuous and satisfies C(IflP+l),c>0,p> 1. Then
cIFF
0.
(4.13)
Various normalizations are possible. Multiplication by A -1, polar decomposition and diagonalization show, for example, that it suffices to consider A = (10 ~),
B=
(~
P0)'
0 1 then there are no rank-1 connections in K; (ii) if A = 1 (and A r B) each matrix in K is rank-1 connected to exactly one other matrix in K;
139
(iii) if .~ < 1 each m a t r i x in K is rank-1 connected to exactly two other matrices in K . T h e o r e m 4.11 Suppose that K given by (4.13) contains no rank-1 connections. Then every Young measure ~ " f~ -+ M ( M 2• with s u p p l . E K is a constant Dirac mass. Moreover [(lc = Krc
= Kqc = Kpc
= [4]
(4.15)
Remark. It is not known whether the same result holds for K = SO(3)ALJ S O ( 3 ) B C M3• some special cases are known ([Sv 93a], [Ma 92]). Proof. T h e crucial observation is that det(F-G)
>0
VF, G E K ,
Fr
(4.16)
By s y m m e t r y and SO(2) invariance it suffices to verify this for G = Id. T h e inequality clearly holds for F = B (by the above exercise) and hence by connectedness and the absence of rank-1 connections for G E S O ( 2 ) B . Similarly det(Id - ( - I d ) ) > 0 and hence by connectedness (4.16) holds also for all other G E SO(2). To determine K qe consider first a homogeneous gradient Young measure u s u p p o r t e d in K and let ~ = @,id) denote its barycentre. We have for F, G E M 2• d e t ( F - G) = det F - c o f F : G + d e t G , where F : G = t r F t G = ~ i , j FijFij. T h e minors relations yield 0
0 and ~ is affine on the line t --9 F(t). A short calculation shows t h a t ~g(F(t))l~=o = (Qa, a) and the quadratic form -
Q = (det B - 1)Id + ~ ( / ~ - k ) [ ( c o f F ) P + P ( c o f F ) T] is indefinite and hence has a nontrivial kernel. Consider thus the rank-1 line F(t) = F + t ( # A)a| on which z2 and g vanish. Let to < 0 be defined by z~(to) = 0. Since g(F(to)) = 0 we deduce t h a t F(to) = (y(0), 0) E K . On the other hand f(F(O)) < 0 and using the fact t h a t g vanishes on F(t) we have f ( F ( t ) ) = (det B - 1 ) ( l y ( t ) l + I z ( t ) [ - 1 ) -+ ec as t --+ oc. Hence there exist t~ > 0 such t h a t f ( F ( t l ) ) = g(F(tl)) = 0 and therefore F(t~) E K Ir by the considerations above. Thus /~ = F ( 0 ) E K l~ and the p r o o f is finished. [] 4.6
A r e all m i c r o s t r u c t u r e s l a m i n a t e s ?
T h e o r e m s 4.7 and 4.10 completely classify gradient Young m e a s u r e s j~qc(K) and quasiconvex hulls [4]qc and thus lead to an abstract solution of problems 2 and 3 in Section 1.4. The catch is t h a t very few quasiconvex functions are known and t h a t the abstract results are therefore of limited use to u n d e r s t a n d specific sets K . A manageable necessary condition is given by the minors relations (4.7). In this section we discuss the issue of sufficient conditions, i.e. constructions of (homogeneous) gradient Young measures s u p p o r t e d on
143
More generally (see e.g. [BM 84]) if h " R ~ -+ R is locally integrable and periodic with unit cell [0, 1]" and zj is defined by (3.5), then zj generates a homogeneous Young measure ~, given by
/gd~= f R
g(h(y))d>
[0,1] n
For a Borel set B C R one has
~(B) = I(0, i)~ n t~-'(B)l. b) Let
j
I(~) =
1
' ( ~ _ i): + u 2 d z ,
0
let uj be a sequence such t h a t
I(~j) + o ,
~(o) =~j(1) =o,
(3.6)
and let zj = (Uj)x (cf. Example 2 in Section 1.2). Then zj is bounded in L 4, a subsequence generates a Young measure t, and bl~xbl= 1 a.e. If we let g(p) = min((p 2 - 1) 9, 1) we deduce from (3.6) t h a t = 1 - 2k(x) (3.7) and
ujk(a ) =
Y
zj~dx --+
0
/
(1 - 2A(x))dx.
(3.8)
0
By (3.6) uj -+ 0 in L 2 and thus A(x) = 1/2 a.e. Hence zjk generates the unique (homogeneous) Young measure 1.
16
By uniqueness the whole sequence zj generates this Young measure.
144
a given set K. The simplest case is K = {A,B}. connected every convex combination =A6A+(1-A)SB,
If A and B are rank-1
AE [0,1],
is a (homogeneous) gradient Young measure. It arises as a limit of a sequence of gradients D~tj arranged in a fine lamellar pattern (see Fig. 13).
\
(1 - A ) I j
Figure 13: Fine layering of the rank-1 connected matrices A and B generates the homogeneous gradient Young measure A54 + (1 - A)6B.
We saw in Section 2.5 t h a t this construction can be iterated for larger sets K . More precisely let C be a matrix t h a t is rank-1 connected to A A + ( 1 - A ) B . Then every convex combination ~' = #(A6A + (1 - A)6u) + (1 - #)5c
(4.18)
is a (homogeneous) gradient Young measure (see Figure 8). This construction can be iterated and motivates the following definition. D e f i n i t i o n 4.14 (IDa 89]) For a finite family of pairs (Ai, Fi) C (0, 1) x 31 m• tile condition (H1) is defined inductively as follows.
145
(i) Two pairs (/~1, F1), (-~2, f2) satisfy (H2) if rk(F2 - F1) < 1,
A1 + A2 = 1.
(ii) A family {(Ai, Fi)}i= f ( ( u , id))
for all rank-1 convex functions f : M "~• ---+R . In other words, the laminates supported on a compact set K are given exactly by J M ~ ( K ) . The question raised in the title of this subsection may now be stated more precisely: Are all gradient Young measures laminates? In view of Theorem 4.16 this may be concisely stated as M,,~ ? jt4q~. This would clearly be true if rank-1 convexity implied quasiconvexity. Conversely if JM rc = Ad qc then rank-1 convexity would imply quasiconvexity in view of the definition of Jtd r~ and the fact that fqc(F) = inf{@, f ) : u C M qc, (u, id} = F} (one equality follows from the definition of 3//% for the other use Theorem 4.5 (iii) for ft = (0, 1) ~, extend c2 periodically, let ~gk(x) = k - ~ F ( k x ) and note that {D~gk} generates a homogeneous gradient Young measure). In the next section we discuss SverSk's example that shows that rank1 convexity does not imply quasiconvexity if the target dimension satisfies m>3.
4.7
Sver~ik's counterexample
T h e o r e m 4.17 (Sverdk [Sv 92a]) Suppose that m > 3, n >_ 2. Then there exists a function f : M mx~ -+ R which is rank-1 convex but not quasieonvex. Using this result Kristensen recently showed that there is no local condition that implies quasiconvexity. This finally resolves, for m >_ 3, the conjecture carefully expressed by Morrey in his fundamental paper [Mo 52], p. 26: 'In fact, after a great deal of experimentation, the writer is inclined to think that there is no condition of the type discussed, which involves f and only a finite number of its derivatives, and which is both necessary and sufficient for quasi-convexity in the general case.' To state Kristensen's result let us denote by ~ the space of extended realvalued functions .f : M m• --+ [-oc, oc]. An operator 7) : C ~ ( M "~• --4 ,~
148
is called local if the implication f = g in a neighbourhood of F ~
P ( f ) = P ( g ) in a neighbourhood of F
holds. T h e o r e m 4.18 ([Kr 97@ Suppose that rn >_ 3, n > 2. There exists no local operator :P 9 C ~ ( M m• -+ 5c such that
:P(f) = 0 ~
f is quasiconvex.
By contrast, the local operator
:Prc(f)(F) = inf{D2 f ( F ) ( a | b, a | b) 9 a C R "~, b ~ R n } characterizes rank-1 convexity. At the end of this subsection we will give an argument of Svergk that proves Theorem 4.18 for m > 6. Most research before Sver&k's result focused on choosing a particular rank-1 convex integrand f (e.g. the Dacorogna-Marcellini example given by (4.2)) and trying to prove or disprove that there exists a function u E W01'~176 R "~) and F E M m• such that
[ f(F + Vu)dx < lf(F)dz" t /
t /
f~
f~
(4.22)
gver&k's key idea was to first fix a function u and to look for integrands f that satisfy (4.22) but are rank-1 convex. He made the crucial observation that the linear space spanned by gradients of trigonometric polynomials contains very few rank-1 direction and hence supports many rank-1 convex functions. To proceed, it is useful to note that quasiconvexity can be defined using periodic test functions rather than functions that vanish on the boundary. P r o p o s i t i o n 4.19 A continuous function f : M mxn -+ R is quasiconvez if and only if
f f(F +
>_f ( f )
Q
for" all Lipschitz functions u that are periodic on the unit cube Q and all F C M m•
149
P r o @ Sufficiency of the condition is clear since it suffices to verily condition (4.1) for Q (see Remark 2 after Definition 4.2). To establish necessity consider a periodic Lipschitz function u and cut-off functions ~k C C~((-k,k)) ~ such that 0 _< cpk _< 1, ~% = 1 on ( - ( k 1), ( k - 1)) ~ and IDol _< C. If we let vk = ~kU, Wk(X) = l v k ( k x ) then quasiconvexity implies that P
Ckn-1 .j
( k,k)'~
Q = k '~ f
f(s
+ D , w k ) ~ - O k '~ ~ >_ k ' V ( f )
- C k ,~
1,
, J
Q Division by k n yields the assertion as k ~ oo.
[]
P r o o f o f T h e o r e m 4f. 17. Consider the periodic function u 9 R 2 --+ R a
1 ( sin 27czl sin 27cx2 sin 27c(x I
)
u ( x ) = 27~
Then Du(x) =
q- X2)
COS271-321
0
0
cos 2rex 9
)
COS27r(a?i +2 ,2) COS27C(2,l + x 2) and L:=span{Du(2,)}x~R 2=
0
s
t
t
"r,s, tER
} .
The only rank-1 lines in L are lines parallel to the coordinate axes. In particular the function g ( F ) = - r s t is rank-1 convex (in fact rank-1 affine) on L. On the other hand S g(D~(2,)) = - 7 1 < 0 = g(0). (0,i) 2
(4.23)
To prove the theorem it only remains to show that 9 can be extended to a rank-1 convex function on M a• Whether this is possible is unknown. There is, however, a rank-1 convex function that almost agrees with g in L and this
150
is enough. Let P denote the orthogonal projection onto L and consider the quartic polynomial
L , e ( F ) = c j ( P F ) + 41FI 2 + IVl 4) + k l F - PFI 2. We claim t h a t for every e > 0 there exists a k(e) > 0 such t h a t f~,e(~) is rank-1 convex. Suppose otherwise. T h e n there exists an e > 0 such t h a t f~,e is not rank-1 convex for any k > 0. Hence there exist Fe C M '~x~, a e r R ~, be E R ~, lakl = [bel = 1 such t h a t
D2 f ,,e (Fk ) (o,k | be, o,e | bk) < O. Now
D2I~,(f)(X, X) -
D2g(PF)(PX, P X ) + 2elX[ 2 + e(41FI2[Xl = + 8 I F : Xl 2) + k i n - PXI 2. T h e t e r m D2g(PF) is linear in F while the third term on the right hand side is quadratic and positive definite. Hence Fe is b o u n d e d as k ~ cxD, and passing to a subsequence if needed we may assume Fe -+ F , a.e --+ a, be --+ b. Since D2f<e >_ D2f<j for k _> j we deduce
Dgg(PF)(Pa|
Pa|
+ 2~+jla|
P a Q b l 2 < 0Vj.
(4.24)
Thus P ( a Q b ) = a | i.e. 6 | E L. Therefore t ~-~ g(P([ c + t a | is affine, and the first t e r m in (4.24) vanishes. This yields the contradiction e
,I)3
for all periodic (Lipschitz) functions u : R 3 -+ R 2. We m a y assume t h a t (F + Du) r C L a.e. Since f(o,1)3 Du = 0 by periodicity we deduce t h a t F r C L and (Du) T C L a.e. Thus 02U 1 = 01 u2 = 0,
03(U 1 -- U 2) = 0.
T h e r e f o r e ~t 1 is independent of x 2, while u 2 is independent of x 1, and differentiation of the second identity yields 0103u 1 = 020au 2 = 0. Thus =
=
Du
=
(a'(z~) 0
+ d(x3),
0 b'(x3) ) c'(:~"2) d'(z 3) "
and an application of Fubini's theorem in connection with the rank-1 convexity of f yields the desired estimate. By a more refined argument one can show t h a t the function f
f~,k(f) = I ( P F ) + c ( I f l 2 + I f l 4) + k l f - PFI 2 considered above provides a finite-valued counterexample if c > 0 is small enough and k _> k(e). To show t h a t
o
,1) 3
/~,k(F + Du) - ]r
+ Df~,k(F)Dudx > O,
one introduces v = (?21,v2, v 3) and w = (w 1, w 2, w 3) by
P(D~)
z ~-
0
722
V3
V3
, (D(IQ) r -- P ( D ( t g ) T ~-
w I
0
,~U3
_W 3
and observes t h a t the differential o p e r a t o r
A(Dv) = (02?21 , 03?2 1 , 017) 2,
03722, 01?2 3, 02?2 3)
155
can be expressed as a linear combination of derivatives of w. Hence IIA(Dv)ltw 1.,(Q) < CII(D~) z - P(D~)TpIL~(Q) and the crucial ingredient in the p r o o f are the estimates
o,1) v~v2v3 dz f(o
,1) 3
_liminf[/'f(Dvj)dx+ f ( f ( D u jJ ~) - f ( D v j ) ) f~\~' d x j _ _ , ~ > ]f~lf(F) - 2MIf~ \ f~'l. Since f~' C C f~ was a r b i t r a r y the assertion follows for u = Fx and similarly for piecewise affine u. For a r b i t r a r y u E W 1'~ (fl, R m) the result is established by a p p r o x i m a t i o n as follows. For c o m p a c t l y contained subdomains f~' C C fY' C C fl there exist vk such t h a t vk is piecewise affine in [~', u = vk in f~ \ f~", IOvkl < C, Dvk -~ Du in measure (and hence in all LP,p < oc). To construct such vk first a p p r o x i m a t e u in f~" by a C 1 function and then consider piecewise linear a p p r o x i m a t i o n s on a sufficiently fine (regular) triangulation. Let Uj,k = uj + vk -- u. T h e n
uj,k -- vk
in WI'~(~, R ~) as j ~ oc,
(4.26) (4.27)
IDUy,kl < C Hence, by the previous result and the d o m i n a t e d convergence theorem
f f(Duj,k)dx
limk_~ liminf j-+er
f~,
>_l i m k ~ f f(DVk) dx f~
= ff(Du)dx
> ff(Du)-Clf~\~2' I
On the other hand by (4.27), the uniform continuity of f on compact sets and the convergence of Dvk in measure lim sup
k --+oo
j
f If(Duj,k) - f(Duj) I dx = O.
,] fl'
Hence lim inf/,_~ f~
f(Duj) dx > f f(Du) d x - 2Clf~ \ f~'l, f~
157
[]
and the assertion follows since ~ / w a s arbitrary.
Proof of Theorem J.5(iii) (formula for fqc). Let
Qf(F,U) :=
infI,oo ~ f f ( F + DT))dz. lul J u
r EW o
We have to show that f q c ( F ) = Qf(F, U). A simple scaling and covering argument shows that Qf is independent of U. By the definition of quasiconvexity Qf >_ Qfqc = fqc. To prove the converse inequality Qf _ ~
1/
I ( F + DO + D7){) dz - e
on U{.
gi
Set 7) = 0 + Y~ 7){ c W~'~(U, Rm). Rearranging terms we find
Qf(F + Dg) dz >_ f f ( F + Dp) dx - ~lUI u
u
>_ Qf(F) - ~lgl, and assertion (4.28) follows as e > 0 was arbitrary. Now (4.28) is enough to conclude that Qf is rank-1 convex and therefore locally Lipschitz continous (see Remark 3 after Definition 4.2). Hence Q f is quasiconvex by (4.28) and density arguments and therefore f q c = Q f . So far we have assumed that Q f does not take the value - o c . If Qf(F + DO) = - e c on Ui then an obvious modification of the above argument shows that (4.28) still holds. Hence Qf is rank-1 convex (see the proof of Lemma 4.3) and one easily concludes that f q c = Qf - -oe since the rank-1 directions span the space of all matrices. []
158
4.9
Proofs: classification
The main point is to show that Jensen's inequality for quasiconvex functions characterizes homogeneous Young measures (see Lemma 4.23). The proof relies on the Hahn-Banach separation theorems and the representation (4.5) for fqc. The extension to nonhomogeneous Young measures uses mainly generalities about measurable maps, in particular their approximation by piecewise constant ones. An important technical tool of independent interest is a truncation result for sequences of gradients sometimes known as Zhang's lemma. (Closely related results were obtained previously by Acerbi and Fusco based on earlier work of Liu.) It implies that every gradient Young measure supported on a compact set K C M "~• can be generated by a sequence {Dvj} whose L ~ norm can be bounded in terms of K alone. For the rest of this section we adopt the following conventions: K
is a compact set in M m•
U, f~ are bounded domains in R '~, IO~l : 10uI = 0. L e m m a 4.21 (Zhang's lemma). Let IKl~ = s u p { I l l : f ~ K}. 71,1 (i) Let uj E141oc ( R n ., R
rn
) and suppose that
dist(Duj, K) -+ 0 in L ~(R'~).
(4.29)
R m) such that Then there exists a sequence vj c 14~I'~(R"; ~o~
IDvjl _a
Vu E Ho(K),
implies (#, f) > c~ V# E .hdoq~(K).
(4.34)
162
Fix
f c C(K), consider a continuous extension to C0(M "~•
and let
fk(F) = I(F) + kdist2(F, K). We claim that lira fff(0) > a.
(4.35)
k--~ oo
Once this is shown we are done since by definition every # E
Mqor
satisfies
(#, f) = (#, fk) >_ (#, f~c) >_fqr Suppose now (4.35) was false. Then there exist 6 > 0 such that /~(0)
_< ~ - 26,
By Theorem 4.5(iii) there exist ~k C
v k.
WJ'~(Q; R m) such that
f fk(Duk)dy _liminf f
.]
0
k-~oo
d Q
fj(Dvk)
/ ( ~ , fj)d.~ = (Aw, fj) _> ~. Q This contradicts (4.36) as
fk >_f; if k > j, and (4.35) is proved.
[]
163
Proof of Theorem ~.~. Necessity of conditions (i)- (iii) was established in Section 4.3. To prove sufficiency we first consider the case t h a t the underlying deformation vanishes. Let
A = {~ c L~(a, M(Mm•
~ Mgc(K) a.e.}
denote the set of maps t h a t satisfy (i) - (iii) with Du = 0. We have to show t h a t every element of A is a gradient Young measure. To do so we use some generalities about measurable maps to approximate the elements of A by piecewise constant maps. First note t h a t the set of subprobability measures M~ = {# C A,I(M mxn) : v _> 0, I1~11_< 1} is weak* compact in 3d(MmX~). Hence the weak, topology is metrizable on M1. To define a specific metric let {f,} C Co(M m• be a countable dense set in the unit sphere of C0(M m• and let oo
d(~, ~') = ~
2-'1l.
i=1
The space (.hall,d) is a compact metric space. Since d induces the weak, topology, a map u : ft --+ M ( M m• t h a t takes (a.e.) values in . ~ 1 is weak* measurable if and only if u : f2 --+ (3/ll, d) is measurable. The set {L, C L ~ ( f t ; .h4(Mm• - u(z) E .M1 a.e.} is also weak, compact in L ~ (~; A,4 (Mm• (cf. the proof of Theorem 3.2). A metric ct t h a t induces weak* convergence on t h a t set may be defined as follows. Let {hj} be a countable dense set in the unit ball of L l(f~) and let
d(., .') = ~
2-~-Jl i} k I
0 then there exist xi r such t h a t ~'i := ~(xi) E M. There exist disjoint compact sets K~ C Ei such t h a t IEi \ K~[ < 1/k; (4.38) if IEil = 0 we take K~ : 0. The Ki have positive distance and thus there exist disjoint open sets Ui D Ki with ]0Ui[ = 0 (consider e.g. suitable sublevel sets of the distance function o f / ( / ) . Now Ei D /)i D Ki and thus d(,(x), ~i) < 1/k in Ki. The assertion follows from (4.38). []
165
5
Exact solutions
Approximate solutions are characterized by the quasiconvex hull I ( qc and Mqc(K) of Young measures. The construction of exact solutions is more delicate. In view of the negative result for the two-gradient problem (see Proposition 2.1) it was widely believed that exact solutions are rather rare. Recent results suggest that many exact solutions exist but that they have to be very complicated. This is reminiscent of rigidity and flexibility results for isometric immersions and other geometric problems (see [Na 54]; [Ku 55]; [Gr 86], Section 2.4.12). set
To illustrate some of the difficulties consider again the two-dimensional two-well problem (see Section 4.5) DuEK
a.e. i n f t ,
u=Fxon0ft,
(5.2)
I< = SO(2)A u s o ( 2 ) B , A=
Id,
B=diag(,~,p),
0 0 , 1 e t V = ( - 1 , 1) ~-~ x ((A - 1)e, Ae) and define v : V -+ R m by
-Aax~ v(x)=-eA(1-A)a+
(1-A)ax,~
if xn < 0 , if x n _ > 0 .
T h e n Dv E { A , B } and v = 0 at xn = e(A - 1) and x~ = cA, b u t v does not vanish on the whole b o u n d a r y OV. Next let n--1
h(z) = cA(1 - k)a E
Ixi["
i=1
T h e n h is piecewise linear and
IDhl =
(A(1 - A ) v ~ -
"gt=v+h.
1. Set
168
Note t h a t ~ > 0 on 0V and let u =
9 v 9 a ( x ) < 0}.
Then ~lu
is piecewise linear dist(Da, {A, B})
, ~lou = O, _< e•(1 - ~ ) v / g - 1, _
0 such t h a t the sets Ui = xi + r i U
are m u t u a l l y disjoint and If~ \ UiU~[ = 0. Define u by { rd2(~) u(x) = 0
if else.
xEgz,
Note t h a t D u ( x ) = D S ( "~: - x i ) ,
ifx C f~i.
ri
It follows t h a t u is piecewise linear, t h a t Ulon = 0 and t h a t dist(Du, {A, B}) < for a suitable e > O. Moreover bv choosing r'i < 1 one can also obtain the estimate for ]u - F x I. [] L e m m a 5.1 can be easily iterated, and using the notion of the lamination convex hull of a set (see Section 4.4) one obtains the following result. L e m m a 5.2 Suppose that U C M "~•
is open. Let v : f~ -+ R m be piecewise affine and Lipschitz continuous and suppose D v E U l~ a.e. T h e n there exist u : f~ -~ R m such that D u C U a.e. in f~,
u=vonOfL
The crucial step is the passage from open to compact sets K C M "~xn. Following Gromov we say t h a t a sequence of sets Ui is an in-approximation of K if (i) the Ui are open and contained in a fixed ball
(ii) gi C a~c~l
169
(iii) Ui --+ K in the following sense: if Fik E Ui~ , ik -+ oc and Fik --+ F, then FEK. T h e o r e m 5 . 3 ([Gr 8@ p. 218; [MS 9@. Suppose that K admits art inapproximation {Ui}. Let v E C I ( f , R "~) with Dv E UI. Then there exists a Lipschitz map u such that Du E K E f~ a.e.,
u=von0f.
Pro@ The proof uses a sequence of approximations obtained by successive application of L e m m a 5.2. To achieve strong convergence each approximation uses a much finer spatial scale t h a n the previous one, similar to the construction of continuous but nowhere differentiable functions. This is one of the key ideas of convex integration. We first construct a sequence of piecewise linear maps ui t h a t satisfy Dui E Ui a.e, sup ]ui+ 1 -- Uil < 5i+1, suPlul--VI_ c(~') > 0 for all z E ~'. Hence it is easy to obtain u l l ~ ' by introducing a sufficiently fine triangulation. Now exhaust ~ by an increasing sequence of sets ~i C C ~. To construct ui+l and (5i+ 1 f r o m u i and 5i we proceed as follows. Let f~ = {x E f : dist(x, O f f ) > 2-~}i Let p be a usual mollifying kernel, i.e. let p be smooth with support in the unit ball and f p = 1. Let p~(x) = e - ~ p ( x / 4 . Since the convolution Pc * Dui converges to ui in L l(~i) as e -+ 0 we can choose ci E (0, 2 -i) such t h a t IIP~i * Du~ - Du~I[L~ 1. Then the two-well problem ( 5 . 1 ) - (5.3)
has a solution if F E i n t K It,
where
K ~ C = { F = ( y ' z ) ' ] y I < A # - d eAt F# - I
' Izl -< detF-1}~_p__l .
Remark. A similar result holds if A# = 1 provided t h a t in the definition of an in-approximation and interior one considers relatively open sets subject to the constraint det F = 1. One only needs to use the remark after L e m m a 5.1 to achieve det Du = 1, provided t h a t det A = det B = 1. A more detailed analysis shows t h a t in the definition of an i n - a p p r o x i m a t i o n one can replace the lamination convex hull which is based on explicit rank-1 connections by the rank-1 convex hull defined by duality with functions (see Section 4.4). This has a striking consequence for the four-gradient example K=
•
03
01
discussed in Section 2.6, see in particular Figure 4. For any m a t r i x
and any open n e i g h b o u r h o o d U D K there exists a m a p u : ft --+ R 2 such that DuEU a.e. i n f t , u = Fx on Oft. This is true despite the fact t h a t small n e i g b o u r h o o d s contain no rank-1 connections so at first glance there seems to be no way to start the construction. This obstacle is overcome by first constructing a (piecewise linear) m a p t h a t satisfies Dv E U rc a.e. and Dv E U except on a set of small measure. One can then show t h a t the exceptional set can be inductively removed. T h e m a j o r o u t s t a n d i n g problem is whether in the definition of an ina p p r o x i m a t i o n one can replace the lamination convex hull (or rank-1 convex hull) by the quasiconvex hull. One key step would be to resolve the following question.
172
Figure 18: Structure of solutions with finite perimeter. The normals nl, n2 are determined by (5.4).
C o n j e c t u r e 5.5 Let K be a compact quasiconvex set, i.e. I ( qc = I ( and let u E ~/~qc([(). Then for every open ,set U D K there exists a sequence uj : (0, 1) ~ --4 a m such that D u j generates ~ and D u j 9 U a.e.
The conjecture is true for compact convex sets [Mu 97a]; this refines Zhang's Lemma (see Lemma 4.21) which implies the existence of uj such that D u j 9 B(0, R) for a sufficiently large ball.
5.2
Regularity and rigidity
The construction outlined above yields very complicated solutions of the twowell problem (5.1) - (5.3). This raises the question whether the geometry of the solutions can be controlled. Consider the set
E = {x e
D
(x) 9 S O ( 2 ) A }
where D u takes values in one connected component of K (or one phase in the applications to crystals). The perimeter of a set E C f~ C R ~ is defined as
For smooth or polyhedral sets this agrees with the ( n - 1) dimensional measure of OE. T h e o r e m 5.6 ([DM 95]). If u is a solution o f ( 5 . 1 ) - (5.3) and i f P e r E < oc then u is locally a simple laminate and OE consists of straight line segments that can only intersect at OfL
173
The proof combines geometric and measure-theoretic ideas. The geometric idea is that the Gauss curvature K(g) of the pull-back metric g = (Du)rDu should vanish (in a suitable sense). Since g only takes two values this should give information on E. One key step in the implementation of this idea is a finite perimeter version of Liouville's theorem on the rigidity of infinitesimal rotations (cf. Theorem 2.4). In this framework connected components are replaced by indecomposable components. A set A of finite perimeter is indecomposable if for every A1 C A with PerA = PetAl + PerA \ A1 the set A1 or A \ A1 has zero measure. It can be shown that each set of finite perimeter is a union of at most countably many indecomposable components.
T h e o r e m 5.7 S u p p o s e t h a t u 9 f~ c R ~ ~ R belongs to W I ' ~ ( f ~ ; R ~) and that det Du >_ c > O. Suppose further that E C f~ has finite perimeter and Du C SO(n)
a.e. in E.
Then Du is constant on each indecornposable component of E. To finish the proof of Theorem 5.6 one can decompose D u as ei~ 1/2 (where g = (Du) rDu E {ArA, B r B } ) and analyze the jump conditions at the boundary of each indecomposable component to deduce that (9 only takes two values and solves (in the distributional sense) a wave equation with characteristic directions nl and rz2. B. Kirchheim recently devised more flexible measure-theoretic arguments, and combining them with algebraic ideas he established a generalization of 3 Theorem 5.6 to the three-well problem K = U s o ( 3 ) u i in three dimensions i=1
with U1 = diag(A1, A2, A2), U2 = diag(At, A2, A~), U3 = diag(A2, A2, ~1), Ai > 0. A major additional difficulty in this case is that the gauge group SO(3) is not abelian and one cannot hope to derive a linear equation for a quantity like O in the two-dimensional situation.
174
6
Length
scales and
surface
energy
Minimization of the continuum elastic energy is a drastic simplification, in particular if a very fine mixture of phases is observed. It neglects interfacial energy as well as discreteness effects due to the atomic lattice. It is therefore not surprising that elastic energy minimization often predicts an infinitesimally fine mixture of phases (in the sense of a nontrivial Young measure), whereas in any real crystal all microstructures are of finite size. Nonetheless elastic energy minimization does surprisingly well. It often correctly predicts the phase proportions and in combination with considerations of rank-1 compatibility the orientation of phase interfaces. It recovers in particular the predictions of the crystallographic theory of martensite. In fact one of the major achievements was to realize that the predictions of that theory can be understood as consequences of energy minimization. This allows one to bring to bear the powerful methods of the calculus of variations in the analysis of microstructures. The problem that elastic energy minimization does not determine the length scale and fine geometry of the microstructure remains. It can be overcome by introducing a small amount of interfacial energy or higher gradient terms. One expects these contributions which penalize rapid changes to be small since otherwise a very fine structure would not arise in the first place. The most popular functionals are
+ / e2pD~ul2dx f~
(6.1)
f~
and
=
+ f l 2 fdx.
(6.2)
The second functional allows for jumps in the gradient and ID2ul is understood as the total variation of a Radon measure. The small parameter c > 0 introduces a length scale and as e -+ 0 both models approach (at least formally) pure elastic energy minimization. More realistic models should of course involve anisotropic terms in D~u or more generally terms of the form h(Du, eD2u). Even the basic models (6.1) and (6.2) are, however, far from being understood for maps u : f~ C R 3 --4 R a. In the following we discuss briefly two simple scalar models which already show
175
some of the interesting effects generated by the interaction of elastic energy and surface energy.
6.1
S e l e c t i o n of periodic s t r u c t u r e s
As a simple one-dimensional counterpart of the two-well problem consider the problem 1
Minimize I ( u ) = / ( u ~
- 1) 2 + 'u2 dx
(6.3)
0
subject to periodic boundary conditions. Clearly I ( u ) > 0 since the conditions u = 0 a.e. and uz = +1 a.e. are incompatible. On the other hand i n f I = 0, since a sequence of finely oscillating of sawtooth functions uj can achieve ujx E {=kl}, uj ~ 0 uniformly. For any such sequence uj~ generates 1 1 the (unique) Young measure u = ~5-1 + 561 (see Section 3.2b)). Note t h a t there are m a n y 'different' sequences t h a t generate this Young measure. Minimizers of the singularly perturbed functional 1
l~(u) = / ~2u 2= + ( u ~ - l ) 2+ u2 dz 0
yield a very special minimizing sequence for I.
T h e o r e m 6.1 If e > 0 is sufficiently small then every minimizer of I ~ (subject to periodic boundary conditions) is periodic with minimal period p~ = 4(2e)i/s + 0(e2/3). A more detailed analyis shows t h a t the minimizers u e look approximately like a sawtooth function with slope :t-1 and involve two small length scales: the sawtooth has period ~ el/3 and its corners are rounded off on a scale e (see Fig. 19). The heuristics behind the proof of Theorem 6.1 is simple and relies on two observations. First, the condition I~(u ~) ~ 0 enforces t h a t u ~ is almost a sawtooth function with slopes -t=1. Second, a key observation of Modica and Mortola is t h a t the first two terms of the energy combined essentially count (~ times) the number of changes in the slope from 1 to -1 and vice versa. Indeed the arithmetic geometric mean inequality yields for any interval (a, b) C (0, 1)
176 E
~x
E1/3
1
-1
Figure 19: Sketch of u ; for a m i n i m i z e r of I f
over which ux changes sign b
b
f 2 2~ = + ( ~ - 1) ~ dx ~ f 2 ~ 1 ( ~ - 1 ) ~ 1 d~ a
a
b >
e I f H'(u=)&:l ~ e IH(u~(b)) -H(u~(a))l a
~ fH(~) - H ( - 1 ) ] , where H ' ( t ) = 2It 2 - 11. On the other h a n d the above e s t i m a t e s can be m a d e s h a r p if one choose u as a solution of the O D E e U x x = ( u ~ - 1) 2, e.g. u~ = t a n h z -E. o T h e two observations strongly suggest t h a t (6.3) is essentially equivalent to the following "sharp-interface p r o b l e m "
177
1
Minimize
cAoN + / u 2 &c , /
(6.4)
0
among periodic function with luxl = 1. Here N denotes the number of sign changes of ux and A0 = H(1) H ( - 1 ) -- 8/3. For fixed N (6.4) is a discrete problem and a short calculation shows that in this case periodically spaced sign changes of u~ are optimal and the second term in the energy becomes 1 N - 2 . Minimization over N yields the assertion. The actual proof of Theorem 6.1 uses the expected analogy between (6.4) and (6.3) only as a guiding principle and proceeds by careful approximations and estimates for odes. Nonetheless it would be very useful to relate (6.4) and (6.3) in a rigorous way, also as a test case for higher dimensional problems where the fine ode methods are not available. Conventional F-convergence methods do not apply since the problem involves two small length scales and the passage from (6.3) to (6.4) corresponds to removing only the faster one (i.e. the smoothing of the sawtooth's corners). Recently G. Alberti and the writer developped a new approach that allows one to do that. One of the main ideas is to introduce a new variable y that corresponds to the slower scale and to view as a map V ~ from (0, 1) into a suitable function space X via V~(z) = v~(~c, .). One can endow X with a topology that makes it a compact metric space and study of the Young measure ~ generated by V ~. For each z c (0, 1) the measure ~'x is a probability measure on the function space X. If u ~ is a sequence of (almost) minimizers of I ~ then one can show that ~,~ is supported on translates of sawtooth functions with the optimal period 4 21/3. One easily checks that the asymptotic behaviour is the same for minimizers of (6.4) and this gives a precise meaning to the assertion that (6.3) and (6.4) are asymptotically equivalent. This approach is inspired by the idea of two-scale convergence ([A1 92], [E 92], [Ng 89]). A crucial difference is that two-scale convergence usually only applies if the period of the microstructure is fixed and possible phase shifts are controlled. This is the case if, for example, the solutions are of the form ~;(z, ~@-~)where ~ is periodic in the second variable.
178
6.2
Surface energy and domain branching
Consider the two-dimensional scalar model problem (see [KM 92] for the relation with three-dimensional elasticity) I
L
+ (u2y - 1)2dx dy 4 min 0
0
u = 0 on z = 0.
(6.5)
The integrand is minimized at Du = (ux, uy) = (0, =t=1). The preferred gradients are incompatible with the boundarv condition. The infimum of I subject to (6.5) is zero but not attained. The gradients Duj of any mini1 ~ 1 mizing sequence generate the Young measure ~b(0,-1) + ~5(0,1). One possible construction of a minimizing sequence is as follows (see Fig. 20). Let Sh be a periodic sawtooth function with period h and slope • and let u(z, y) = sh(y) for x > 5, u(x, y) = ~sh(y) for 0 < x < 5. Then consider a limit h -+ 0, 5 --~ 0 such t h a t h/5 remains bounded. Similar reasoning applies if we replace (6.5) by the condition t h a t u vanishes on the whole boundary of [0, L] x [0, 1].
u=0 h linear
interpolation
try = - - 1
J
i
h/2
u~=l
L Figure 20: Construction of a minimizing sequence.
To understand the influence of regularizing terms on the length scale and the geometry of the fine scaIe structure we consider
//,2 1
z
L
~X
0
0
+ (u~ - 1)2 + ~2u2yydx dy,
179
Figure 21: T h e self-similar construction with 1/4 < (9 < 1/2. generations of refinement are shown.
Only two
subject to (6.5). Instead of the second derivatives m y one can consider other regularizing terms, e.g. ID2ul 9. The derivatives in y are, however, the most i m p o r t a n t ones, since we expect t h a t fine scale oscillations arise mainly in the y direction. It was widely believed t h a t for small e > 0 the minimizers of I ~ look roughly like the construction uh,a depicted in Figure 20 (with the corners of the sawtooth 'rounded off' and optimal choices d(e), h(e)). This is false. Indeed a short calculation shows t h a t c~(e) ~ (eL) 1/2, h(e) ~ (eL) 1/2 and I~(u~h,L) ,.o e l / 2 L 1 / 2 . On the other hand one has Theorem
6.2 ([Sch 94]) For 0 < e < 1 there ezists constants c, C > 0 such
that ce2/aL */a < --
rain u~O
I~
< Ce2/aL ~/a.
a.t x ~ O
T h e u p p e r b o u n d is obtained by a s m o o t h version of the self-similar construction depicted in Figure 21. T h e m a t h e m a t i c a l issues become clearer if we again replace I ~ by a sharp interface version L
1
§ 0
subject to
0
duvyIdydx
(6.6)
180
Iuyt = 1 a.e.
(6.7)
Thus y ~-~ u(x, y) is a sawtooth function and fo luy~ldy denotes twice the number of jumps of uy. Minimization of (6.6) subject to (6.7) is in fact a purely geometric problem for the set E =
y).
y) = 1}.
The first term in d~ is a nonlocal energy in terms of E, while the second is essentially the length of 0 E (more precisely its projection to the z-axis; as before we consider this to be the essential part since oscillations occur mainly in the y direction). The functional and the constraint are invariant under the scaling which suggests a self-similar construction with @ = (89 T h e o r e m 6.3 ([KM 94]). For 0 < ~ < 1 one has c(2/3LI/3
R(F) -1, by Brezis & Coron [BC]. We refer to [Str2], [Str3], [Str4], [Wa], [BR] for further progress in this non-uniqueness problem. An intuitive idea to produce "large" H-surfaces with constant H is to minimize area in the class of surfaces x: U -~ R3 satisfying the Plateau boundary condition and a volume constraint (i.e. the volume enclosed by x and the cone over F is prescribed). The solutions x, whose existence was shown by Wente [Wen2], [Wen3], are solutions to "P(g, F) with a constant H which is, however, not prescribed but determined by the Lagrange multiplier associated with the volume constraint. Since these surfaces x are "large" if the prescribed volume is big, one can infer the existence of "large" H-surfaces with boundary F for all values of the constant H which do occur as Lagrange multipliers for big volumes. In [Ste2] it was shown that the set of these values accumulates at 0 from above and below, but it is not clear (and may not be true, in general) that it contains a punctured neighborhood of 0 aud, hence, Rcllich's conjecture could not be proved completely in this way. In the context of geometric measure theory F. Duzaar [Du3] has used the same approach to prove some results on "large" H-surfaces in general dimensions. With regard to unstable H-surfaces we mention that there is also an existence theory for "small" unstable surfaces of prescribed mean curvature (see [Heh], [Strh], [Str3], [ST]) which extends part of the extensive corresponding theory for parametric minimal surfaces.
216
In anothcr direction tile existence theory for the Plateau problem has been generalized replacing the unit disc U by a multiply connected domain in ]R2 or by an oriented compact surface with boundary. This is called the general Plateau problem or Plateau-Douglas problem, and the principal difficulty is that the conformal structure on the domain cannot be fixed a priori. In the energy minimization process the conformal structure therefore has to be varied and it may degenerate in the limit of a minimizing sequence. (Geometrically speaking, tile surfaces in the minimizing sequence may break up into a system of surfaces of simpler topological type.) With appropriate assumptions (so-called Douglas conditions) such a behaviour can be excluded, however, and the Plateau-Douglas problem then has a solution. Another variation of the theme is to replace the Plateau boundary condition by a free boundary condition with two degrees of freedom, i.e. the surfaces are required to have (part of) their boundary on a given 2-dimensional supporting manifold. Since the additional difficulties in all these generalizations occur already in the minimal surface case H = 0, we refer to the monography [DHKW] which also contains many discussions, hints to the literature, and bibliographical entries related to H-surfaces. Finally, we note that many of these results of the 2-dimensional parametric theory for H-surfaces, which we have just mentioned and which will not be discussed in these lectures, have also been treated ill the setting of geometric measure theory and thereby extended to g-hypersurfaces in p , + 1 (see [DF1], [DF2], [Du2]-[Du4], [DS1]-[DS4]).
2
The
method
of bounded
vector
fields
One convenient way to define the volume functional mentioned in Section 1 is to represent the prescribed (continuous, bounded) mean curvature function H as the divergence H = div Z of a vector field Z on ~ a and let VH(X) = / u ( Z o x ) . x u A Xv dudv. To see tile geometric meaning of VH(X) we consider the 1-form w on Ra which is dual to Z and note that div Z = H is equivalent with dw = H ~ where f~ is the Euclidean volume form on ]Rz. (It may be preferable to work generally with w instead of Z; we wanted to follow the historical development, however.) Then Vt4(x) is just the integral fv x#co of co over the parametric surface x. If x satisfies the Plateau boundary condition for the given boundary curve P and if Z, co are of class C 1, then Stokes' theorem tells us that, up to a constant depending on H and F only, VH(X) equals the H-weighted volume enclosed by x and the cone over F. Since we fix the boundary P and different choices of Z corresponding to the same function H will change the H-volume of surfaces spanning F only by an irrelevant constant, we have written VH(X) instead of V z ( x ) , abusing slightly the notation. Note that we have to assume boundedness of Z on the image o f x to secure the existence of the integral above for x e WI'2(U, p3). Of course, if we restrict our considerations to surfaces contained in a subset A of ]R3, then we need a bound for Z only on A. Assuming this, the above integral and hence also the energy functional EH(X) = D(x) + 2VH(X) (more precisely denoted E z ( x ) ) is defined for all surfaces x E Wt'2(U, A), i.e. for all x E WI'2(U, ]R3) mapping (almost all of) U into A. In the sequel we will in fact need the bound SUPA ]Z[ < 1 in order to have coercivity of the energy functional on Wx,2(U, A).
217
It is now routine to compute the first variation of EH, and one obtains, as is well-known and expected in view of the discussion in Section 1: 2.1 P r o p o s i t i o n (first v a r i a t i o n ) . Suppose A is closed in ~-s Z is a bounded C 1 vector field with div Z = H on A, and x E WI'2(U, A).
(i) If ~ e W~'2(U, [{3) is bounded with x + t~ e W"2(U, A) for 0 < t 0 for ~EW~'2NL~176 IRa) with x+t~EWI'2(U,A), 0 < t 4< 1.
220
For instance, if A is the closure of a C = domain in ~3, u is any C 1 extension to ~3 of the inner unit normal field along OA, V is a neighborhood of OA, and ~ C WIaN L~176 IRa), then we have 5Ei1(x;~) > 0
if ~.(uox) > 0 ahnost everywhere on x - l ( V ) .
'ib see this we choose 0 < 0 E :D(~ 3, IR) and observe that (0 ox)(c + E l~lv o z) with r > 0 is admissible in the variational inequality. Letting r tend to 0 and then 0 to the constant 1 in appropriate fashion we deduce the assertion. Note that x -I(V) is well defined up to a set of measure zero. Now, to obtain a variational equation instead of an inequality for the minimizers x of EH on $(F, A), one further ingredient of the theory is needed, namely a g e o m e t r i c i n c l u s i o n p r i n c i p l e asserting that x maps U into the interior of A. For the case of a ball A of radius R0 in R3 centered at the origin such a principle can be deduced from the maximum principle for subharmonic functions. For this one introduces the function f = Ixl2 ~ w'a(u, ~) and computes, for 0 _< 7/E WI'2(U, JR), (Vz)u 9 z u + ( V z ) v . z v = ~rluf,, ~ 1 + ~r/vf~ + r](]x=[2 + [x~12),
12(Hoz)~x.x= Ax, I
R and extend H continuously to the concentric ball A of 3 -I radius R0 such that suPA [HI < 5R0 and [al[H(a)l _ 0
for all ~ e W~ '~ n L~~ assertions hold:
rt 3) with x + t~ e W',~(U,A), 0 < t 0, where dox is the distance of x to OA and r is a cut-off function concentrated near a. A careful discussion of the terms appearing then in the variational inequality and the inequality trace Dv < - 2 I H [ near a, which is implied by the assumption [H(a)[ < ItoA(a) in (iii), then give the assertion E2(x-l(V)) = 0 for some neighborhood V of a. D It can be seen from the proof that x actually has its image in an interior parallel set A~ = {a E A : dist(a, OA) > ~} if the following conditions are satisfied: A has bounded principal curvatures, smooth global inner parallel surfaces to OA exist up to distance ~, the conformal solution x E WI'2(U, A) to the variational inequality has boundary trace x[ov with values in A~ and [HI _< HA holds pointwise on A \ A~, where HA(a) denotes
224
the mean curvature of the parallel surface to OA at a 9 A \ Ae. If the boundary mean curvature HOA is replaced by the minimum of the principal curvatures in all the statements of Proposition 2.6 then the conformality assumption for x may be dropped. Combining Proposition 2.6 with Proposition 2.3, Proposition 2.1, and Corollary 2.2, we now immediately obtain the following general existence theorem of Gulliver & Spruck [GS3] (with somewhat weaker assumptions on A and OA here; see also Hildebrandt & Kaul [HK]). The contractibility of F in A ensures ,.q(F, A) ~ 0. 2.7 T h e o r e m ( G u l l i v e r gr S p r u c k ) . Suppose A is the closure of a C 2 domain in ~-(3, the prescribed mean curvature H and the boundary mean curvature HOA of A satisfy IH[ < HOA pointwise on OA,
and there exists a continuous vector field Z with div Z = H on a neighborhood of A in the distributional sense such that 1 sup Iz[ < ~. A
Then, for every Jordan curve F C A which is contractible in A the Plateau problem P(H, F) has a weak solution in WI,~(U, A). Moreover, if [H(a)[ < HoA(a) holds at some point a 9 (OA) \ F, then each solution surface omits a neighborhood of this point. [] Choosing A as a ball or a rotationally symmetric cylinder and Z correspondingly as before we recover Theorems 2.4 and 2.5 as special cases of the preceding general theorem. To give an application not covered by Theorems 2.4 and 2.5 consider A contained in a slab I - R , R] x ]R2 of width 2R in ]R3. Here we can take Z l ( a ) = al
(/0
H(ta,,a~,a3)
dt
)
,
Z~(a) = Z3(a) =
O,
1 -1 . If also IHt < HoA is valid along and we have suPA Izl < 89 provided suPA IHI < ~R OA, then Theorem 2.7 applies. Further examples like ellipsoids or rotationally symmetric bodies bounded by Delaunay surfaces have been discussed in [GS3], [Hi6]. Another idea of Gulliver & Spruck [GS3] is to use solutions to the n o n p a r a m e t r i c mean curvature equation div
Vf =2H ~/1 + ]Vft 2
on D C ~ 3 ,
in order to obtain on A = D a C 1 vector field Z=
t
V/
~/1 + IVfl ~ satisfying d i v Z = H on A and suPA IX[ < 1, provided f has bounded gradient V f on A. The geometric meaning of this differential equation is that the graph of the scalar function f has mean curvature H(a) at the point (a, f(a)) for each a 9 D. Indeed, the vertically constant unit vector field orthogonal to the graph N ( x , y ) -- ( - V f ( x ) , 1)
~/l+lVf(~)P
225
satisfies, on account of OAf/Oy = O,
-Vf(x)
divAf(x, y) = div
~/l+[Vf(x)l 2' and, in view of Af. OVA/" = 0, - 89
y) = mean curvature of the graph of f at (x, f ( x ) ) .
The nonparametric mean curvature equation was solved (in general dimensions) with arbitrary continuous Dirichlet boundary data by Serrin [Se] for bounded C 2 domains D and bounded C l functions H satisfying [HI _< HA
on A .
Herc HA(a) denotes the mean curvature of the parallel surface to OA through a whenever this is defined (i.e. whenever a has a unique nearest point in OA and the principal curvatures of OA there are smaller than the reciprocal distance from a to OA), while we sct ItA(a) = cc otherwise. We call HA the p a r a l l e l m e a n c u r v a t u r e f u n c t i o n of A. Serrin's result was extended to unbounded domains D with finite inner radius (the supremum of radii of balls contained in D) and with global inner parallel surfaces by Gulliver & Spruck [GS3]. Assuming a uniform bound for the C 1 norm of H on A they also established that the solution f to the nonparametric mean curvature equation with zero Dirichlet boundary conditions has a bounded C 1 gradient on A = D so that SUPA [Z[ < 89 is valid for Z above and Theorem 2.7 is applicable. The hypotheses in the following theorcm arc slightly weaker than those needed in [GS3] for the reasoning just described. For a proof of this stronger version of the Gulliver & Spruck result we refer to [DSh]; we will come back to this in Section 3. 2.8 T h e o r e m ( G u l l i v e r & S p r u c k ) . Suppose A is the closure of a C 2 domain in ~3
with finite inner radius and with smooth global exterior parallel surfaces. If we have [HI ]H I when ]HI < HA on A. An isoperimetric condition with c = eXrA(1 + e2ArA)-1/2 < 1 follows for H. In fact we can allow ]HI < x/1 q-E2HA on A with c > 0 so small that /~r A < x/1 + c21og [cl, and we still obtain an isoperimetric condition with constant c < 1. Thus, we may admit in Theorem 2.8 values of ]H I (slightly) larger than HA in the interior of A insisting, however, that ]H I _ 0 that none of these components is bounded. Unbounded components of finite measure can be excluded with suitable uniformity assumptions on A at infinity, e.g. the existence of global exterior parallel surfaces for 0A. With these observations and Theorem 3.3 we have proved all the assertions made in Theorem 2.8 (and in fact stronger statements). To conclude this Section we briefly indicate how to prove the r e g u l a r i t y o f w e a k s o l u t i o n s x E S(F, A) to the Plateau problem 79(F, H) which are obtained from Theorem 3.3. The main point is to show continuity of x on U, because we then use a representation H = div Z with Z bounded on a contractible neighborhood W of the compact set x(U) and we have, with a constant V0 depending on F, Z only, V~t(X, yr) = fv (Zo~)'hcu A ~vdudv - Vo = V z ( ~ ) - v0 for ~ e S ( F , A ) with image in W. (The constant is just V0 = Vz(Yr).) Moreover, for a given w0 E U we can make ]Z] as small as we like near x(wo). It follows that all the results on analytic and geometric regularity mentioned at the end of Section 2 can be applied to the present situation. (There is one exception related to the exclusion of false branch points. In this connection various authors have used the global condition sup [Z] < 89 An inspection of the proofs in [All], [A12], [Gu2], IGOR] reveals, however, that an isoperimetric condition for H with constant c < 1 is actually sufficient, and this was also shown in [SW].) Now, to verify continuity of x we employ the energy minimizing
238
property and repeat the arguments from the end of Section 2 which lead to the Dirichlet growth condition for x. The only property of the energy functional needed there was the inequality, with a constant 0 < c < 1, l+c
Do(x) _ 0. (B n+l (0, R) denotes the ball of radius R centered at the origin in ~:~n+l.) However, we can dominate this amount of H-volume disappearing at infinity by the corresponding amount of mass disappearing at infinity, i.e. by liminf limsup M (TkL_ (~:~n+l \ Bn+I(0, R))) . R~oo
k~oo
This can be proved by using standard slicing techniques of geometric measure theory to construct from the Tk n-currents supported in ~n+l \ B , + t ( 0 , R), which may be thought of as "bubbles" near infinity, and by then applying the isoperimetric condition with constant c _< 1 to these currents. For details we refer to [Du2] and [DS4]. In any case, it is clear that, having carried out this technical point, we can again conclude lower semicontinuity of the energy En on the sequence Tk --+ T, and (i) follows. For (ii) the proof of corresponding statement in Theorem 3.3 can be repeated almost verbatim. Finally, (iii) is based on an inclusion principle analogous to Proposition 2.6 for solutions to the variational inequality which one obtains instead of the H-hypersurface equation if one minimizes EH on T(F~A) with A ~ 1%~+1. Note that the extra constraint in the definition of T(F, A; a) is not effective on account of the strict inequality M (T) < a M (To) derived in (ii). Since we have already omitted the details of the proof for Proposition 2.6 it would not make sense, for the purpose of these lectures, to now discuss the modifications necessary in the present context. We therefore refer to [Du2] and [DS4] for a complete proof. [:3 The currents T with prescribed mean curvature H produced in the preceding theorem have a variety of special properties, see [Du2], [DS4]. For example, they are indecomposable in a certain sense, and they have compact support if their boundary F is supported in a compact set. For applications of the theorem we note that all the arguments used in Section 3 to prove an isoperimetric condition for H are valid also in ~n+l, hence we can state 4.2 Corollary. All the existence theorems of Sections 2 and 3 for parametric H-surfaces
in A C ~3 with given boundary curve P have analogues valid for integer multiplicity rectifiable n-currents in A C ~:tn+l with prescribed mean curvature and with a given closed ( n - 1)-current as boundary. [] For example, Hildebrandt's existence result, Theorem 2.4, in balls B of radius R is valid in Rn+l with the conditions sup B
IH I
2 the mean curvature of spt T is the prescribed value H(a) at every point a E spt T \ sing T. We note that the case of positive multiplicities m and m + l near a boundary point can really occur, i.e. the boundary of the H-hypersurface can pass through an "interior leave" of the same surface. This can be seen already in the case H -- 0 from the example where P is represented by two concentric circles in a plane in ~3 with the same orientation and with
244
multiplicity 1. Tim mass minimizing 2-current T for this boundary configuration is the sum of two discs with equal orientation and with multiplicity 1, and T has multiplicities l and 2 near each point of the inner boundary circle. However, if spt T is connected then one can conclude from the indecomposability of the EH minimizing hypersurface T that it is represented by an oriented n-submanifold with boundary C and multiplicity 1 locally at each boundary point. Moreover, discarding components of spt T \ spt cOT with even multiplicity one always obtains an oriented n-submanifold of ~ + 1 \sing T with prescribed mean curvature H and with boundary C -- spt OT, but the boundary orientation of T is possibly not compatible with the orientation prescribed on C by F (i.e. T does not solve the Plateau problem for P, but for a current obtained from F by reversing the orientation on some components of C). For n _< 6 the solutions to the Plateau problem "P(H, F) obtained here are completely free of singularities, so that one obtains smooth embedded codimension 1 submanifolds in ]R~+1 with prescribed mean curvature and with given smooth boundary. This is interesting even for n -- 2, because we do not know complete geometric boundary regularity for energy minimizing parametric surfaces of prescribed mean curvature with Plateau boundary conditions due to the unsolved problem of boundary branch points (see thc discussion at the end of Section 2). However, in contrast with the parametric theory where we have fixed the topological type of the admitted surfaces in advance, the solutions to the Plateau problem coming from geometric measure theory have a priori undetermined topological type. They cannot be discs if the boundary curve is knotted, for instance, but there are also examples of unknotted curves in ]Ra which cannot bound embedded discs of prescribed mean curvature H -- 0, cf. [AT].
5
Isoperimetric
inequalities
in Riemannian
manifolds
In the final Section 6 of these lectures we discuss the Plateau problem with prescribed mean curvature in a Riemannian manifold. Since the method of isoperimetric inequalities, employed in Section 3 for 2-dimensional parametric surfaces in ]R3 and in Section 4 for u-dimensional integer multiplicity rectifiable n-currents in ~n+~, is of geometric nature, it will work as well in an ambient Riemannian manifold, provided we can prove isoperimetric conditions for the prescribed mean curvature under reasonable assumptions. Such conditions will depend in turn on isoperimetric inequalities in Ricmannian manifolds, and wc therefore review here some simple facts from the corresponding theory. We refer to [DS4, Scc.2] for a more complete treatment of the material that is needed here and to [BZ] for general information. We assume that N is a smooth, connected, oriented and complete Riemannian manifold of dimension n + l , and we denote by # the Riemannian measure on N. It is no essential restriction, by the embedding theorems of Nash (for N compact) and Gromov & Rohlin (for N complete), to assume that N is isometrically embedded as a closed subset of some Euclidean space ~n+l+p and then # : .l/n+1 [_N is just the (n§ Hausdorff measure on N. We also consider a nonempty closed subset A of N (in which we will try to find our hypersurfaces of prescribed mean curvature later).
245
The isoperimetric inequalities we will discuss are of two types: A linear i s o p e r i m e t r i c inequality is one of the form #(E) _< c P ( E ) , while by a n o n l i n e a r i s o p e r i m e t r i c i n e q u a l i t y we mean #(E) < o,P(E) 1+1/" Here E denotes a set of finite perimeter P ( E ) in A, i.e. a # measurable subset with ;~(E) < oo which has finite boundary area P ( E ) in the distributional sense (the distributional gradient field of the characteristic function XE of E is a vector measure of finite total variation P ( E ) on N). We usually require that these inequalities hold for a certain class of such sets E, and tile smallest possible constants c or "), will then be referred to as i s o p e r i m e t r i c c o n s t a n t s . Smoothing XE with a standard procedure one can see that it suffices, under appropriate conditions on A (e.g. a smooth uniform neighborhood retract in N), to verify such isoperimetric inequalities for smooth subsets E of A where P ( E ) = 7-l'~(OE) is the classical n-area of the boundary. We also introduce the i s o p e r i m e t r i c f u n c t i o n s CA(s) and "/A(S) of A, defined for 0<s 0 : # ( E ) < c e ( E ) for all E c A with P ( E ) _ < s } , "/A(S) = inf{? > 0: #(E) < -yP(E) ~+~/" for all E C A with P ( E ) < s}, where in a formula involving the expression P ( E ) we always understand that E is a set of finite perimeter. Wc then have, for all sets E C A,
#(E) < CA(P(E))P(E), #(E) < 7A(P(E))P(E) '+1/" , and CA(S), 7A(S) are the smallest nondecreasing functions with this property. Related to the isoperimetric functions is the i s o p e r i m e t r i c profile bm($), defined for 0 < t < oo by
bA(t)=inf{P(E):EcA
with #(E) > t } .
Then bA(~(E)) ___P ( E ) holds for all sets E C A with finite perimeter, and bA(t) is the largest nonincrea,sing function with this property. We emphasize that the value oo is allowed for CA(S), "/a(s), bA(t). 5.1 E x a m p l e . Let N. be the simply connected (n+l)-manifold of constant sectional curvature a2, i.e. N~ is the Euclidean (n+l)-sphere with radius a-1 if tr > 0, N~ = ~ + l if e; = 0, and N~ is the hyperbolic (n+l)-space with its standard distance function scaled by a factor I~,[-1 if a ~ 0 is purely imaginary. Denote by c~(r) the volume of a ball of radius r in N~ (with I~[r < 7r if a2 > 0) and by ~ ( r ) the n-area of its boundary sphere. Then we have
=
246
where w(n) = /~0(1) = (n+l)~0(1) is the area of the standard n-sphere and, of course, a-1 sin a r = I~1-1 sinh ]~lr for ~2 < 0 and a-1 sin a r = r for ~ = 0. From the isoperimetric property of balls in N . (sce [Sch], [DeG], [BZ, 10.2]) we have the following o p t i m a l i s o p e r i m e t r i c i n e q u a l i t i e s in N~, where c~(r) denotes the quotient c~(r)/p~(r):
p(E) < c~(r)P(E)
if #(E) < a n ( r ) ,
tz(E) _< ,,l@iP(E)
if ~2 < 0,
#(E) __nh in the sense of distributions. Now, from the description of first variation of n-area in terms of the mean curvature vector field we infer that nh(x) is the quotient of the area change at x by the volume change under geodesic homotheties of spheres with center a. Therefore, for x = exp~ ( the mean curvature h(x) can be expressed in terms of the Jacobians J( exp~ of the exponential map at points r 6 T~N, d nh(x) = (J~ expa )-1 ~ t=oJe,~exp a o
247
Note that these Jacobians are equal to the corresponding tangential Jacobians for spheres in TaN centered at the origin, because D~ exp~ preserves length on rays and orthogonality to these rays by the Gauss Lemma. If the sectional curvature of N satisfies Seen _< ~2 on A we can therefore apply the comparison theorem for Jaeobians [Gii], [BZ, w33] to infer, in the situation above, h ( x ) ___ h ~ ( 0 ( ~ ) )
= ~ cot ~0(~:),
where h~(r) = ~cot ~cr is the constant mean curvature of a sphere of radius r in N~ and t)(x) is the length of the geodesic from a to x in the star-shaped region A, as before. In geometric terms the comparison theorem just says that the mean curvature of a sphere in N with SccN _< ~c2 is not smaller, at each point of the sphere, than the mean curvature of a sphere with the same radius in the manifold N~ of constant sectional curvature ~2. Denoting by R = sup=e A O(x) the m a x i m a l r a d i u s of A we have proved:
Suppose A is geodesically star-shaped in N as above and A has finite maximal radius R. Then, i/SccN < a2 on A, the linear isoperimetric inequality
5.2 T h e o r e m .
#(E) nH holds in the sense of distributions on A for every locally integrable function H with H _ c -1 for some c > O, then the linear isoperimetric inequality #(E) < c P ( E ) []
holds for all sets E C A with finite perimeter.
With regard to the behaviour of isoperimetric functions the following observation for product manifolds N = M x IR is of interest. Clearly CN(S) < (x~ can hold only for values of s smaller than twice the n-area 7-/"(M) of M, because Et = M x [0, t] has volume it(Et) = tTt~(M) --+ co as t-~oo, while P ( E t ) = 27-/"(M) for all t > 0. Using slicing arguments one can show that the isoperimetric constant is indeed bounded for sets E with P ( E ) < 27/~(M) (see [DS4, 2.11]): 5.6 P r o p o s i t i o n . Suppose N = M x ~ where M is a compact n-manifold without boundary. Then there exists 0 < e(M) < co such that
CN(S)
=oo 2 7 { a ( M ) , for 0 < s < 27-{"(M).
We now turn to nonlinear isoperimetric inequalities in the Riemannian manifold N. Here we note that such an inequality always holds for sets E C N with sufficiently small perimeter and with volume It(E) < 89 (if it(N) < oo), provided N satisfies some uniformity condition. A suitable condition is that N is h o m o g e n e o u s l y r e g u l a r in the sense of Morrey, i.e. there exists A 6 [1, oo[ such that each point in N has a neighborhood which can be mapped to the unit ball in ]l=t"+t by a biLipschitz mapping ~ with biLipschitz constant max{Lipq~, Lip~5-1} < A. (This is similar to the condition of quasiregularity explained at the end of Section 2.) Using the homogeneous regularity of N and relative isoperimetric inequalities on balls in ]Rn+l (cf. [Fe, 4.4.2 (2)], [Fe, 4.5.2 (1)]) one proves the following result and its corollary (see [DS4, 2.2 and 2.3]): 5.7 P r o p o s i t i o n . If N is homogeneously regular, then there exist ~ > 0 and 0 < 7 < oo such that the nonlinear isoperimetrie inequality #(E) < "TP(E) 1+1/" holds for all sets E C N with perimeter P ( E ) < 5 and with measure #(E) 0 and 0 < "7 < oo such that the isoperimetric function of A satisfies CA(S) 4 the best constant so far (which is, however, larger than "Y~+Iand presumably not the optimal one), and in the case n = 2 the conjecture has recently been verified by B. Kleiner (see [Cr], [K1]). Of course, the conjecture is also true in the spaces N~ of constant sectional curvature n 2 < 0. For n = 2 Kleiner proved in fact that the isoperimetric profile of N, assumed simply connected with SecN < n 2, is not smaller than that of N~. To derive nonlinear isoperimetric inequalities in general by comparing N with spaces of constant curvature one needs S e c N < n 2 for the comparison of boundary areas, but also a lower bound Ricg > ~2 on the Ricci curvature for the comparison of volumes. This does not give the optimal constant. We refer to [DS4, 2.4] where further references are given. [] We will need the isoperimetric inequalities not only for sets E with finite perimeter in A C N but also for integer multiplicity rectifiable (n+l)-currents Q on N with support in A. Such currents are represented by integer valued functions iQ E LI(N, #; ~ ) of bounded variation, i.e. the distributional gradient of iQ has finite total variation M (OQ). In fact, by the decomposition theorem [Fe, 4.5.17] (which is valid also in Riemannian manifolds) one can write iQ as an L 1 convergent combination of characteristic functions of sets Ek, k E 2E, with coefficients in {1, - 1 } , such that the perimeters P(Ek) add up to the boundary mass M (OQ). It is clear from this description that a linear isoperimetric inequality for sets with finite perimeter in A immediately implies a corresponding inequality for integer multiplicity rectifiable currents Q in the top dimension, i.e.
M (V)