This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
n, where n = dim E. Now let 1 r n. It is a consequence of the universal property that if e,, ... , e" is a basis of E then the elements
it'= + 0 if f(tx) = tk f (x) for all t e R and x e E. Show that if f is Ck then it is necessarily a polynomial F) (and hence f is C map, i.e. f e Hint: By Taylor's formula we get
f(h) = k! Dkf(O).h(k) + o(Ihlk)
since Dif(0) = 0 for j < k. Replace h by t - h, divide both sides by tk and let t-1. 0. 13.
Define the function f: 682 - I8 by 2)
f(x,Y)=
xY(x2 - y
if (x, Y) # (0, 0)
x2+y2
if (x, y) = 0, 0)
0
Show that f is of class C'. Conclude from Exercise 12 that f is not of class C2.
A linear differential operator .d of degree k on an open set U in R" has the form ,d (x, a) = I=Y k az(x)
aa alt ... ax." ax"
where A, are Cm real-valued (or matrix) functions on U. For short we write sd = Y,1,1 ,k a,(x) sa, and the function
sd(x. S") = Y a,(x)S2,
a polynomial in
e R",
I2IEk
is called the complete symbol of .d. Note that ml(x, ) = e where denotes the usual inner product of vectors in R". 14.
sd(x, a) e" {
Let sd(x, a) be a linear differential operator on U c I8". Prove
Hormander's generalized form of the Leibniz rule: sd(x, a)(uv) = Y (sd'°`'(x, a)u) a=v(x),
u,
v e C°'(U)
where sd(")(x, S) = 04W(x, ). Hint: If we fix v then each side of this formula can be regarded as a differential
operator in u; show that both have the same complete symbol (use Taylor's formula). 5.2 Manifolds
A locally Euclidean space M of dimension n is a topological space M for which each point has a neighbourhood homeomorphic to an open subset of 61.". A coordinate map is a pair (U, x) where U e M is open and K is a homeomorphism of U onto an open set in R". (Often it is convenient to assume that U is connected.)
114
Boundary value problems for elliptic systems
When dealing with more than one coordinate map we use the notation K: U" -+ V" to indicate the domain U. and range V" c 11" of the coordinate map.
Definition 5.11 Let r be an integer > 1 or oo. A C' structure on a locally Euclidean space M is a family 5 of coordinate maps x: U - V" such that (i) The domains U. cover M: U"E U,, = M. (ii) If K, K' C- a and U. n U,, # 0 then the overlap map U".)
is C'
(iii) The family a is maximal with respect to (ii): if KO is a coordinate map such that Ko o K-' and K - KO ' are C' for all K E t5 then KO E
.
Note: Since K is a homeomorphism the set K(U,, n U",) is open in R". Also, if the overlap map K- K-' is C' then so is its inverse K o K'-' by Theorem 5.8. If ddt = {(U1, K,)}
(i ranging in some index set) is any collection of
coordinate maps satisfying (i) and (ii) then there is a unique C' structure j5 containing 91; namely, let a- = (K-, K is a coordinate map and K1 K - ' is C' for all i}. Then j5 21, and it is easily checked that it satisfies (i) and (ii). j5 is maximal with respect to (ii) by construction. A C' structure can thus be defined by an arbitrary family 21 satisfying (i) and (ii) only. Such a family is called a C' atlas and two atlases are called equivalent if they define the same C' structure. Clearly, two atlases 211 and 212 are equivalent if and only if 91t v 9I2 is an atlas. Definition 5.12 A C' manifold is a pair (M, a) where M is a locally Euclidean,
Hausdorff, second countable space (i.e. there is a countable basis for the topology), and a is a C' structure on M. Usually we refer to the underlying space M as the manifold. The maps in car are called admissible coordinate maps, or charts. If K is a chart with domain
U n x, we say that (U, K) is a chart at x. It is known that any Cr atlas (r 1) on a set contains a C°° atlas; see [Hi]. Thus, it is no loss of generality to assume that r = co, which we do from now on. Examples (1) An n-sphere is the set S" = {x a R"+'; IxI = 1 } with the relative topology as a subset of 18"+'. Let N = (0,. . . , 0, 1) and S = (0, ... , 0, -1) denote the
north and south poles. Then a C' structure on S" is defined by the atlas {K1, K2} where K1: S"\N -+ 18" is defined by
(x1,....X"+1)-+(x1/(1 - x"+1)....sX"/(l -Xn+1)) and K2: S"\S -+ 1%" is defined by (X1,....X"+1)+"+(x1/(1
+X"+1))
Manifolds and vector bundles
115
The overlap map is ice o K1 ': R"\O - R"\O and is given by K2 o Ki '(y) _ y/Iy12, y e R"\O which together with its inverse is C°° so condition (ii) is satisfied.
Remark The charts K1 and K2 are essentially "stereographic projection" from the north and south poles, respectively. For example, the stereographic
x, Fig. 1. Stereographic projection from the south pole.
projection of the point x = (x,, . . . , x"+ ) from the south pole to the plane x"+, = I is the point with coordinates (2x,/(1 + x"+ 1), ... , 2x"/(l + x"+ 1), 1).
(2) R" is a manifold with an atlas formed by a single chart (R", identity). This is called the standard C°° structure on R". Further, if E is a finite dimensional vector space then, by virtue of a linear isomorphism x: E - R" defined by any basis of E, there is an atlas {K} for a C' structure on E. Any two such linear isomorphisms K, K' define the same C°° structure because the overlap map K' a K-': R" - R" is a constant matrix, hence is C.
(3) It is possible for a locally Euclidean space to possess non-equivalent atlases. For instance consider M = R and the following coordinate maps on R: K(t) = t and w(t) = t3. Then 91, = {K} and 212 = {w} are atlases on R but w-' , x is not differentiable at the origin. In other words 211 and 212 define distinct C' structures on R (but they are diffeomorphic, see Exercise 3 at the end of this section.
(4) An open subset U of a manifold M with C'O structure a is itself a manifold with C' structure RU = {K e a; domain of K C U). Note that if K E a and the domain of K intersects U, then its restriction to U also belongs to a by virtue of the maximality condition (iii).
Next we define the concept of a smooth map between manifolds and the local representation of a map.
Boundary value problems for elliptic systems
116
Definition 5.13 Let M and N be manifolds. We say that f: M -+ N is of class C'° (or smooth) if
(i) f : M -+ N is continuous
(ii) for each chart (U, (p) of M and chart (V, 4) of N such that f(U) e V, the local representation, 41, = l, o f - cp - ', is of class C".
The set of all smooth maps f: M -+ N is denoted C'(M, N).
Proposition 5.14 Let 91M be an atlas on M and 21N an atlas on N. Let f: M -+ N be continuous. Then f is of class C°° if and only if for each m e M, there exist charts (U, (p) E `BM and (V, i/i) E 91N such that
m e U, f(U) a V, and the local representation f,, is C'. Proof The necessity is clear. Conversely, if this condition holds, let (U, (p)
and (V, i/i) be any charts of M and N, respectively, such that f(U) c V. We must show that the local representation f 0 is C°°; it suffices to show this in a neighbourhood of each m e U. Now, by hypothesis, there exist charts (U', q,') e 21M and (V', 016 2IN such that m c- U', f(U') c V', and the local representation f.... is C'°. By restricting the domains of gyp,
and 0' we may assume that U = U' and V = V'. Then f,, = (0 o Ji'-') ((P ° V -') -' is C'° since it is a composition of C`° functions defined on open sets in R" (Proposition 5.1). If we recall that the composition of C'* maps on open sets in Euclidean space is C° we obtain the following proposition.
Proposition 5.15 If f e C°°(M, N) and g E C°°(N, P) then go f EC'(M, P).
A map f: M - N where M and N are manifolds is called a diffeomorphism if f is C°°, bijective, and f -': N -, M is C'°. If a diffeomorphism exists between two manifolds, they are called diffeomorphic.
The product of two manifolds is defined as follows. If M, and M2 are manifolds with Cm structures a, and 52 then M, x M2 is a manifold with atlas {K = K1 X K2; KJ E RJ,.l = 1, 2}
where if KJ: U,;j -
c R"' then K, X K2: U,,, X U12 -+ V,,, X VX2 c R"''", is
the product map. The C'° structure for M, x M2 is the maximal family containing this atlas.
An example of a product manifold is the n-torus T" = S' x
x S1 (n
times).
We conclude this section with a study of the axioms for an atlas. Often it happens that an atlas is constructed on a set S which is then used to define a topology on S.
By an atlas on a set S we mean a collection of pairs 91 = {(U;, K;)} (i ranging in some index set) satisfying the following conditions:
Manifolds and vector bundles
117
AT I The Ui's cover S
AT 2 Each Ki is a bijection of U; onto on open set K1(Ui) in R", and for any i, j, Ki(U, n Uj) is open in 18".
AT 3 If Ui n Uj # 0, the overlap map KI
Kj ': K,.(Uj n U;) - K;(Uj n U;)
is C°D
Conditions AT I and AT 3 are the same as (i), (ii) previously. It is necessary
to add condition AT 2 since S is just a set so that the K, are not (a priori) homeomorphisms. Proposition 5.16 Let 21 = {(Ui, Ki)} be an atlas on a set S. Then the collection of subsets of S,
= {W; 3Kj E 21 such that W e Uj and Kj(W) is open in R"},
forms a basis for a topology on S. This topology makes S into a locally Euclidean space and the family 91 is an atlas for a C" structure on this space.
Proof The key point in the proof is the following: if W e `., then Ki(U, n W) is open in 11" for every Ki E'2I
(15)
By definition there exists K E 21 such that W c U,, and K(W) is open in R". Then, for another K, E 21, we have
Ki(Uin W)=Ki°K-'(K(Uin W)) which is open in 68" by virtue of the Inverse Function Theorem since Ki c K is a diffeomorphism and K(U, n W) = K(Ui n UK) n K(W) is open in R".
Let W, W' e G. Then for some Ki, Kj a 21 we have W c Ui, W' (-- Uj and Ki(W) and K/ W') are open in R". If we can show that W n W' e S, then G is a basis for a topology on S. But this follows since W n W' c Uj and
K,(Wn W')=K Uin Uj)nKj(W')nK,{Ujn W) which is an intersection of three open sets in R" (see (15) for the third set) and hence is open itself. We claim that with this topology each K e 21 becomes a homeomorphism. By the definition of C it is immediately obvious that K is continuous. It remains to show that K(U) is open in R" for every open set U in the domain of K. But since U is a union of sets in Z it follows that
K(U)=U{K(W); WE C., We U} is open since it is a union of sets K(W) which are open by (15). Hence K is a homeomorphism. Thus every point in S has a neighbourhood homeomorphic to an open set in R". Remark The set S with the topology in Lemma 5.16 is not necessarily second
countable, or even Hausdorfl (see Exercise 8 below). In other words, we cannot conclude that S is a manifold in the sense of Definition 5.12.
Boundary value problems for elliptic systems
118
Exercises
Show that the upper hemisphere S+ = {x E 18'; IxI = 1, x3 > 0} of the 2-sphere is diffeomorphic to the open disc D = {z e R2; JzJ < 1), where S+ and D are given the C'° structures as open sets in S2 and 182, respectively. 1.
2.
Let U,-' _ {x E S"; ± x; > 0} and define x; : U; -+ {x' a 68"; Jx' J < 1 } by
XI-- X =(XI,...,xi-1,X,+,...... n+1) Show that 21 = {(U; , xi is an atlas of 2n + 2 charts for S", and that the C' structure defined by 21 is the same as that defined by the atlas in Example (1). 3. Consider the two C" structures on F in Example (3). Show that the map t -+ t`f3 is a diffeomorphism from (68, a,) to (R, j2).
Let B,(xo) denote the open ball in U8" of radius r and center x0. Show -jxJ2)-12 is a diffeomorphism from B1(0) to Ui". Conclude that any manifold has an atlas {(U,, h,)} for which all h,(U,) = ff8". 4.
that the map 5.
(a) For fl: M --> Ni, i = 1, 2, define f: M --' N, x N2 by f(m) = (f1(m), f2(m)).
Show that f e C°`(M, N, x N2) if and only if both f e C'°(M, Ni), i = 1, 2. (b) For g,: M, - Ni, i = 1, 2, define g, x g2: M, x M, --+ N, x N2 in the obvious way. Show that g, x g2 is of class C' if and only if g, are of class C°`,
i=1,2.
Quotient topology
An equivalence relation on a set S is a binary relation x, y and z e S:
such that for all
(i) X - X
(ii) x-yandy-x
(iii) x -- yandy - zimpliesx - z The equivalence class containing x is [x] _ {y e S; x - y}. The set of equivalence classes is denoted S/ - and the map n: S -, S/ defined by x )-+ [x] is called the canonical projection. If S is a topological space, the collection of sets
{U c S/-'; n-(U) is open in S} satisfies the axioms for a topology, called the quotient topology on S/ -. Also see the text at the end of §5.4 for a discussion of quotient manifolds. 6.
Let - be an equivalence relation on a topological space S and
n: S - S/-i the canonical projection. Let T be another topological space. Prove that a map q : S/- T is continuous if and only if W e n: S T is continuous.
Manifolds and vector bundles 7.
119
The set r _ {(x, x'); x - x'} c S x S is called the graph of the equival-
ence relation -. Note that f = or x ir) '(A) where A is the diagonal of
(S/-) x (S/-).
(a) If S/ - is a Hausdorff space, show that r is closed in S x S. (b) Conversely, if r is closed in S x S and it is an open map (i.e. the image of any open set in S/- is open in S), show that Si- is Hausdorff.
(line with two origins) Let S = (R x 0) u (R x 1) and define the equivalence relation - on S which identifies (t, 0) with (t, 1) for t 0 0. Let 8.
h;:Rxi-+6Fbedefined asx;(t,i)=t for i = 0, 1. (a) Show that 'U = {(R x i, h,)},v0., is an atlas on Si-. (b) Show that the induced topology on Si-. (by Proposition 5.16) is locally Euclidean but non-Hausdorff. (c) Show that the topology in (b) is the quotient topology.
The real projective space RP" is defined as S" (the unit sphere in R"+') with antipodal points x and - x identified for all x e S". The canonical map n: S" -+ RP" is given by x t- [x] = {x, -x}. (a) Show that RP" with the quotient topology is a compact, connected and Hausdorff space. (b) Show that the restriction of the canonical map n to any open hemisphere in S" is a homeomorphism onto an open set of RP". Conclude that RP" is locally Euclidean. (c) Show that there is a unique C' structure on RP" such that a restricted to any open hemisphere is a diffeomorphism, and with this structure RP" is a manifold. 9.
5.3 The tangent bundle Let M be a manifold and m e M. In this section we define the tangent space, 7,,M, at the point m and show that the union, TM, of all these tangent spaces
itself forms a manifold, called the tangent bundle. Then we show that a smooth map between manifolds induces in a natural way a map between their tangent bundles. Let 1 denote an open internal in R with 0 E 1. A curve at m is a C r map c: I -+ M such that c(O) = in. We say that two curves c, and c2 at in are tangent at m if they have identical tangent vectors in every chart, i.e. (K ° c 1)'(0) = (k ° c2)'(0)
(16)
for every chart (U, K) with m e U (we denote the derivative of a function g: I -+ R" at t = 0 by g'(0) e R"). The intervals 1 j on which c j are defined are of course taken sufficiently small so that c fflj) lies in the domain of a, j = 1, 2. Tangency of curves at m e M is a concept that is independent of the chart
used, i.e. it suffices that (16) holds for one chart K at m. Indeed, let is be another chart at m. Then if we multiply (16) by the matrix D(i: ox'')(>;(m)), it follows by the chain rule that (i: 0 c,)'(0) = (ic o c2)'(0) also.
120
Boundary value problems for elliptic systems
Tangency at m e M is an equivalence relation among curves at m. An equivalence class of such curves is denoted [c]m, where c is a representative of the class. Definition 5.17 The tangent space to M at the point m is the set of equivalence classes of curves at n TmM = {[c]m; c is a curve at m}
The disjoint union of the tangent spaces is the tangent bundle
T.M = U TmM meM
To emphasize that this is a disjoint union we often write elements of TmM
in the form (m, t) where t = [c]m. The map n: TM -+ M defined by n([c]m) = in is the tangent bundle projection. If U is an open subset of M, it inherits a CI structure and we identify TmU = TmM for m e U. Then TU = n 'U = (J,, TmM with projection nu: TU -+ U. The set of smooth real-valued functions f: M - R is denoted C'(M, R), or just Cm(M) for simplicity. If f e C'°(M) we define the differential of f at m e M to be the map df(m): TmM -+ R
defined by [c]m i--+ (f o c)'(0)
which is well-defined, independent of the representative c, by virtue of the chain rule. Then we define the map df: TM -+ I8 such that its restriction to TmM is d f (m), that is, d f I T.,,At = d f (m). We will often write d f Im instead of d f (rn), using whichever notation is convenient.
Remark
(i) If U is open in M then d(f l u) = df I ru (ii) Leibniz' rule holds for f, g e C"(M) d(f-g) = f-dg +
that is, d(f
f (m) dg(m) + g(m) d f (m) for all in e M.
If P U -+ l is vector-valued, then df is defined in the same way as above.
Note that if we write f = (f,..... fk) then df = (df,,...,dfk): TU
R".
Theorem 5.18 TmM has a unique vector space structure such that, for any function f e C x (U, R), where U is an open neighbourhood of m, the map
df: T,M
1k is linear. (Note: When no confusion is possible we write d f instead of d f(rn).)
Proof If (U, K) is a chart at m e U, then K: U -+ fl" and we have a map dK: TmM -+ iB". By the definition of dK and of tangency of curves it is clear that dK is injective. We claim that dK is bijective. For v a R" define the curve c(t) = K''(K(m) + tw), t e (-S, S); b is sufficiently small. Then dK[c] = (K - c)'(0) = w which shows that dK is surjective. Since dK is bijective
we can use it to impose a vector space structure on TmM; in other words
Manifolds and vector bundles
121
such that
dK([cl]m ®[c2]m) = dK[C1]m + dx[C2]ml
(17)
a 68 J = This definition of the vector space operations ® and O in T,M is inde-
dK(i.O [ c ]
pendent of the choice of chart at in, for if (V, c,) is another chart at n: then multiplication of the equations (17) by the map D(4po K-')(K(m)): Q8" - 88, which is linear, shows via the chain rule that drp([c1]m ® [c2]m) = do[c1]m + dq [c2]m
(17')
d4 (%O [C]m) _ %-dw[c]m, A E R
Thus the equations (17) hold if and only if (17') holds and the vector space operations on T,M are well-defined. Now let f: U -+ I8' be a smooth map. We wish to show that df: T,M R't is linear. Let [ct]m e TmM, j = 1, 2 and let Cc]., = [c,]. ® [c2]m. Then by definition of df and applying the chain rule it follows that dJ ([c1]m ® [c2],,) _ (f -c)'(0) = D(f c K -')(K(m)) (K ° c)'(0)
By virtue of the definition (17) of [c]m = Cc,]. ®[c2Jm, we have (K o c)'(0) = (K o cl)'(0) + (K o c2)'(0) for any chart K at in, hence the linearity of D(f K-')(K(m)) implies df([C1]m (D[C2]m)
= D(f ° K -')(K(m)) '(K ° C1)'(0) + D(f °
K-')(K(m)).(K
° c2)'(0)
= (f °cl)'(0) + (f °c2)'(0) = df[C1]", + df [C2]m
Similarly, one shows d f(;. O [c]m) = ). (f o c)'(0) = ). d f [c]m for all i. 6 R. Hence df: TTM -. R1 is a linear map.
The differentials, dK, make it possible to define a topology and a C' structure on the tangent bundle. If (U, K) is a chart, we have the map TK: TU - K(U) x R" given by [C], h- (K(m), (K ° c)'(0))
that is, TK = (K o n, dK) where n: TU - U is the tangent bundle projection. As shown above, dK(m): T,M W' is a bijection for each in, whence TK is a bijection from TU to K(U) x 18". Now we can use Proposition 5.16 to prove the following theorem. Theorem 5.19 The family 91 = {(TU, TK); (U, K) is a chart on M }
is an atlas for a C' structure on TM. making TM into a manifold. With this structure, then for any C' map f: M -+ R*, the map df: TM - l is also C. The tangent bundle projection n: TM -+ M is a Cm map. Also M e TM
Boundary value problems for elliptic systems
122
if we identify m e M with the zero vector 0. E T,"M, and this makes M into a C°° submanifold of TM.
Proof The conditions AT I and AT 2 obviously hold (Proposition 5.16). To verify AT 3, let (U, K) and (V, (p) be two charts for M. If x e ic(U n V), w E R",
we have (TK)-'(x, w) = [c]," whence c is a curve at m = K-'(x) such that (K a c)'(0) = w. Then Tcp a (Tic)-'(x, W) = Tq)([c].) = ((P(m), ((P a c)'(0))
=
(4P(K-1(x)),
D((o a K-1)(Y) - w)
which is a C°° map from K(U n V) X R" to q (U n V) x R". This shows that 91 is an atlas for a C°° structure on TM. It is not hard to show - with the topology on TM as defined in Proposition 5.16 - that TM is Hausdorff and has a countable basis since this is true of M and R n. The proofs of the last three statements in the theorem are left to the reader. Remarks (i) If U is an open subset of a manifold M then it is itself a manifold in the obvious way, and T.U and T,,M can be identified for each m e U.
(ii) There is an identification TR" - R'` x Rk given by the collection of maps {x} x Rk defined by [c]., )-4 (x, c'(0)). from
Definition 5.20 Let f: M - N be a C°` map. We define Tf: TM -p TN by Tf([CJm) = If aCJf( )
Tf is called the tangent of f. The notation T. f is also used for the restriction of Tf to the tangent space T",M; we regard T,,f as a map T",M - Tf(M)N.
TA C). = If ° C)l(m)
Fig. 2. The tangent map Tf.
Manifolds and vector bundles
123
Note: df is defined only if f is real-valued (or vector-valued) on M. whereas the tangent map, Tf, is defined for any C map f between two manifolds and Tf is a map between the tangent bundles. By virtue of Remark (ii) above we see that if f: M -+ Rk is a C°o map then Tf([c]m) = (f(m), df [c]m)
Cc]m a T,M
(')
for any f: M - N. Indeed, if (U, K) and Locally, Tf has the form (V, gyp) are charts on M and N, respectively, such that f(U) c V then Tip o Tf o (TK) -' is given by
K(U)x R"
rp(V) x R°
(n =dimM,d= dim N)
(x, W)-(((P-f° K-')(x), D(q o f o K-')(x)' w), for if c is a curve at m = K-'(x) such that (K o c)'(0) = w then
(18)
Top- TI c(TK)-'(x, w) = Trp° TfCC]., = Tco[f oc]t(m) = (w(f(m)), ((P of oc)'(0))
= (((pof oK-')(x), D(cpof oK-')(x)-w) Theorem 5.21
(i) 1ff:M-+NisaC"'
(ii) Suppose f: M - N and g: N - P are C" maps of manifolds. Then gof: and Tigof) = TgoTf (iii) 1ff: M -+ M is the identity map, then TI: TM -+ TM is also the identity.
(iv) 1f f: M - N is a diffeomorphism, then Tf: TM - TN is a bijection and
WY' = 7 f ').
Proof The statement (i) follows from (18) since lp o f o K-' is smooth by definition (see Definition 5.13). By using local representatives it is easily seen
that g o f is C°° if f and g are C°° (see Proposition 5.15). Moreover, and
T(gof)[c]., = [gof oc]gafirm
(Tgo Tf)[c]m = Tg([f °C]t(m,) _ [gof whence T(g o f) = Tg o Tf.
(iii) is obvious, and (iv) follows from (ii) and (iii).
Remark The following chain rule holds for smooth functions g: N -+ R and
f:M-+N,
d(gof)=dgoTf
(19)
Exercise Let M, and M2 be manifolds and pl: M, x M2 - M, and P2: M, X M2 _+ M2 the canonical projections. Show that the map (7p Tp2) is a diffeomorphism
124
Boundary value problems for elliptic systems
of the tangent bundle T(M, x M2) with the product manifold TM, x TM2.
With this identification, show that if g,: N -+ M, and g2: N - M. are smooth maps then T(91 x 92) = T91 X T92-
5.4 Submanifolds
Let us write R" = R"-" x R" and denote points of R" by x = (x', x") where
x' c- R", x" a R". The notation R"-" x 0 indicates the subspace of R" consisting of points (x', 0, . . . , 0) (k zeros). An (n - k)-dimensional submanifold of an n-dimensional manifold M is a
subset Y c M endowed with the topology induced from M and with the property that for each y e Y there is a chart (U, K) of M with y e U such that SM
K(U n Y) = K(U) n (R"-" x 0)
(the submanifold property for K). The number k is called the codimension of Yin M.
An open subset of M is a submanifold of M of codimension 0. A submanifold of codimension I is called a hypersurface of M.
Proposition 5.22 Let Y be a submanifold of a manifold M, and give Y the relative topology induced from M. Let {(UU, Ki)}ie, be a family of charts satisfying the SM property and such that the sets U, n Y cover Y. If we identify
R"-" with the subspace R"-" x 0, then the family of pairs {(U1 n Y, Kilu,ny)}iet
is an atlas for a C" structure on Y, making Y into an (n - k)-dimensional manifold. The inclusion map i : Y - M is C'. Proof Any coordinate map K: U - K(U) (i.e. a homeomorphism to an open U n Y -, K(U n Y) set in R") on M restricts to a homeomorphism is a coordinate map on and, further, if K satisfies property SM then Y (i.e. a homeomorphism to an open set in R"). We may let K = Ki for any i e 1, and the open sets U, n Y cover Y, thus it follows that Y is locally Euclidean. To complete the proof we must show that the overlap of two maps, (Ui n Y, Kilu,ny) and
(Uj n Y, Kjl ujnY),
is C''. But this is clear since K j o Ki ' is a C'° function defined on an open set in R" and
Kjlu,r,r°(K"lu,nr)-' = Kj0Ki 'IKi(Ui n Y)
is a restriction of this C' function to an open set in R"-". Hence (ii) of Definition 5.11 holds and 91 is an atlas for Y. Also Y is Hausdorff and has a countable basis since this is true for M.
Manifolds and vector bundles
125
The inclusion map i is C" for if (U, K) is a chart on M satisfying the SM property then the local representation of i with respect to charts K and fl" and is hence smooth. KIU,,r is a restriction of the inclusion 12"-k Example Consider the set Y c RI as in the figure below:
p
Fig. 3. A manifold which is not a submanifold of R2.
The arrow means that the line approaches itself arbitrarily close at p, without touching. It is clear that there is a bijection R - Y. By means of this bijection
one can define a topology and C'° structure on Y; however, Y is not a submanifold of RI because the topology on Y is not the relative topology induced from 182. (To show this, examine neighbourhoods of the point p.) Submanifolds have the following universal property. Proposition 5.23 Let Y be a submanifold of M. Given another manifold N, and
a C' map f: N - M such that f(N) e Y, let fl: N - Y denote the induced map. Then f, e C'°(N, Y).
Proof Note that in the simplest case when Y = 68"-` x 0 and M = R", the proposition is obvious (see §5.2, Exercise 5). The same is true when Y = W n (Q8"-" x 0) and M = W is some open set in R". The SM property enables us to reduce the general case to this case. Suppose that f is C'. Then f is continuous, and since Y has the relative
topology in M it follows that f, is continuous. To show that f, is C" it suffices to show it is smooth in a neighbourhood of each x e N. By property SM there is a chart (U, K) at y = f(x) such that K(U n Y) = K(U) n (R"'" x 0). Then there is a neighbourhood V a x such that f,(V) c U n Y. The function Kof,Iv maps into K(U) n (Q8"-I x 0), so it is smooth since K o f I,, is smooth.
Corollary The universal property in Proposition 5.23 uniquely characterizes the C' structure on Y.
Proof If ji, and j52 are any two such C' structures on Y then this property implies that the (set-theoretic) identity map Y - Y is C°° from (Y, &) to (Y. a2). Hence O2 a a,, for by composing a chart in a2 with the identity it becomes a chart in &. Similarly, a, a j52, so that Fr, = j52. The structures are the same, not just diffeomorphic.
126
Boundary value problems for elliptic systems
For examples of submanifolds, see the Submersion Theorem below.
Let us now consider the tangent space to a submanifold Y c M. The inclusion map 1: Y - M is smooth, and we claim that the tangent map T.i : T.Y -+ T,.M,
y c Y,
is injective; in this way the tangent space, T, Y, can be identified with a subspace of T,.M.
To prove that the tangent map is injective it suffices to work in local coordinates. Let (U, K) be a chart having the SM property. With respect to charts Klur,y and K, the local representation of : is K e o (Klu,,r)-' and is just a restriction of the inclusion f8"-k - R". By Theorem 5.21 we see that the local representation of the tangent map Ti: TY - TM is
TKoT1o(TKIur,r)-' and is a restriction of the inclusion R' -k x R"-k R" x R" (also see (18)). It follows that the tangent map Ti: T,. Y TM is an inclusion for each
yeY. Remark With this identification (of T,. Y as a subspace of T, M), we have
T.Y = (T
K)-'(R"-k
x 0)
for any chart (U, K) at y satisfying the SM property. In the following lemma, E and F denote finite dimensional vector spaces.
Lemma 5.24 Let f: U - F be of class C' where U e E is open. Suppose that, for some uo E U, Df(uo) is surjective. Let E, = ker Df(uo). Then there exists an open subset U' of U containing uo and a C' di,Q'eomorphism 0: W, x W2 --. U'. where W, a E, and W2 e F are open, such that (f °W)(x1, x2) = x2
for all (x1, x2) E W, x W.
Proof Let E, = ker Df(uo) and choose a complement E2 such that E = E, ® E2. Then the partial derivative D2 f(uo): E2 -. F is a linear isomorphism. Thus, we may assume that E2 = F and E = E, x E.
Define the map g: U - E1 x E2 by g(xt, x2) = (x j(x x2)) and note that the derivative of g at uo is the linear map E1 x E2 Dg(uo) =
E, x E2 given by
0
IEt
Dlf(uo) D2f(uo)
which is an isomorphism. By the Inverse Function Theorem, g is a diffeomorphism of some open set U' containing uo onto an open set W1 x W2 containing g(uo). Let 0 = (glu.)''. If (x1, x2) a W, x W2 we have
(x,, x2) = g(0(x,, x2)) and hence 'I'
(x1, x2) = (x,, f(W(x,, x2))) Thus f(W(x1, X2)) = x2
Manifolds and vector bundles
127
Let f e C'(M, N). A point v e N is called a regular value if for each x e f -'(y), the tangent map Tf is surjective. A critical point is a point x e M where Txf is not surjective. If Tx f is surjective for each x in a set S, we say that f is a submersion on S. Theorem 5.25 (Submersion Theorem) Let
f: M -+ N be of class
C°`
and let yo e N be a regular value. Assume that the level set f -'(yo) _ {x; x e M, f(x) = yo} is non-empty. Then f -'(yo) is a submanifold of M with tangent space at x e f -'(yo) given by ker Tx f.
Proof Let xo e f -'(yo). Let (V. (p) be a chart at yo and (U, K) a chart at xo such that f(U) c V. Consider the local representation ffK = 0-f- K_'. We have that T(fQK) = TV c Tf a (TK) -' is the map K(U) x R" - O(V) x Rd given by (x, w) H (fvK(x) D(ffK )(x) - w)
(see (18)). Since Tx"J is surjective, it follows that the derivative is a surjective map. Hence we can apply Lemma 5.24. Let n = dim M and d = dim N. By shrinking U and V, if necessary (and replacing K by 0 -' 0 K), we may assume that K(U) = W, x W2, [p(V) = W2 where W, c R' -d and WZ Rd are open, such that ffK is the projection W, x W2 -+ W. on the second factor.
We also assume that K(xo) = 0. With this choice of charts. we have K(U (I f -'()1o)) _ (f,.)-'(0) = W, x 0, hence
;(U r) f-'(yo)) = ti(U) n (R"-d x 0), which is the submanifold (SM) property. Hence f -'(),o) is a submanifold of M of codimension d = dim N. We also have for u e W, x 0 R"-d x 0 = ker(T"f9K)
where c(x) = u, implies
which in view of TK(f'K) = Tf,x)(p o TXf
that T-x(f -'(3b)) =
(Txn)-'(f8"-d x 0)
= ker T;J.
Example Let f: IB"+' - R be defined by f(x) = x12. By identifying 7 W' OB"' x 68"" 1, we get Tf(x, v) = (Ix12, 2<x, v>)
where < , > is the usual scalar product of vectors in f1"+'. Let x e S", i.e. f(x) = 1. Then TxJ: Tx08"+' - T,08 is the map v i- 2<x, v> and is surjective since x # 0. Hence I is a regular value of f, so the n-sphere S" = f ()) is a submanifold of R"+', and for any x e S" we have Tx(S") = {v;veR"+',<x,v> =0}. We conclude this section with a study of quotient manifolds (see also §5.2, Exercises 6 to 9).
Boundary value problems for elliptic systems
128
Definition 5.26 An equivalence relation - on a manifold is called regular if the quotient space Ml- has a manifold structure such that the canonical projection n: M - M/- is a submersion.
If - is a regular equivalence relation, then M/- is called the quotient manifold of M by -. Proposition 5.27 Let - be a regular equivalence relation on M.
(1) A map f: M/- - N is CD if and only if f' n: M N is C'. (ii) Any C" map g: M -+ N which factors through the equivalence relation
-, that is,
x-y
g(x) = g(1'),
defines a unique C" map g: M/- -, N such that g o n = g. Proof (1) If f is C", then so is f o a by Proposition 5.15. Conversely, let f o it be C'. Since n is a submersion, it can be locally expressed as a projection,
i.e. (as shown above) there exist charts (U, K) at m e M and (V, tp) at n(m) a MI- such that K(U) = W, X W2, qp(V) = W2 where W, a R"-d and W2 c R' are open, and
n,x: W, x W. - W2 is the projection on the second factor. Hence if (V', qi) is a chart at f on(m) in N satisfying f o n(U) c V', then
=O°(f°n)°K_'1(OIxW2.
Therefore f,,, is C°°, being the restriction to {0} x W2 of the C' function
(f
(ii) It is evident that g is uniquely determined; and it is C' by (i). Corollary 5.28 The manifold structure of Ml- is unique.
Proof Let (M/-), and (M/-')2 be two manifold structures on M/^- having it as a submersion. If we apply Proposition 5.27(ii) to the map n: M -+ (M/-)21 it follows that the set-theoretic identity map (M/ - ), (M/-)2 is C'. But of course the roles of (M/-), and (M/-)2 can be reversed, so the identity map induces a diffeomorphism. There is in fact a bijective correspondence between surjective submersions and quotient manifolds; see Exercise 6. Exercises
(a) Show that if Y, is a submanifold of M;, i = 1, 2, then Y, x Y2 is a submanifold of M, x M2. (b) Show that if f: M N is a diffeomorphism and Y c M is a submanifold then f (Y) is a submanifold of N. 1.
Manifolds and vector bundles
129
Let h: M -+ N be C°`. Show that if Y is a submanifold of M then the restriction hl r: Y -+ N is C'. Hint: The inclusion map i: Y - M is C'. 2.
Let Y be a submanifold of M. Show that f e CO'(Y, k) if and only if for every y e Y there is an open set U in M and a function f e C'(U, N) such that y e U and 3.
Ilvr-r=fIvr-r 4.
Verify that the C' structure on S" defined in the example above (by
means of Proposition 5.22) is the same as the one defined by stereographic projection.
Let f e C'(M, N). The graph off is defined to be graph(f) = {(x, f(x)); XEM}. (a) Show that graph(f) is a submanifold of M x N of codimension k = dim N. 5.
(b) Show that the inclusion map, graph(f) -, M x N. induces a diffeomorphism ftmuu(graph(f )) = graph(T," f ). Hint for (a). First consider the case N = 68`. Define the map gyp: M x N -
M x N by (p(x, y) = (x, y - f(x)) and use the Inverse Function Theorem to show that (p is a diffeomorphism of a neighbourhood of (x0, f(xo)) onto a neighbourhood of (x0, 0). 6.
Let f: M -' N be a submersion and let - be the equivalence relation
defined by f, i.e.
x - y. f(x)=I(Y) Show that - is regular, M/- is diffeomorphic to f(M), and f(M) is open in N.
Let M and N be manifolds, and R and Q regular equivalence relations on M and N, respectively. If f: M - N is a C" map compatible with R and Q, that is, if 7.
xRy - f(x)Qf(y), then f induces a unique C°° map J: M/R - N/Q such that nN, f = J O ltM.
A C" map f: M -+ N is called an immersion at m if the linear map T", f. T,,M - T",N is injective. If f is an immersion at each m, we just say that f is an immersion. A C' map f: M - N is called an embedding if it is an immersion that is a homeomorphism of M onto the image f(M) with the relative topology induced from N.
Let f: M -, N be a C°° map. Show that the following are equivalent: (a) f is an immersion at m. (b) there are charts (U, K) and (V, 0) on M and N, respectively, with 8.
f(U) c V, m E U, K(m) = 0, K: U -+ U' and 0: V - U' x V' such that
0 o f e K-1: U' - U' X V' is the inclusion x i-4 (x, 0). (c) there is a neighbourhood U of m such that f(U) is a submanifold in N and f restricted to U is a diffeomorphism of U onto f(U). 9.
Find an example of an injective immersion which is not an embedding.
130
Boundary value problems for elliptic systems
10. Let N be a manifold. Show that a subset S c N is a submanifold if and only if S is the image of an embedding.
5.5 Vector fields
A vector field on a manifold M is a C'O map X : M - TM such that X. a T. M
for all m e M (i.e. a smooth section of the tangent bundle TM). The set of vector fields on M is denoted by 1(M). C'(M) is a commutative ring under pointwise addition and multiplication of functions, and X(M) can be given a C"'(M)-module structure as follows: if X, Ye f(M) and f e C'(M) then X + Y E AC(M) and f X e Y(M) are defined by and (fX)(m) =
(X + Y)(m) = X(m) + Y(m)
To facilitate calculations in local coordinates, we introduce some notation.
Let i.': R" -. R denote the projection on the ith component. If (U, K) is a chart for M, we write K =
K: U - R are the component functions of K. Note that xi is that is, x; Cm since A' and K are C°°. Hence dx;: TU - R is C1, and, by Theorem 5.18,
dx,)m a (T,,M)*, a linear function on T,"M for each m e U. We have dK = (dx1, ... , dx") so the definition of TK takes the form TK(m, v) = (K(m), dx1(v),... , dx"(v))
for v e T.M. Let e1, ... , e" denote the standard basis of R" and let c/axjI," denote the unique vector in T",M such that dKIm(c/exjlm) = ej, that is, dxd.(a/ax j1",) = b j
i, j = i, ... , n c/cx"l," form a basis of T,,M and
For each me U, the vectors
dx1I" .... dx"J. is the corresponding dual basis in (T,,M)*. Note that c/11x...... c/ex" are vector fields on U since TK o a/ax j o K -1(x) = (x, e j ).
X E K(U ),
(20)
is C' on K(U), whence c/cxj: U - TU is smooth. Proposition 5.29 Let (U, K) = (U, x 1..... x") be a chart on M. Then X(U) is generated by c/11x1, ... , e/11x" as C°`(U)-module, i.e., a map X: U - TU is a vector field on U if and only if it has the form
X=X1
0 ex1
+...+Xn ex"
where X' e C'(U), i = 1, ... , n.
Proof Let X be a vector field on U. Since X(m) a T, M, the definition of
Manifolds and vector bundles
131
a/ax,l. implies that X(m) _ P ( M ) ax
I
+...+ X '(M)
Im
(21)
8x "
where X'(m) = dx,(X(m)). The X' are smooth since X' = dx; a X where v: U - TU and dxr: TU - R are smooth. Conversely, any map X: U -' TU of this form is obviously smooth and X(m) E T",M for all m e U; thus it is a vector field on U. Let X e X(M), a vector field on M. The directional derivative of f E C°°(M) along X is defined to be Xf := d f o X, that is,
(Xf)(m) = df(m)(X(m)).
Since d f and X are smooth then so is Xf. See Exercise 2 at the end of this section for properties of the directional derivative. Now let (U, K) = (U, x ... , x") be a chart on M. For the vector fields
0/ax; E 1(U), it follows by the chain rule that the directional (partial) derivative (a/ax;) f= d f o a/ax; is given by a,{ f o K- I)(K(m)),
ax f (m) =
ME U,
where f e C'(U) and a; is the jth partial differentiation operator for functions defined on open sets in R". Then we have
f
d f (m) = o
(m) dx I (m) +
- + az (m) dx"(m),
meU
(22)
"
1
Now we turn to the definition of pull-back and push-forward for functions and vector fields. If cp: M - N is a diffeomorphism and X E f(M), a vector field on M, the push forward of X by rp is defined by (p*X = Tcp e X ° cp- I E 1(N)
(23)
It is the vector field that makes the diagram
T(M) T*
T(N)
xtl
M commutative. Note: If x e N then ((p*X )fix a TTN, so the push-forward of X is indeed a vector field on N. We also define a pull-back map qp* for vector fields by replacing (P by gypthat is, (p*X = Tcp- I o X a cp.
For functions, the pull-back map rp*: C'(N) - CX(M) is defined by (p*(f) _ f o gyp,
.f e C"(N)
Similarly, the push-forward for functions is defined by cp.(f) = f a cp-
Boundary value problems for elliptic systems
132
Remark (i) We have cp* = ((p-')* for both functions and vector fields. (ii) The pull-back of functions, cp*(f) = f ^ cp, is defined for any smooth map
(p, but the pull-back of vector fields makes sense only for diffeomorphisms (but see Exercise 4). The push-forward and pull-back maps are U8-linear on both functions and vector fields. Also, on functions, the pull-back and push-forward maps are algebra homomorphisms, that is, (p*(fg) = (p*(.f).qp*(g)
f, g c- CA(N)
W*(fg) = W*(f)'w*(g)
f, g E C°°(M)
and
The local representation of a function f on M with respect to a chart (U, K) is
f= fok The local representation of a vector field X on M with respect to the charts (U, K) on M and (TU, Tic) on TM is
X=TKoXoK-' (compare with (20)).
Integral curves
Recall that we say c: I - M is a curve at m if I is an open interval, 0 e I, and c(0) = m. Since I is an open set in R we can identify TI = R for each t e I in a canonical manner, and the tangent map Tc: R - T(,,M enables us to assign a tangent vector to the curve at each point c(t) by c'(t) = T,c(1)
Note that if cp: M - N is a smooth map, then
0 or a = oo,
and a C' mapping
F: U0xI.-+R8", where 1" is the open interval (- a, a), such that F(x, 0) = x and the curve 1" -+ R" given by t -+ F(x, t) satisfies the differential equation (*).
This theorem is local in nature; it can be applied, essentially without any further proof required, to manifolds.
Theorem 5.31 Let M be a manifold and X e '(M). For each mo e M there exists a neighbourhood Uo of mo, a real number a > 0 or a = oo, and a C°° map
F: such that F(m, 0) = m and the curve 1" - M given by t i-+ F(m, t) is an integral curve of X. If F': Uo x 1,. - M is another map with the same properties then
F and F' are equal on (U0 n Uo) x (I. n IQ ).
Proof Let (U, x) be a chart at x0, and let X : K(U) - R2" be the local representative of X, given by X = TK o X c K -'. By Theorem 5.30 there exists
a neighbourhood Uo c rc(U) and a smooth function F: Uo x I -+ R" such that F(x, 0) = x and the curve t i- F(x, t) satisfies the equation (24).
By shrinking Uo and I we may assume that F(UO X I),e U. Now let U0 = rc-'(U0) and define F: Ua x I -+ M by F(m, t) = h-'(F(x(m), t)). Since c is an integral curve of X if and only if K o c is an integral curve of X, it is clear that F satisfies the required properties.
134
Boundary value problems for elliptic systems
Now let F be another function with the same properties as F. Due to the uniqueness of integral curves proved below in Theorem 5.32, for each x e Uo n Uo we have Fl,,. F'I;x, x 1 where 1 = la n IQ.. Hence F = F' on
(U0c Uo)x1.
Observe that solutions c: I - M of c'(t) = X(c(t)) remain solutions under translations; that is, if t, e I then a(t) = c(to + t) is defined in a neighbourhood of t = 0 and 7'(1) = c'((o + t) = X(c(to + t)) = X(x(t))
This fact allows us to extend the uniqueness part of Theorem 5.31 to get a global uniqueness theorem on manifolds. It is important here that we have required that all manifolds be Hausdorff. Theorem 5.32 If c, and c2 are integral curves of X at in e M then c, = c2 on the intersection of their domains. Proof It is not sufficient just to refer to Theorem 5.30, since we do not assume that the domains of c,: I, -+ M and c2: 12 . M lie in a single chart. However,
observe that the intersection, I = 1, n I2, of the domains of c, and c2 is a connected set. The set J = IS a 1; c,(s) = c2(s)) is certainly closed since M is Hausdorff. On the other hand, J is also open, for if s e J then the curves t'- c,(t + s) and t F-+ c2(t + s) are integral curves of X passing through c,(s) = c2(s) when t = 0, so by virtue of Theorem 5.30 they coincide in some
neighbourhood oft = 0. Hence c, = c2 on a neighbourhood of s. Thus J is both open and closed in I, which implies J = I since I is connected. Corollary Let c,: 1, - M and c2: 12 - M be integral curves of X and suppose that c,(t,) = c2(0) for some t, e I. Then the curve t F- c,(t + t,) is defined on the open interval 1, - t, (which contains 0) and must coincide with c2 on the intersection (11 - t,) n I2 since both are integral curves for X at e2(0). Hence c(t) --
CIO),
fc,(t - t,),
t e 11
t e t, + 12
is also an integral curve of X.
The proof of the corollary is obvious, but the fact that any two integral curves of X meeting at some point can be pieced together is an important idea to keep in mind when proving the global results in Theorem 5.33. In view of Theorems 5.31 and 5.32 there exists an integral curve at in for
X with maximal domain. We denote this integral curve by c,,, and, for emphasis, we mention once more that c' (t) = the domain of c. is an open interval containing 0, and we have cm(0) = in. The right- and left-hand end-points of the domain of c,,, are denoted by t+ (in) and t -(in), respectively.
Manifolds and vector bundles
135
Let 2x be the subset of M x P consisting of the points (m, t) such that C(m) < t < t+(nt)
and define the map F: 2x - M by F(nt, t) = cm(t). We call F the (global) flow of X. Note that 2x is the set of (m, t) E M x P such that X has an integral curve
c: I -+ M at m with t e 1. In particular 2x z) M x 0.
The vector field X is said to be complete if 2x = M x P. Thus, X is complete if and only if each integral curve can be extended so that its domain
becomes (- cc, oo); that is, t+(m) = oc and C (m) = - oo for all m e M. Examples (1) On M = P2, let X be the constant vector field, X(m) = (0, 1). The integral curve through m = (x, y) is t i-- (x, y + t). Each integral curve of X is defined for all t e P. so X is complete.
(2) On M = 082\0, let X be the same vector field. The integral curves are the same as before, except for the fact that any curve that passed through (0, 0) in Example (1) cannot now be extended to infinity; thus X is not complete. For example, t-(0, 1) = - I and t+(0, 1) = co.
(3) For M = Plet X E .x'(08) be defined by X(x) = _X2 . The integral curves of X are the solutions of do/dt = -[c(t)]2. Solving this differential equation
gives c(t) = (t + c(0)-')-' if c(0) # 0, and c(t) _- 0 if c(0) = 0. X is not complete; for example, t+(1) = cc but t-(1) = -1. Theorem 5.33 Let M be a manifold and X a £1(M). Then
(i) 1x is open in M x P and contains M x 0. (ii) The map F: !?x - M is smooth. (iii) If (m, t) E 2x then (F(m, t), s) E Zx f (m, t + s) a 2x; in this case F(m, t + s) = F(F(m, t), s)
Proof Let (m, t) E 2x. Both s i- cm(t + s) and s " cF,m,,j(s) are integral curves of X with the same initial condition at s = 0. Hence they must coincide and have the same (maximal) domain of definition. Taking s = t', (iii) follows immediately.
It remains to show that 2x is open in M x P and that F is smooth on 2x. It is clear from Theorem 5.31 that F is smooth in a neighbourhood of M x 0, but an additional argument is required for the other points of 2x. Let mo a M, and let J be the set of points t in the domain of c,,,0 such that F is C'° at (mo, t), that is, for which there exists a number b > 0 and a neighbourhood U of mo such that the product U x (t - b, t + b) is contained in 2x and the restriction of the flow F to this product is a C°° map. To show that 2x is open and F is C'° everywhere on 2x we must show that, for any mo e M, every point in the domain of belongs to J. We do this by a connectedness argument. It is clear that J is open. It remains to show that J is closed in the domain of c.0. Let s be in the closure
136
Boundary value problems for elliptic systems
of J; then by Theorem 5.31 there is a neighbourhood V of cma(s) and a number a > 0 such that the flow F is smooth on V x (-a, a).
(25)
Since c, is smooth (hence continuous) at s there exists t, e J such that Cmo(ti)
lies in V, c V. Since t, e J, the flow F is C' at (mo, t,). By the continuity of F there is a number b > 0 and neighbourhood U of mo such that
F maps the product U x (t, - b, t1 + b) into V
(26)
and F is C°° on this product. We can also suppose that It, - sJ < a. Thus for any in e U we have integral curves for X,
cm: (t, - b, t, + b) - M and
(-a, a) - M.
and, in view of the corollary to Theorem 5.32, every t e t1 + (-a, a) lies in the Cm(t) =
domain of the integral curve cm when in e U and we have t1). In other words, the product U x (t, - a, t, + a) is
contained in 2X and F(m, t) =
t 1), t - t 1)
for all (m, t) in this product. Hence in view of (25) and (26) the restriction
of F to this product, U x (t, - a, t, + a), is a composition of two C°° functions, so it is C°° itself. Since U x (t, - a, t, + a) is a neighbourhood of (mo, s) we have s e J. Thus J is both open and closed. By connectedness, J is the whole domain of Cma.
From now on we write the domain of the flow as 9 instead of 9X. Let 2, c M be the set of points in e M such that (in, t) e 9 = 9'r. Since -9 is open in M x R it is evident that 2, is open in M. The restriction of F to -9, defines a map F: 9, - M by F,(m) = F(m, t). Note that (iii) of Theorem 5.33 says that F, - F, = F:+, wherever it is defined. Corollary 5.34 Let X be a vector field on M and F: -9 - M its flow. Then .9, is open in M for each t e M, and F, is a dii feomorphism of.9, onto an open set in M. In fact, F,(9,) = 9_, and F,-1 = F_,.
Proof The fact that 1, is open has already been proved. Now let m e 9,. Then (m, t) e 2 and (iii) implies (F,(m), - t) e 1, since (in, 0) e 9 always holds. Hence F(m) e.9-, and F_,(F,(m)) = in
(inc 9,)
Similarly, if in e.9-, then F,(F_,(m)) = in.
Proposition 5.35 Let X e 5t'(M). Let c be an integral curve of X with a maximal domain of definition. If for every finite open interval (a, b) in its domain, c((a, b)) lies in a compact subset of M, then c is defined for all t e R.
Manifolds and vector bundles
137
Proof It suffices to show that the end-points a and b of the interval (a, b) lie in the domain of c. Let t e (a, b) with t. b. By passing to a subsequence converges to a point x e M. Now, since the domain we can assume that of the flow contains a neighbourhood of (x, 0), there is a neighbourhood U of x and a number r > 0 such that integral curves starting at points in U e U and persist for a time longer than T. If we choose n such that b - t < r, we can extend c to a time greater than b. Hence b is in the domain of c, since c has maximal domain. A similar argument holds for the other endpoint, a. The support of a vector field X defined on a manifold M is the closure of the set (m a M; X(m) # 0}.
Corollary 5.36 A vector field with compact support on a manifold M is complete. In particular, if M is a compact manifold then all vector fields on M are complete.
Note that when the vector field X is complete, the maps F, are diffeomorphisms M -+ M for each t e R and F, o F, = F+, for all s, t e R. The set (F; t e R) is called a one-parameter group of dii feomorphisms. Exercises 1.
Let c be a curve on M. Show that for all f e C°°(M)
2.
Let X e T(M), a vector field on M. Show that the directional derivative
X: C'(M) -+ C" (M) defined by Xf =df oX has the following properties
for f,geC°°(M)andi.aR:
X(f +g) = Xf + Xg, X a vector field on M, and let F denote its flow. Show that Xf = (d/dt)Fr(f )I,=0, that is, for all m e M, (Xf)(m) = d f(F,(m))I,=0
(f e C'O(M))
Let q: M -, N be a smooth map of manifolds. A vector field X e 1(M ) is said to be (p-related to Ye 1(N) if 4.
TipoX = Yore.
For smooth maps cp: M - N and 0: N --+ P show that if X is (p-related to Y and Y is cp-related to Z then X is (.y o q,)-related to Z.
Even when the vector field X e ((M) is not complete, the flows, F are local diffeomorphisms in the sense of the following definition (see [AMR]).
Boundary value problems for elliptic systems
138
Definition A flow box of X at m E M is a triple (U0, a, F) such that
(i) U0cMisopen,mEU0,andaeR,wherea>Oora=+ooandFisa smooth map UO x I. - M where I, = (-a, a). (ii) for each x e UO, the curve la - M given by t --+ F(x, t) is an integral curve
of X at the point x. (iii) if F,: UO - M is defined by F,(x) = F(x, t), then F,(Uo) is open, and F, is a diffeomorphism onto its image,
for all t e I". For each m e M, there exists a flow box of X at m, as the reader is asked to show in the next exercise. 5. Let X be a vector field on the manifold M. If the triple (U0, a, F) satisfies (i) and (ii) in the definition above, choose an open set Vo c UO and a positive
real number b < a such that F,(V0) c UO for all t c (- b, b). Show that (Yo, b, Flva,(-b,b)) is a flow box at m.
Let cp: M -+ N be a C' mapping, and let X e X(M) and Y e .E(N) be vector fields on M and N, with flows denoted FX and FT, respectively.
6.
(a) Show that X is q,-related to Y if and only if cp o F' = Fl o 4p. (b) If cp is a diffeomorphism, show that Y is the push-forward of X (i.e. Y = (p,X) if and only if the flow of Y is cp o FX o tp -'.
Hint for (a). For any curve c on M, we have ((p o c)'(t) = T(,)cp(c'(t)). 5.6 Partitions of unity
Before we can prove the existence of a partition of unity on a manifold M, we need the following lemma in R" which guarantees a plentiful supply of C°` functions.
Lemma 5.37 Let r, < r2. There exists a C°° function h: R" - [0, 1] such that h(x) = 1 if IxI < r,, 0 < h(x) < I if r, < IxI < r2 and h(x) = 0 if IxI > r2.
Proof By a scaling transformation we can assume r, = I and r2 = 3. The function cp: R -+ [0, 1] defined by
exp(-1/(1 - r2)),
Irl < 1
0,
Irl >, 1
(p(r) =
is easily shown to be C. Now let k
h(x) =kI
cp(r) dr > 0 and
2-IxI
(p(r)dr,
xeR".
Then h(x) equals 0 if IxI >, 3 and equals I if IxI < 1, and h is strictly between
0 and I if I < IxI < 3. Clearly h is C' when x 0 0. But h is constant near x = 0 so it is C' everywhere.
Manifolds and vector bundles
139
Corollary 5.38 Let M be a manifold and m e M. Then given any neighbourhood V of m there is a function A e C`°(M), 0 < A < 1, such that /i = 1 in a neighbourhood of m and k = 0 outside V. (We call A a "bump function".) Proof There exists a chart (U, K) at m such that U c V, K(m) = 0, and K(U) is the ball lxi < 3. By the lemma there exists a function h e C'(R"), 0 < h < 1, equal to 1 on lxi < 1 and vanishing when Jxi > 2. Let
A=
h-K on U 0
on CU
Since the support of h is a compact subset of K(U), it follows that A e C'(M).
Let M be a manifold. Recall that the support of a function f e C'(M) is the closure of the set of points x e M such that f(x) # 0. A C°° partition of unity on M consists of a system of smooth real-valued functions p j e C°°(M), j e J, satisfying the following conditions:
(a) for all x e M we have p,{x) > 0 (b) every point has a neighbourhood in which Ej pj is a finite sum (c) for each point x e M we have Y J pj(x) = 1 If { V, } jE j is an open covering of M we say that the partition of unity { p j } is
subordinate to { V J if supp p, e i' for all j e J. A collection T of subsets of a manifold M (or any topological space) is said to be locally finite if each point m e M has a neighbourhood U such that U n C = 0 except for finitely many C e W. Condition (b) says that the supports of p, are locally finite. It turns out that there exists a partition of unity subordinate to any given open cover. The open cover Y' = {V} is not assumed to be locally finite; however, the first step in the proof of existence of a partition of unity is to find a refinement of 'V which is locally finite. By a refinement of *- we mean another open cover {U;} such that for every i e 1, U, e V for some j. If the manifold M is compact one can just choose a flinite subcover, but in the non-compact case the following lemma is needed. Recall that a set in
a topological space is said to be a-compact if it is a countable union of compact sets.
Lemma 5.39 Any manifold M is a-compact. In fact, there exists a sequence of open sets G1, G2, ... such that
G" is compact, G" c G"+1, U G. = M. "=1
Proof In a countable basis for the topology of M, select those sets which have compact closure. The sequence U1, U2,. .. of open sets obtained in this way covers M, by local compactness of M. The G. Is can now be defined inductively as a union of these sets. Let G1 = U1, and suppose
G"= U, u U2u...uUj.
140
Boundary value problems for elliptic systems
has already been defined such that G" is compact for n -< N and G" a G"+1 for n < N. Since GN is compact it is covered by a finite number of the sets U,, U2. .... Then we let GN+, = U, j. u Uj,,, , where JN+ 1 is the smallest positive integer greater thanjN such that U, v v Uj.,,,, GN. The sequence of open sets G,, G2, ... defined in this way covers M, since j" - oo. A topological space is said to be paracompact if it is Hausdorff and every open cover has a locally finite refinement. Corollary 5.40 Any manifold is paracompact.
Proof Let G G2,... be a sequence of open sets with properties as in the lemma. Also let Gj = 0 when i , 2 + i.
In the statement of the next theorem, B(0, r) denotes the open ball in I8" with center at the origin and radius r. Theorem 5.41 Let M be a manifold. Any open cover of M has a locally finite refinement consisting of a countable number of charts (Uj, rcj), i e N, such that i:j(Uj) = B(0, 3) and such that the open sets is '(B(0, 1)) cover M. There is a partition of unity { pj } subordinate to { Uj } with supp pi a compact subset of Uj.
Proof Let { V, } be a given open cover of M. For each x e M we can find an arbitrarily small chart (Ux, a,r) at x such that Kx(Ux) = B(0, 3). The proof of Corollary 5.40 shows that there is a countable number of charts (Ui, icj) such that { Uj } is a locally finite refinement of { Vj },
ic,(Uj) = B(0, 1) for each i, and the open sets W = is '(B(0, 1)) cover M.
Indeed, for each point x e A", we can find a chart (Ux, icx) at x such that icx(Ux) = B(0, 3), Ux c G"+1\G"_2 and U. c Vx for some jX. By compactness of A", finitely many of the sets W = icx'(B(0, 1)) cover A. We let {(Ui,'c )}
be the collection of charts obtained in this way for n = 1, 2, .. . By Lemma 5.37 there is a bump function h with h 3 0 everywhere, h = 1 on B(0, 1) and supp h e B(0, 3). Transporting h to the manifold
- h(Ki(x)), 0, {x)-j(
X e Uj
C0U,
we obtain a smooth function ipj e C'°(M ), 0 < 4/j M is the projection on the first factor, is referred to as a product bundle. More generally, a vector bundle E over M is called trivial if it is isomorphic to a product bundle, i.e.
E M x R' for some r. Lemma A map f: M x68' - M x R° is fibre-preserving and linear on each fibre if and only if it has the form f(x, v) = (x, Ax) 1)
for some map f: M -+ L(R', Rd). If
(`)
holds then f is smooth if and only if
is smooth.
f
Proof The first statement is obvious. For the second statement, the "if" part is clear. Conversely, suppose that f is smooth. Let pr denote the projection
M x Rd .+ Rd on the second factor. This is a linear map, hence C. Let e...... e, be the standard basis of R'. Taking v = e; we find that f(x)e, = pr ° f(x, ei)
is smooth in x. But f(x)e4 is just the ith column of the matrix f(x), Remark As usual we identify L(R', Rd) with the space of d x r real matrices.
144
Boundary value problems for elliptic systems
If f e HOM(E, F) and (U, fi) and (U, r/') are local trivializations of E and F, respectively, then, since f is fibre-preserving, the map
f",s=0°f°f-1:U x l'
U X Rd
(d = dimension of the fibres of F) has the form (x, v) i - (x, AX) v),
where f: U -+ L(R', Rd) is smooth. We refer to f as the local representation of f with respect to given trivializing maps. The following properties hold: (i) The local representation of the identity map f = idE e HOM(E, E) is the r x r identity matrix; (ii) if f c- HOM(E, F) and g c- HOM(F, G) and (U, Ii), (U, 0) and (U, ip) are local trivializations of E, F and G, respectively, then (g o f ).p = g.,,. c This means that the local representation of g o f is the map x i - g-(X) f (x) (matrix multiplication).
Lemma 5.46 Let E, F be vector bundles over M. If f e HOM(E, F) and is bijective, then f is a vector bundle isomorphism.
Proof Since f is fibre-preserving we have f(Ex) c Fx for all x c- M. But f is
also bijective so it must be that f(Ex) = Fx. The set-theoretic inverse g: F -+ E is therefore fibre-preserving and it is linear on each fibre since f is. To prove that g is smooth we can work locally. The local representation of f has the form f: U -. GL(R'), and then the local representation of g is the map U -+ GL(fE') given by g(x) = [ f(x)] '. In the notation of Lemma 5.2 this means that g = J o f, so g is C°° and hence g is C'. Let E be a vector bundle over M, and choose an open cover { U; },E l of M and trivializing maps /3, as in Definition 5.45. The maps h,;: U; n UI GL(f8') defined by
hy(x) = (I 1 1)x are called the transition matrices (or functions) for E, and we have h!, =
(hi,)-1
in L
U,
in U,nUjnU,.
hit' hjk = hik
these properties being known as the cocycle conditions. Given a system (h,,)
of r x r matrix functions with C' entries satisfying the cocycle conditions, one can define a vector bundle having hip as its transition matrices by forming
the disjoint union of all U, x f8', i e 1,
Q=U{i}xU,xli', iel
and by identifying
(i,X,v)EU, X f8'-(j,x,v)eU, x l' if v' = h f,(x) - v. It follows from the cocycle conditions that this is an equivalence relation - on Q, and it follows from Proposition 5.47 below
Manifolds and vector bundles
145
that the quotient set S = Q/- is a vector bundle with projection p: S M defined by [(i, x, v)] i--' x and trivialization f3,: p-1(U,) -' U; x68' given by [(i, x, v)] r+ (x, v). Also (S, p, M) is isomorphic to E if h,, were obtained from local trivializations of E as explained above.
The following proposition states the axioms by which a collection of trivializing maps on an arbitrary set S define a vector bundle structure on S.
Proposition 5.47 Let M be a manifold, and p: S - M a mapping from some set S into M. Let {U1} be an open cover of M and, for each i, suppose we are given a bijection /3,: p-1(U,) -+ U, x 68' satisfying the following properties: VB I The map /3, is fibre-preserving, i.e. the diagram
p-1(U,)
-,,
U, x 68'
.+ /f
U,
commutes. Consequently, there is an induced map on the fibres, which we denote by fl, : p-'(x) - 1W. VB 2
If U, and U, are two members of the cover such that U, n U, 0 0, then for each x a U, n U,, the map (/3, f; 1)x a L(68') is a linear isomorphism, and the map of U, n U, into GL(R') given by
is C'. Then there is a unique topology and C' structure on S, and a unique vector space structure on each fibre p''(x), such that fl,., is a linear isomorphism and (S, p, M) is a vector bundle with /3, as trivializing maps.
Proof For x e U; n U;, let h,;(x) = (f, By hypothesis, h;; is a smooth map U, n U; - GL(68'). It follows that the overlap maps
(U;nU;)x68'->(U,nU;)x1W are given by (x, v) - (x, h,,4x)v) and are therefore smooth. These maps are in fact diffeomorphisms since hi;' = h;, is also smooth. We claim that by Proposition 5.16 there is a unique C' structure on S such that the #i's are diffeomorphisms. The collection {(p-'(U,), /3;)) is not quite an atlas for S since the image of 1, is not a subset of Euclidean space. But we can argue as follows: each U, is a union of chart domains so we may assume without loss of generality that we are given an open cover (U,) with charts (U;, K,) for M. We then let Xi = (ic, x id) o /3; to obtain a bijection X;: p -1(Ui) - K,(U;) x 68' c 68" x 68'
which satisfies AT 1, AT 2 and AT 3 of Proposition 5.16. Hence there is a
unique C' structure on S such that the X,'s (and hence the /t; s) are diffeomorphisms.
It is clear that p is smooth, since p: S - M is locally obtained as a composite of smooth maps, i.e., pI e, = p; ° P; where pi: U; x 18' - U; is the
146
Boundary value problems for elliptic systems
projection on the first factor. If x e U, we can transport the vector space structure of R' to the fibre p-'(x) by means of fi,,,. The result is independent of the choice of Ui since (Yi fi! '),, is a linear isomorphism.
Remark When applying Proposition 5.47 it often happens that the fibres p'(x) already possess a vector space structure such that fix is linear. The vector space structure defined in the proof will agree with the given structure.
Let (E, p, M) be a vector bundle over M. A smooth map v: M -. E is called a section of E if p o v is the identity on M; in other words,
v(x) a E., = p-'(x)
for all x e M.
The space of all sections of E is denoted C'°(M, E). If U is an open set in M we refer to a section s e C°°(U, El v) as a section of E over U.
Let u e C'(M, E) be a section of E. If we have a covering {U,} of M as above with local trivializations /3,: EIv, - U, x R', then ui = u e C I(U1, R') and we have
in U, n Uj,
u, = hi j ut
(27)
where h,1 are the transition matrices introduced above. Conversely, any system u, a C'°(Ui, R') satisfying (27) defines a section of the vector bundle. Lemma 5.48 A collection of sections v1,. .. , v, a C°°(U, E) is called a local basis for E if for every point x in U, the vectors v,(x), ... , v,(x) form a basis in the vector space E, E has a local basis over U if and only if EJ , is trivial.
Proof If fi: EIv -+ U x R' is a trivializing map and e, denote the standard unit vectors in R' then the sections i = 1, ... , r
vi(x) = (Q:)-1(ei),
are a local basis on U. Conversely, if vi, ... , v, is a local basis on U, then the map ip from U x R' to Elv defined by (x,
i=1
(28)
Vi(x)
is a vector bundle isomorphism (it has a smooth inverse by Lemma 5.46). Hence fi = cp -' is a trivializing map. The following corollary is the analogue of Proposition 5.29. Corollary 5.49 Let E be a vector bundle over M, and let v1,. .. , v, be a local basis on U. If v is a section of E over U, that is, v e CI(U, E) then we have r
f(X)-VAX),
v(X) _
xeU
i=1
f o r unique functions f e C'"(U), i = 1, ... , r. Conversely, any map v: U -' E defined in this way is a section of E over U.
Manifolds and vector bundles
147
Proof Let fl: El - U x R' denote the trivializing map defined as the inverse of the map (28). Then
s\,-t L Si
l\`
ss
zz
V1(x))=(x,S1,...,Sr)
XEU,i;ER'
Now, /3 v is a section of the trivial bundle U x R', so we have
fi(v(x)) = (x, P0 for a unique function f e C' (U, R'). This proves the first statement since f (x) = Ei =1 fi(x) - e; and vi(x) = (/x) -' (e; ). The second statement is obvious.
If E = M x R' is a trivial bundle the space of sections of E can be identified with the smooth vector-valued maps M - R' and we write C'(M, R') rather than C'°(M, E). If r = 1 we write C'(M). Note that E is trivial if and only if there exists a global basis, that is, a basis defined on all M. Examples
(1) Let S' denote the unit circle lxi = 1 in R2. Since S' is a submanifold of R2 the tangent space to S' at any point is identified with a subspace of R2; at the point x = (x,, x2), we have
T.(S')_ {I;ER2;x1s, +x2 2 =0} We say that RS') is a line bundle because its fibers are one-dimensional. Note that T(S') has a non-vanishing section: the section v: S' - T(S') defined by (X1, x2) i-' (-x2, x1). Hence T(S') is trivial because there is the vector bundle isomorphism S' x R - T(S 1) defined by (x, A) i- A v(x).
(2) Let T" = S' x . . . x S' (n copies) be the n-torus. Then
71S') x ... x T(S') and, since T(S') = S' x R, we have T(T") = T" x R". Thus the tangent 7IT")
bundle to the n-torus is trivial. (3) Let S2 denote the unit sphere lxi = 1 in R3. At any point x = (x1, x2, x3) we have T;(S2) = {S E R3; x161 + x22 + x3c3 = 0)
Using degree theory it is possible to prove that T(S2) does not have a non-vanishing section (see [AMR], §7.5). This implies that T(S2) is non-trivial.
(4) (Mobius band) Next we give an example of a non-trivial vector bundle
dl over S'. On the product manifold R x R, consider the equivalence relation defined by
(t, v)-(t',v')-t'=t+ k,v'=(-1)kv
for some keZ and denote by n: R x R - !l the quotient topological space...!! is called
Manifolds and vector bundles
149
end points must match under the twist, and hence s must cross the 0 section at some point. Now we verify this fact analytically. Suppose that there is a non-vanishing section s: S1 -* Al. Let c be the closed curve in Al defined by c(t) = s(e2n"), 0 < t 5 1. Recall that 3, defines a trivialization of l(I v, and let f: (0, 1) -+ R be defined by f(t) = second component of fl,(c(t)).
A simple argument shows that f can be continuously extended to [0, 1]. In fact, near t = 0, f(t) is equal to the second component of /32(c(t)), and near t = 1 it is equal to minus the second component of f2(c(t)). Now let c: [0, l] R x R be defined by c(t) = (t, f(t)). Then c is a "lifting" of c in the sense that c = n o c, that is,
c(t)=[t,f(t)],0,t,1 f(0). The Since c(0) = c(l) we find that [0, f(0)] = [1, f(l)]. Hence f(l) continuity of f and the fact that [0, 1] is connected implies that f(to) = 0 for some to. Then c(to) = [to, 0] = 0, which contradicts the assumption that s is non-vanishing. Sub-bundles
Let (E, p, M) be a vector bundle over M. A subset F c E is called a sub-bundle if for each x e M there is an open set U containing x and a local trivialization fl: p-1(U) -+ U x R' such that SB
/3(p-1(U) n F) = U x (08'-k x 0).
Proposition 5.50 Let (E, p, M) be a vector bundle over M. If F is a sub-bundle of E, then (F, pIF, M) is a vector bundle over M such that the inclusion map F -+ E is a vector bundle homomorphism.
Proof Let 1, Ui} be an open cover of M such that there exist local trivializations /3i: p- 1(U;) -+ U; x 11' satisfying the SB property relative to F. The conditions VB 1 and VB 2 of Proposition 5.47 are easily verified so (F, pI F, M)
has a vector bundle structure. For instance, note that if hi,(x) _ (/1,
are the transition functions for E. then F has transition functions h;,{x) = I1,,01 R..11 ,,1 which are smooth from U, r Uf to GL(f3'-"). The inclusion map F -' E is obviously smooth because if /3 is a trivializing map satisfying the SB property then its local representation with respect to
#IF and $ is the inclusion U x (R''* x 0) - U x R r. Sub-bundles have the following universal property. Proposition 5.51 Let F be a sub-bundle of E, and G another vector bundle over M. Let f: G -+ E be a vector bundle homomorphism such that f(G) e F. Then the induced map f,: G - F is a vector bundle homomorphism.
Boundary value problems for elliptic systems
150
The proof is similar to Proposition 5.23. Also note that the vector bundle structure on F is uniquely determined by the requirement that the inclusion map F -+ E be a vector bundle homomorphism.
Proposition 552 Let E and F be vector bundles over M. If f: E -+ F is a vector bundle homomorphism let f,: Ex -+ Fx be the restriction off to the fibre over x e M, and define the kernel and image off by
ker(f) = U ker(f)
and
im(f) = U im(fx)
xEM
XEM
If the rank of fx is locally constant then ker(f) is a sub-bundle of E and im(f ) is a sub-bundle of F.
Proof Let U be an open set in M such that there exist trivializing maps (3: Ev -+ U x R' and ki: Fu - U lFl". The local representation
ft,=0°.f °$-':U fl-: X68' -+U X Rd has the form (x, v) -+ (x, f(x)v), where f: U L(Il ', II") is smooth. Fix xo e U. If we let R' = 'V1 ®Y'2 where I' is the kernel of f (xo) and Rd = -YVI (j)'# z where 7f is the image of f (xo), then f (x) has the block matrix representation
f(x) =
[aii(x) a21(x)
a12(x)1
E L(Y'l 0 12,
1 (D Y12)
a22(x)
where aii: U -+ L(Yr, Wi) is smooth. The choice of Y'; and W' implies that a12(xo) is an isomorphism. Therefore, a12(x) is an isomorphism for all x in a neighbourhood of x0; by shrinking U we can assume that this holds for all
x e U. Note also that a11(xo) = 0, a21(xo) = 0 and a22(xo) = 0. Since the rank of f(x) is locally constant we can shrink U further, if necessary, so that f(x) has constant rank for all x e U. Since aI 2(x) is an isomorphism and the rank of f(x) is equal to dim YW, it follows that a2t(x) = 0 and a22(X)=O for all x E U Now if we let A(x)
=[
I - a12(x)-''a11(x)
0
I
C_ GL(Y e I-, Y- D I-)
then
1(Y) A(r) =
C0
a,2(x)
0
0
(29)
Replacing the trivializing map P by A-'#, we may assume that f(x) is equal to the matrix on the right-hand side of (29). Then we have ker f(x) = Y', and im f(x) = Y!; for all x e U. Finally, by a linear isomorphism (change of basis) in 68' and I we can assume that Y, = R'-k x 0 and 'Y! i = 08" x 0, where k is the rank of f(x). Now we have (fl'"k x 0) and ifi(FF n im(f )) = U x (Ilk x 0) fl(Ev n ker(f )) = U x which is the SB property for ker(f) and im(f).
Manifolds and vector bundles
151
Remark From the proof of the proposition we see that if f: E F is any homomorphism then rank(fx) rank(ff0) for all x near x0. Thus, rank(f) is an upper semi-continuous function of x. Definition 553 A projection operator P for a vector bundle E is a homomorphism P: E - E with p2 = P. If P is a projection operator for E, then im(P,,) a im(I - P.) = E., and
rank(Px) + rank(I - P,) = dim E. Since both rank(P) and rank(I - P.) are upper semi-continuous functions of x, they are locally constant. Thus by Proposition 5.52 both ker(P) = im(1 - P) and im(P) are sub-bundles of E and E = im(P) ® im(1 - P). Proposition 554 Let (E, p, M) be a vector bundle over M, and F e E a sub-bundle. Let EJF be the (disjoint) union of the quotient vector spaces E/F,, x e M. Then ELF has a unique vector bundle structure over M for which the canonical map n: E ELF is a vector bundle homomorphism.
Proof Let P: E/F --* M be the map induced from p: E - M. By the SB property, there is an open cover { U; } of M and local trivializations Jar: Eau, -+ Ur x R' such that /J1(Flu,) = U, x (R'-" x 0). Then l; induces a unique map
A:P-'(Ui)
Uix(0xRk)
by the condition fir o n = AI0 1(v, _ (o Rk) . Note that /3, is a bijection and property VB 1 holds. To verify VB 2, let h denote the transition function x (-+ (fir c / ')X for E. We claim that the transition function h for E/F given by x F- (/3, /j 1)s is also smooth. Writing h as a block matrix with respect (Rr-k to the direct sum 68' = x 0) ® (0 x Rk) we have h(x)
h12(x)1
= h2,(x)
h22(x)
where h,1, h12, h2, and h22 are smooth matrix functions on Ur n U, of dimensions (r - k) x (r - k), (r - k) x k, k x (r - k) and k x k, respectively. Since 18'-k x 0 has to be carried into itself by h, we have h21 = 0. Since h(x) is invertible, then so is h22(x) for each x. The proof is complete since h = h22. By Proposition 5.47, (E/F, P, M) is a vector bundle. It is evident from the
local representations that the canonical map it is a vector bundle homomorphism (i.e. smooth). Exercises 1.
Let Y c M be a submanifold.
(a) Show that TY is a sub-bundle of TMI y.
Boundary value problems for elliptic systems
152
(b) The
bundle of Y is defined to be the quotient bundle v( Y) _
(TMI).)/TY. Let k denote the codimension of Y. Show that v(Y) is trivial if there are smooth maps X;: Y - , TM, i = 1, ... , k, such that X;(y) a T,.M and X, (y), ... , Xk(y) span a subspace Vr satisfying T,. M = T. Y ® 1 . for all y e Y. 2.
Let S" c It"+' be the sphere. Show that the normal bundle of S" is trivial.
Let E be a finite dimensional vector space and let F c E be a subspace. The annihilator of F, denoted F°, is the set of i. E E* such that = 0
for all v e F.
There is a natural isomorphism F° -_ (E/F)*. 3. Let Y c M be a submanifold. The conormal bundle, p(Y), is defined to be the union of the annihilators (T,. Y)° c Ts M, y e Y, i.e.
p(Y)={SeT,*.M;<S,v>=0forall veT.Y, yc- Y} (a) Show that p(Y) is a sub-bundle of T*MI,.. (b) Show that p(Y) = v(Y)*, i.e. the conormal bundle is isomorphic to the dual of the normal bundle. 5.8 Operations on vector bundles
Let Y be a submanifold of M. If E is a vector bundle on M we can restrict it to Y
Ely= U rE,.
(=p-(Y))
For the sake of simplifying the notation we also write Ey = Ely. Each fibre of Ey is a vector space (being a fibre of E) and we would like to show that it is a vector bundle. The crucial step is to show first that Ey is a submanifold of E.
Now, for each y e Y there exists an open set U containing y and a local trivialization /3: p-'(U) -+ U x R'. By shrinking U we may assume that it is the domain of a coordinate map A satisfying the submanifold property SM:
k(U n Y) = K(U) n (IF"-k x 0)
(k = codimension of Yin M)
Then the composite map y = (K x id) /1, that is, y: p
-'(U) B
U x 98'
x(U) x Q8' c G8"'',
satisfies the following condition:
y.(p-'(U) n Ey) = !.(p-'(U)) . (R"-' x 0 x R') Thus (p-'(U), X) is a chart on E that satisfies the SM property for E,. in E; this shows that Ey is a submanifold (also of codimension k) in E.
Manifolds and vector bundles
153
Proposition 5.55 Let E be a vector bundle with projection p: E -+ M, and let Y be a submanifold of M. Then
Er=UEy yer
is a submanifold of E and a vector bundle on Y with projection ple,: Er - Y.
Proof We have already shown that Er is a submanifold of E. Now, plE, is a C°° map since it is the restriction of a C'° map to a submanifold (see Corollary 5.23). For the same reason, if we have a collection of local trivializations of E,
p-'(U;)
U, x R r'
such that { Ui } cover M, and restrict them to E, we obtain a collection of local trivializations of Er,
p- '(U;) n Er -' (U; n Y) x such that the sets UJ n Y cover Y. (The restriction is C'° and the same holds
for the restriction of the inverse since (U; n Y) x R' is a submanifold of U;x1W.)
Let N and M be manifolds and (E, p, M) a vector bundle over M. Any
smooth map f : N - M induces a vector bundle f '' E on N, called the pull-back of E by/. The pull-back / - 'E is the subset of N x E consisting of pairs (n, e) e N x E with p(e) = f (n); it is the unique maximal subset of N x E that makes the following diagram commutative:
-
Ipl
Iv
NfM
where the vertical map / `p on the left is given by (n, e) 1-4 n and the top horizontal map is (n, e) F-+ e. The fibre off `E over a point n e N is
V-'E) = {n} x which we identify with E,,(.,, and thus has a vector space structure. We also give/ - 'E the relative topology as a subset of N x E. Example Suppose that E = M x 1W, and p the projection on the first factor.
Then if f: N - M we have
f -'(M x R')= {(n, f(n),v);neN,ve1W) =
r x 118'
where r is the graph of f. By virtue of Exercise 5 in §5.4, f ''(M x 1W) is a submanifold of N x (M x 1W).
154
Boundary value problems for elliptic systems
Suppose E is trivial, that is, there is an isomorphism rp: M x68' -+ E. Then
the diffeomorphism idN x rp: N x (M x68') -' N x E maps f `(M x P') to/ -'(E). Hence/ - `(E) is a submanifold of N x E (see §5.4, Exercise 1(b)).
Since any vector bundle is locally trivial, it follows that / `E is a submanifold of N x E when E is any vector bundle over M. Indeed, let U be an open set in M such that Ev is trivial. Then V =f -'(U) is open in N, and the pull-back of Ev under the map /I v: V --+ U is a submanifold of V x Ev. Note also that this pull-back is equal to
f-'EIv=(VxEu)nf-'E which is open in/ - `E since V x Ev is open in N x E. Thus, every point of
/- ' E belongs to an open set (in f `E) which is a submanifold of N x E, and, consequently,/ -'E is itself a submanifold. Proposition 5.56 Let f : N -. M be a smooth map and E a vector bundle with projection p: E - M. Then f -'E is a submanifold of N x E, and is a vector
bundle over N with projection f `p: f - 'E - N. Proof In fact, if { U; } is an open cover of M such that Ii,: EIv, - U, x 68' are trivializing maps, let A2 denote the second component of f, (the composition of f, with the map Ui x 68' 6t'). Then the sets V =/ -'(U,) are open and cover N, and the maps (p,: f -'EIv, - V xpp 68' (n, e) f-' (n, Nt2(e))
are trivializing maps for/ -'E. Each map 4p, is C'° since it is the restriction of a smooth map on N x E to a submanifold of N x E. The inverse of 4p, is also smooth due to Lemma 5.46. Natural operations on vector spaces carry over to vector bundles. In this section it is convenient to extend slightly the generality of Proposition 5.47. We can replace R' by any r-dimensional vector space E. The bijections fl,: p- `(U,) -+ U, x E satisfying VB I and VB 2 then define a vector bundle structure on S modelled on E. The transition functions h,j are smooth maps
U, n U - GL(E). As the first example of an operation on vector bundles, we consider the direct sum. If E and F are vector bundles over M modelled on vector spaces E and F, then
E®F= U Ex®F has a vector bundle structure over M modelled on E (D F. In fact, there exists an open cover {U,} of M and trivializing maps
F -+ U x F
(30)
Manifolds and vector bundles
155
so that we have a bijection
f#r:(E®flu, - U, x (E(D F) and, for each x e M, fJr: = cpu ® cp,X is a linear map E. ®Fx -, E ®F. Thus VB I and the first part of VB 2 are satisfied; the map p: E ® F -, M
is the obvious one defined by p(x) = E. ®F. for each x e M. As for requirement, let hij(x) = (q', q ')x a GL(E) and c- GL(F) be the transition functions for E and F, respectively. Then E (D F has transition functions U, n UU -+ GL(E ® F) given by the
smoothness ((p,
0
0
J
which are smooth since h,j and h,, are smooth. By Proposition 5.47, E ® F is a vector bundle modelled on E ® F. If E is rank r and F is rank d then E ® F is rank r + d (since R'(D Rd -_ R" 1d). Similarly, we can define other natural operations on vector bundles. For the tensor product E ® F = U E® ® F, given the trivializing maps (30) we have the bijection
Pr: (E ®F)v, - Ur x (E (D F),
(31)
and, for each x e M, it is a linear map E® ® F,, -+ E 0 F defined by A. = (pi. (& (p,.. The transition functions x r-+ hr,.{x) 0 ht(x) E GL(E (& F) are
again smooth, so by Proposition 5.47, E ® F is a vector bundle modelled on E 0 F. If E has rank r and F has rank d then E ® F has rank (since R' (9 Rd - R' d). For the dual bundle E* = U (Ex)*, the transition functions are not quite
so immediate. Recall that for a real vector space E, the dual space is E* = L(E, R), the real-valued linear functionals on E. A linear map f: E -+ F
induces the transpose map f: F* - E* defined by
(f(a),v)_(a,f(v)>,
aeF*,veE,
that is, f(a) = a- f (a c- F*). Note that the transpose operator t is a linear map L(E, F) -+ L(F*, E*). Now, if 0j: Eu6 -+ Ur x E' is a trivializing map for E, then
fir (E*)v, - U, x E* where fiu is a bijection which has transition functions x r-+ ($, Q; 1). = [`h, (x)] ` a GL(E* ) where h,Ax) = (pi ' )x E GL(E). Since the transpose operator t is linear, hence smooth, these transition functions are smooth in view of Proposition 5.2. Thus E* is a vector bundle over M modelled on E*. If E is rank r then
E* is also rank r since (R')* W. These types of constructions of vector bundles can be unified with the language of category theory. Proposition 5.57 and the series of exercises following it are meant to illustrate this idea.
Boundary value problems for elliptic systems
156
Recall that a category 21 consists of a class of objects and for any two objects A and B, a set Hom(A, B) of morphisms from A to B, satisfying the
following properties. If f is a morphism from A to B and g a morphism from B to C, then the composite morphism g o f from A to C is defined; furthermore, the composition operation is required to be associative and to have an identity 1A in Hom(A, A) for every object A. Some examples of categories are:
The class of all topological spaces, with morphisms being the continuous maps. The class of finite dimensional vector spaces, with morphisms being the linear maps. The class of all groups, with morphisms being the group homomorphisms.
For a concise introduction to categories, see [Do]. A covariant functor T from a category 21 to a category S is a rule which to each object A in 91 associates an object T(A) in 23 and with each morphism
f: A
B a morphism t(f): r(A) - r(B) such that
r(f ° g) = r(f) ° r(g) whenever f and g are morphisms such that f o g is defined, and t(14) = 1,(4)
(32)
(33)
A contravariant functor r is defined by reversing the arrows in (32) in the following sense: t is a rule which to each object A in 21 associates an object t(A) in 2i and with each morphism f: A -+ B a morphism t(f ): r(B) - T(A) (going in the opposite direction) such that (33) holds and (32') t(f ° g) = t(g) ° T(f) whenever f and g are morphisms such that f c g is defined. From now on, we always work with the category 21 of finite dimensional vector spaces over O k with morphisms being the linear maps. Let t be a covariant functor on W. Then the functor r defines a map
r: L(E, F) - L(t(E), T(F)); we say that T is a smooth functor if this map is smooth for any real finite dimensional vector spaces E and F. Let VB(M, 21) denote the category of vector bundles over M modelled on vector spaces in IN, with morphisms taken to be the vector bundle homomorphisms. The next proposition shows that a smooth functor on 21 can be extended to VB(M, 21). Proposition 5.57 Suppose that r is a smooth covariant functor on 91, the category of finite dimensional real vector spaces. If E is a vector bundle over M, define the family of vector spaces,
r(E) = U T(Ex) x
(disjoint union)
Manifolds and vector bundles
157
r(F) F is a vector bundle homomorphism, define the map T(f ): r(E) such that its value on the fibre over x is given by r(fS): T(E,,) -+ T(FS). Then, for each E e VB(M, RI), there is a topology and C' structure on T(E) 1f f: E
such that r(E) e VB(M, RI) and the following properties hold: (a) If f : E -+ F is a vector bundle homomorphism then T(f) is a vector bundle homomorphism T(E) -+ T(F). (b) r is a functor: r(f a g) = r(f) o r(g) whenever fog is defined, and r(idL) =
(c) If E = M x E is a trivial vector bundle then T(E) = M x r(E) is a trivial vector bundle.
(d) 1f f : N -+ M is a smooth map between manifolds then r(f -1E) = f 'r(E).
(To be precise in (d) we should write T,(j -'E) =/ -'TM(E); that is, there is a distinct functor r = TM for each manifold M. But (d) shows that there is no harm in dropping the reference to the manifold.)
Proof As a preliminary remark, note that it follows from (32), (33) that if
f: E -+ F is an isomorphism then so is r(f): r(E) - r(F), and in fact
T(f)-' =T(f-').
We assume that M is connected, so that each fibre is isomorphic to a fixed vector space E. Let { U; } be an open cover of M with trivializing maps for E
/J:EIv,-+U,xE Then we have a bijection
T(fi): T(Elv,) - U, x r(E) defined on the fibers over x e U, by the linear isomorphism r(f ix): T(E)
We must show that these bijections satisfy the conditions of Proposition 5.47. There is an obvious map a: r(E) -+ M such that VB I holds for (r(E), n, M). To verify VB 2 note that the transition functions for r(E) are x F-- r(fi ) a r(f%x)- 1
The expression on the right is equal to r((/li o fJj '
T(h,t(x)) where hip are
the transition functions for E. Since h,; is smooth from U, n Uj to GL(E) then r a h,j is smooth from U; n UU to GL(L(E)) due to the smoothness property of the functor r. Thus there is a unique topology and C°' structure on r(E) such that r(fl,,,) is a linear isomorphism on each fibre n-'(x) = r(E) it, M) is a vector bundle with r(fi) as trivializing maps. Now, for each vector bundle homomorphism f: E - F we have the map r(f ): r(E) -+ T(F) which is obviously a vector bundle homomorphism, provided we can show that it is C'. To do so we can argue locally. We have the commutative diagram
EIv, - Flv, 1e,
14,
U,xE -i UixIF where the map in the bottom row is (x, v) i- (x, f(x)v), and by the functorial
Boundary value problems for elliptic systems
158
property of r we obtain another commutative diagram T(f)
T(Elv)
r(F'Iv,)
U; x r(E)
Ui x r(F)
where the map in the bottom row is (x, w) f- (x, r(f(x))w). The map f: U; -+ L(E, F) is the local representation of f_and is smooth, so that by the smoothness property of r, the map x -. r(f(x)) is also smooth. Hence r(f) is smooth. Conditions (b) and (c) are obviously satisfied. We leave the proof of (d) as an exercise.
Examples (1) Let r(E) = nk(E), the kth exterior product (see §5.1). If f e L(E, F) there
is a unique linear map r(f) = nkf e L(AE, AF) such that A ... A f(vk)
V1 A ... A
(34)
Then r is a covariant functor on the category of finite dimensional real vector spaces, that is, A(f - g) = nkf c nkg whenever f - g is defined and Aide) = idME. (2) Let r(E) = E* and for each linear map f e L(E, F) define the linear map r(f) = f e L(F*, E*). Then r is a contravariant functor on the category of finite dimensional real vector spaces. One can also define functors of several variables r(E, F,.. .... ), covariant in some variables, contravariant in others. For instance, a functor r in two variables is said to be contravariant in the first variable and covariant in the second if r(fi `f2, g) = r(f2, g) ° r(f1, g) r(f, g1 "g2) = r(f, g1) ° r(f, g2) and r(idE, idF) = id,IE,FI. Further, r is said to be smooth if the map
L(E', E) x L(F, F') - L(r(E, F), r(E', F')) defined by (f, g)"r(f, g) is smooth. See Exercise 7 below. Exercises
Let Y be a submanifold of M, and e': Y - M the inclusion map. Show that 1E = El r. Hint: The isomorphism is given by (y, e) i- a where y e Y, 1.
e e E, with inverse e F- (p(e), e). 2.
V
(a)
Let f : N -, M and y: Z - N be smooth maps. Show that
°p)-'E
1VIE)
(b) Show that isomorphic vector bundles have isomorphic pull-backs.
Manifolds and vector bundles 3.
159
Show that nk is a smooth covariant functor in the category 21 of finite
dimensional real vector spaces. That is, if f e L(E, F) then there is the induced linear map nkf a L(nkE, nkF) defined by V1 n ... A Vk~f(V1) A ... A f(VL)
(34)
and it has the following properties:
(i) If g e L(F, G) and f e L(E, F) then nk(g f) = nkg, nk (ii) If idE a L(E, E) denotes the identity on E then A (idE) = idnkE, and the smoothness property:
f
(iii) The map nk: L(E, F) -' L(nkE, nkF) is C'. Hint for (iii). If bases are selected for the vector spaces, then / \ becomes a matrix function with polynomial entries.
Let r be a smooth covariant functor on 21. Let E be a vector bundle over M, modelled on E, and let SE C''(M, t(E)) be a section, that is,
a smooth map s: M -, t(E) such that s(x) E r(Ex) for all x e M. If /i: EI,; -' U x E is a trivializing map then we obtain an isomorphism
E, hence an isomorphism t(/3.,): r(E.,) -+ t(E) for each x. Thus, s has a local representation r(E) (') is U where i is smooth. /3x: E.
4.
Consider once again the functor N. Write out in detail what the local
representation (*) of a section s e C'(M, Ak(E)) looks like in this case, given
a basis v...... v, in E. Show that Proposition 5.57 holds for smooth contravariant functors, where in (a) the arrow is reversed. (A contravariant functor gives a map L(E, F) - L(t(F), t(E)) and, as before, t is said to be a smooth functor if this map is smooth for all E and F.) 5.
and for each f e L(E, F) let r(J) = f e E") be the Let T(E) = transpose of f, i.e. t(f )h = h - f. Show that t is a smooth contravariant 6.
functor on the category of finite dimensional real vector spaces.
The functors considered in Proposition 5.57 and Exercises 3 to 6 are functors of a single variable. Similarly, one can define functors of many variables, contravariant in some variables and covariant in others. 7.
Let t be a smooth functor of two variables in the category 21,
contravariant in the first variable and covariant in the second. If E and F are vector bundles over M, define
T(E,F)=Ut(EY,F.), s
and, for each pair of vector bundle homomorphisms f: E' -, E and
Boundary value problems for elliptic systems
160
V. define the map
g: F
t(f, g): t(E, F) -+ r(E', F') whose value on the fibres over x is given by t(fx, gx): t(E, F.) - t(Ex, F."). Show that t is a functor in the category of vector bundles with properties analogous to (a)-(d) of Proposition 5.57. 8.
Let Hom or L be the functor which associates to finite dimensional
vector spaces E and F the finite dimensional vector space
Hom(E, F) = L(E, F) (the space of linear maps from E to F), and associates to each pair of linear
maps f: E' - E and g: F -, F' the linear map Hom(E, F) -+ Hom(E', F') defined by h i-+ g It - f. Show that t is a smooth functor, contravariant in the first variable and covariant in the second. (Note: We can conclude from Exercise 7 that for each pair of vector bundles E, F over M,
Hom(E, F) = L(E, F) = U L(EX, FF) x
is a vector bundle over M. However, this vector bundle should be distinguished from HOM(E, F). HOM(E, F) is the space of sections of Hom(E, F); see the discussion preceding Lemma 5.59.) 9.
If E and F are vector spaces then E* ® F - L(E, F) by a canonical
isomorphism such that i. ® w e E* (& F corresponds to the linear map E -+ F given by v --+ w. Show that this induces an isomorphism
L(E,F)-E*®F for any two vector bundles E and F over M. 10.
Formulate the definition of a smooth functor r of two variables which is
(i) covariant in both variables (ii) contravariant in both variables. Show that in each case t can be extended to the category of vector bundles with properties analogous to (a)-(d) of Proposition 5.57. 11.
Let L2(E) denote the set of bilinear maps E x E -+ R.
(a) If E is any vector bundle over M, show that L2(E) = U= L2(E,,) has a natural vector bundle structure with properties analogous to Proposition 5.57 (a)-(d). (b) Show that the canonical isomorphism L2(E) - E* ® E* of vector spaces induces an isomorphism
L2(E) - E* ®E* of vector bundles over M. 12.
Let E be a vector bundle over M. Let v ... , v, be a local basis for E
on U:
Manifolds and vector bundles
161
(a) At each point x e U define the dual basis i.'(x), ... , i.'(x) e Ex by Q.'(x), vi(x)> = Sr;
Show that ..' is smooth and hence a section of E*lv. Hint: Consider the isomorphism cp defined by (28). What is the image of A (x) under the transpose map '(qps)? (b) As in (b) of the preceding exercise, we identify LZ(E) with E* ® E*. Show
that every section g ofLZ(E) can be written locally in the form gs = E gi,{x)Ai(x) (9 Ax), where gij e C'(U) are unique.
xEU
5.9 Homotopy property for vector bundles Let E be a vector bundle over M. A map s: A - E is said to be a C'° section defined on the closed set A c M if for every x e A there is an open set U = Us containing x and a Cr section s1: U E such that sllu,A = SI vtA. The following lemma, a smooth version of the Tietze extension theorem, shows that s can be smoothly extended to a section ss M - E defined on all M.
Lemma 5.58 (Smooth Tietze Extension Theorem) Let E be a vector bundle over M. Suppose that s: A -+ E is a smooth section defined on the closed set A, that is, s is smooth in the sense defined above and p o s is the identity on A. Then s can be smoothly extended to a section of E.
Proof Consider the open cover { Us; x e A} of A which is given by the definition of smoothness of A. If we include also the open set M \A, we obtain an open cover of M. Let { 1;} be a locally finite refinement of this cover, and let {pi} be a partition of unity subordinate to { 1;} with supp pi compact in V1. Let s, denote a C' section 1; -. El,,, so that si and s coincide on A n Y, (where si = 0 if A n Y = 0). Then we get a C°° section s, of E by defining if x e Vi s,(x) _ Jp'(X)s,{X) 0
if xOVi
Then s = Y, .'s, is a C" section of E, and it is an extension of s, for if x e A then
s(x) _
p,(x)s,(x) _
p,(x)s(x) = s(x)-
Remark In particular, the lemma can be applied when E is a trivial bundle M x R'. Thus if A is a closed set in a manifold M and f: A -+ R' is smooth,
then f has an extension to a smooth function f : U - R' where U D A is open.
For any two vector bundles E and F over M, we let Hom(E, F) be the vector bundle for which the fibre over x e M is the vector space of linear
Boundary value problems for elliptic systems
162
maps from E,, to F,
Hom(E,F)=UL(E.,F,) X
(see §5.8, Exercise 8). If f : M - Hom(E, F) is a section, then for each x e M we have
f
a local representation of the form f: U -+ L(R', R°) where f is smooth (r and d are the fibre dimensions of E and F, respectively). The smoothness requirement on f is the same, therefore, whether it is regarded as a section M - L(E, F) or as a homomorphism E - F. Hence C'°(M, Hom(E, F)) = HOM(E, F), that is, the sections of Hom(E, F) are precisely the homomorphisms from E to F. Lemma 5.59 Let Y be a submanifold of M, and E and F two vector bundles over M. Then any isomorphism cp: El,. - Fly extends to an isomorphism El u - Fl v for some open set U containing Y.
Proof Note that cp is a section of Hom(E, F)jr. Applying Lemma 5.58 we can extend cp to a section g of Hom(E, F). Let U be the subset of M consisting of points x for which gx is an isomorphism. Since GL(R") is open in L(R"),
U is open and contains Y.
Throughout the rest of this section, I = (-a, 1 + a) denotes an open interval containing the unit interval [0, 1].
Definition 5.60 Two smooth maps fo: N M and f,: N -+ M are said to be (smoothly) homotopic if there exists a smooth map /: N x I - M such that I (x, 0) = fa(x) and I (x, 1) = f, (x) for all x e N. Ordinarily, a homotopy is defined as a continuous map/: N x [0, 1] - M. However, if two smooth maps /o and f, are continuously homotopic then they are also smoothly homotopic (see §10.6) so there is no loss of generality
in restricting attention to smooth homotopies. Example Let,?: M -+ M and A: M -+ R be smooth maps. Then (S, A): M M x R is homotopic to the map (y, 0). The homotopy is/(x, r) = (y(x), tf(x)),
XEM, te1. In Lemma 5.61 and Theorem 5.62 we assume that the manifold M is compact. This is sufficient for our purposes, but these two results in fact hold for any (paracompact) manifold. See [Hi] or [Sp 2] for the proof in general.
For a fixed t e I, M x {t} is a submanifold of M x I, so it makes sense to consider the restriction EI v x i,F, and we regard it as a vector bundle over M.
Manifolds and vector bundles
163
Lemma 5.61 Let M be a (compact) manifold. Then for any vector bundle E
over M x I
Elm.o - EIM.l where = indicates an isomorphism as vector bundles over M.
Proof Fix to e I and define z: M x I -+ M x I by t(x, t) = (x, to). Also let i : M x to - M x I denote the inclusion map. The restrictions t-'EIM,,,Q
and
EIMX,o are isomorphic
since they are isomorphic to i -'t -' E and I -' E, respectively, and since toy =i. By Lemma 5.59 this isomorphism can be extended to an isomorphism
z-'Elu ' Elu
(35)
for some open set U z) M x to. Since we are assuming that M is compact, then U = M x (to - e, to + e) for some & > 0, and restricting (35) to the submanifolds M x t we obtain Elm., = EIM X,o
when It - tol < e
(regarded as vector bundles over M). This implies that the set I0 = {t a 1; Elm. o} is both open and closed in I. Since I is connected we have Elm X ,
10=1. Theorem 5.62 Let N and M be (compact) manifolds and let f;: N -+ M, i = 0, 1, be smoothly homotopic maps. Then / i 'E 2t / 0'E for any vector bundle E over M.
Proof Let f : N x I -+ M be a homotopy connecting f, and j0. Since f, = f f where I,: N x t - N x I denotes the inclusion map, we have
Is'E^-'f-'Elv.,
t=0,1
and, by virtue of Lemma 5.61, the restrictions/ -'El,., and/ - 'EINX 0 are isomorphic vector bundles over N. A manifold M is said to be contractible if there exists a point x0 e M and a smooth homotopy f: M X I -- M such that I (x, 1) = x and I (x, 0) = x0 for all x e M. In other words, the identity map on M and the constant map are homotopic. Corollary 5.63 If M is contractible, then any vector bundle over M is trivial.
For example, any star-shaped open set in R" is contractible. Recall that U c R" is star-shaped if there is a point x0 e U such that, for any x e U, the line segment from x to x0 lies in U.
164
Boundary value problems for elliptic systems
5.10 Riemannian and Hermitian metrics Let E be a vector bundle over M. We say that a Riemannian metric has been
defined on E if we have an inner product 0 since p,(x) > 0 for all i. Also p,(x) > 0 for some i, hence 0 if v # 0. The smoothness property holds for < , )x because it holds for each term in the sum and every point has a neighbourhood in which the sum is finite. Let E be a vector bundle over M. In view of §5.8, Exercise 11,
L2(E) = U L2(Ex) xe M
is a vector bundle over M. A section g: M - L2(E) of this vector bundle has a local representation of the form g: U -, L2(f8') where g is smooth (see
§5.8, Exercise 12). Since L2(f8'), the set of bilinear maps on f8', can be identified by a linear isomorphism with the set of r x r matrices, we find
Manifolds and vector bundles
165
that the local representation of g has the form O(x) = [g, (x)]
where g;; e C`(U).
It follows that the set of Riemannian metrics on E is the set of sections gx = < , >x of L2(E) such that gx is symmetric and positive definite on Ex for all x e M. An inner product < , > in a vector space E induces an isomorphism between
E and its dual, E*. We call the map E -+ E* defined by v r. the "index-lowering" operator. Its inverse, E* - E, is called the "index-raising" operator. We denote the index lowering and raising operators by the symbols (subscript) and * (superscript), respectively. That is,
X,, =<X,-> and S=<S",-> The reason for this terminology comes from the Einstein summation convention where the components of vectors X e E with respect to a basis are written as superscripts X', while components of covectors e E* are written as subscripts ; . Let [g;;] denote the matrix of < , > with respect to a basis v...... v, of E, that is, g, = . The index-lowering operator maps the vector X e E with components X' to the covector X,, a E* with components Xi = Y gi;X',
(EX'v1)# =EX;i.' where i.'..... i.' is the dual basis corresponding to v1.... , v,. Now, let [g'J] = [gij]-' denote the inverse matrix. The index-raising operator maps the covector e E* with components i to the vector 1; * e E with com-
ponents S' = 0 4
(E Si)')" = E VVI The inner product on E induces an inner product on E* by <se, p> _ >; it is not hard to verify that it has matrix [g')] with respect to the <S *, dual basis that is, g'' = X1dx',
ax1
where X; = Y g1JXj and g;,{x) = 0 and cp(Y, x") =
(x', -x.) if x" < 0, then the restriction of 9 to each half-space At is of class CX, but (p itself is not C'. A chart (U, K) on M induces a coordinate map is on M as in the proof above, and in the same way the chart (U, (0 o n) on M induces a coordinate map W o is on M, but the overlap map (qp o ic) k -' _ W
is not of class C.
178
Boundary value problems for elliptic systems
To define a C'° structure on Al for an arbitrary manifold, therefore, takes a little more work due to the necessity of verifying the smoothness of overlap maps. In order to define a C' structure on M we use a collar of OM in M. The existence of a collar means that we have a diffeomorphism a: W -* aM x [0, 1)
such that a(x) = (x, 0) for all x e 3M, where W is an open set in M containing 3M. This induces a map d: W - aM x (-1, 1) defined by a(z)
a(x) if x = [(x, 0)] (z(a(x)) if x = [(x, 1)]
and W = a((W x 0) u (W x 1)) is open in Al. Theorem 5.77 Let M be a manifold with OM 0 0, and W an open set in M containing OM on which there is a diffeomorphism a: W - OM x [0, 1) such that a(xj= (x, 0) for all x e M. Then there is a unique C°° structure on the double M such that both inclusions M M are C°°, and such that the map a: W OM x (-1, 1) induced by a is a diffeomorphism. Proof We take as coordinate maps on Al the following:
(U x 0, K) and (U x 1, K), if (U, K) is a chart on M with U c Int M
(x-'(U' x (-1, 1)), (K' x id) o a), if (U', K') is a chart on aM
where id denotes the identity map (-1, 1) -' (- 1, 1). It is straightforward to verify that the overlap of two such maps is of class C. The domains of these maps cover M so we have (an atlas for) a C'° structure on M. Exercises 1.
Suppose that a,: W, - aM x (-1, 1), i = 1, 2 are two diffeomorphisms
satisfying the conditions of Theorem 5.77.
(a) Show that the CO' structures on k determined by a, and a2 are diffeomorphic.
(b) Under what conditions are the two C'° structures the same? 2.
Let M and N be manifolds and f : M -+ N and g: M -+ N continuous
maps such that f(x) = g(x) for all x e M. Then there is an induced map h: M - N given by h([(x, 0)]) = [(f(x), 0)] and h([(x, 1)]) = [(g(x), 1)].
Show that h is continuous. If f and g are of class C', what additional conditions are required in order that h also be of class C'? Let M be a manifold with OM * 0. Show that there exists a vector field X e f(M) that is inward-pointing when restricted to 3M. 3.
Let (X, d) be a compact metric space and A e X a closed subset. Let f: X - S be a local homeomorphism such that f1,, is one to one. (a) Let K be the set of points in X x X such that x y and f (x) = fly). Show that K is compact. 4.
Manifolds and vector bundles
179
(b) Show that there is a neighbourhood U of A such that f lu is one to one. (Hint: Define b: K - R by S(x, y) = d(x, A) + d(y, A); since K is compact, there is an c > 0 such that 6 s on K.) An open set it in R" is said to have C°° boundary if for each xo a dig there is a C" function cp defined on an open set W (in R") containing xo such that cp(xo) = 0, grad cp(xo) 0 0, and Q n W = (x e W; (p(x) > 0} 5.
Let Cl be an open set in .gt" with C' boundary. Show that i2 is a
submanifold of if, with boundary af. Let Cl be an open set in R" with C`° boundary. Note that if r: S2 - R" is a vector field on Sl then it has flow F(x, t) = x + t v(x). Suppose now that ail is compact, and use Exercise 4 to give a short proof of Theorem 5.73 in a tubular neighbourhood in R". this context, i.e. that ahas Cl 6.
6 Differential forms
6.1 Differential forms Before introducing differential forms we need the following result, which is
just a restatement of Proposition 5.57 for the functor N. Recall that N is a smooth functor by virtue of §5.8, Exercise 3. Proposition 6.1 The functor At extends to the category of vector bundles over M. For any vector bundle E over M let NE = l)xe /\k(Ex). If f: E - F is a vector bundle homomorphisms define the map Af: AkE -+ nkF such that its value on the fibre over x is given by the map Ak(fx): nk(Ex) -+ nk(Fx). Then
NE has a natural topology and C' structure making it into a vector bundle with the following properties:
(a) If f E - F is a vector bundle homomorphism then nk(f) is a vector bundle homomorphism nk(E) -- nk(F). (b) nk(f ® g) = At(f) ° nk(g) whenever f o g is defined and nk(idE) = idnkiEi.
(c) If E = M x E is a trivial vector bundle then Ak(E) = M x nk(E) is a trivial vector bundle.
(d) If /: Y - M
is a smooth map between manifolds then
At(f -E) _
t 'nk(E) If E is a vector bundle over M, there is a natural map E= -AtE defined on the fibres over x, which is easily seen to be smooth. Indeed, locally on
an open set U in M where Elv is trivial,
UxEik' - UxNE given by
it
is expressed as the map
map Eiki *AkE is C°` because it is multilinear.
If v ... , vk is a local basis for E on U, then the sections x H Vi1(x) A ... A Vi,(x),
ii < ... < ik,
form a local basis for NE on U; these sections are smooth because they are a composition of smooth maps. In view of Corollary 5.49 applied to the bundle
NE, any section s e C'°(U, NE) can therefore be written in the form s(x) = Y si,...ik(x) vik(x) A ... A Vik(x)
where si, ik e C'(U) and the sum is taken over all indices i, < 180
< ik.
Differential forms
181
Let M be a manifold and consider the tangent bundle TM. We know from the general considerations in §5.8 that the dual bundle T*M = UXEM (TTM)* is a vector bundle over M, which we call the cotangent bundle.
Let us write out explicitly what the trivializing maps look like for T*M. TTM has the Let (U, x) = (U, x,, ... , x") be a chart for M, then TU = trivializing map f: TU - U x I8" defined by v '-. (x, dn(v)). Now let i.'. ... . J." be the dual basis corresponding to the standard basis in R", i.e. i.': IO" - R is the projection on the ith coordinate. Since'(#,)(;.') = i.' oIs = dx1 we see that dx...... dx" is a local basis for T*M on U
and is the dual basis corresponding to a/cal,... , alax". Thus T*M has the trivializing map T*U -- U x I8" given by n
c1,.... bn)
E i=1
A C" section of T*M is called a covector field on M. For example, if f e C »(M) then the differential df can be regarded as a section M - T*M defined by df(x) = df, YM e (TTM)*. It is clear that df e C'"(M, T*M) because locally in the coordinate chart (U, x,, ... , x") we have (see the text following Proposition 5.29)
df=Of
ax,
ex"
dx"
and Of /axi e C'°(U ). Thus df is a covector field. Definition 6.2 We define the set of differential forms of degree r on M to be
r(M) = C r(M, n'(T*M)), the C°` sections of the vector bundle n'(T*M). In other words an dorm w e (2'(M) is a C' map
w: M - n'(T*M)
such that
w(x) a n'(T, M)
for each x e M.
Note that AO(T*M) is the trivial bundle M x Fl, so a 0-form is just a C'° function on M. Also, the 1-forms are the covector fields since A'(T*M) = T*M. We can also define differential forms on any open set by replacing M with U. In view of the comments following Proposition 6.1, if (U, x...... x") is a < ir, n dxi,, i, < chart for M then A'(T*M) has the local basis dx,, n on U. Thus a differential k-form on U can be written uniquely in the form
w(x) =I f ,...i,(x) dxi, n ... A dxi.
where f,...,, e C°°(U) and the sum is taken over indices i, < short we often write w = Y ft dx,.
(1)
< i,. For
182
Boundary value problems for elliptic systems
By identifying the differential form dx; with the corresponding form dx;
on the open set u(U) a R", we see that (1) is the local representation of w e C'(M. A'(T*M)) as defined in Exercise 4 of §5.8. The set (Y(M) of r-forms is a vector space over R, with addition and scalar multiplication defined pointwise, and in fact it is a C (M )-module. The product of differential forms is also defined pointwise: if a and q are forms of degree r and s, respectively, on the manifold M (or any open set) then the wedge product a n ry is defined by (a A r!)(x) = a(x) A n(x)
XEM
Note that a n rl is smooth, which is clear from the local representations of a and ry, whence a A q E SY''(M ). If f is a form of degree 0 (a real-valued function) and a is a form of degree r then f A a = .fa,
where f a is the r-form on M given by (f a)(x) = f (x) a(x). The operation n is bilinear over R and associative since this holds pointwise. In fact n is bilinear over C°°(M). Let f2(M) denote the direct sum of SY(M), r = 0, 1, ... , n together with its structure as a module over C'(M), and with multiplication n extended componentwise to Q(M). We call Q(M) the algebra of differential forms on M, and it is an associative algebra over C'(M). 6.2 The exterior derivative d
We begin with the definition of differential forms on open sets in R" and define the exterior derivative of such forms. The exterior derivative satisfies certain properties which characterize it uniquely, and make it possible to define this operator on manifolds. Then We identify TmR" = R" by the canonical isomorphism dx...... dx" is the dual basis in (R")* corresponding to the standard basis e1,. .. , e" in R". The canonical isomorphism TmR" = R" means that we can also identify /V(T.*R") = A'((R")*) which we do from now on. Recall from §5.1 that A((R")*) is the algebra over R generated by dx,, . . . , dx" with the relations
dx,Adx;=-dxj Adx;. As a vector space over R, A((R")*) has basis 1, dx,, dx, n dxj, dx; n dx1 A dx,,, ... , dx, n
i<j Every
A dx"
i<j n. The wedge product of two differential forms is defined by
2A1 =>f,g., dx,ndxlefr"(U) if
a = Y ff dx, e O.'(U) and q =
(- l)"q n a and
fang=an fq,
y, dx, e SY(U ).
We have a A q =
feC"°(U)
(2)
Then Q(U) = Q°(U) ® i2'(U) ®. ® Q"(U) is an algebra with product A defined in the obvious way (see §5.1).
Remark In the definition of the wedge product n one has to take care
of the ordering: I and J are ordered, but Y is not. For instance, if r.,
at = Yajdx,a(2'(U) and q =YbjdxjeW(U) then a A q= E (ajbj - ajbj) dx; A dxj. j<j For 0-forms (functions) f e i2°(U) = C x(U) we define d( = Y O flax, dxj
(3)
so that df is a one-form on U, dfei2'(U). (This agrees with the definition in §5.3.) The operator d can be extended to forms of degree > I as we show in the next proposition. Proposition 6.3 Let U be open in I8". The operator d extends uniquely to an operator d: S2'(U) . (Y+'(U) for each r = 0, 1, ... , n, satisfying the following properties: (1) d is an anti-derivation with respect to A, that is, d is R-linear and
d(anq)=dxnq+(-1)'aAdq foraefY(U)andgefr(U). (ii) d2=dod=0.
Boundary value problems for elliptic systems
184
Proof For uniqueness, note that if a = Y f, dx, e W U) then property (i) gives
dot =Zdf, n dx,+f,d(dx,) By properties (i) and (ii) we have d(dx,) = d(dxi, n
A dx,,) = 0, so in
view of (3) it follows that dx =
oL dxj n dxj
(4)
axj
To prove existence, if x = f, dx, is an r-form, r >, 1. we simply define doe by the formula (4) and show that d satisfies (i) and (ii). Let q = g, dx,, then
2A11=Yf,g,dx,Adx, I.,
and, by definition of d and properties of A, d(a A ry) _
_
d(f -g,) A dx, n dx, (g,'df, + f,-dg,) A dx, A dx,
dj,Adx,AYg,dx,+(-1)'Yf,dx,AYdg,Adx,
=dl nq+(-])'IAd>f Note that in the third equation we used (2) and the fact that dg, A dx, _ (-1)' dx, A dg,, since dx, = dxi, n A dxi, is an r-form. To verify (ii) we first show that if f e C'(U) then d(df) = 0. This follows since
d(df) = d
Of (Y-
ax,
dxj _ Z
f
oz
ax;axi
dx, n dxi = 0
by symmetry of the second partial derivatives and the fact that dxj A dx, _
-dx, n dx;. Now, if a = f, dx, is an r-form, then (ii) follows from property (i) since f, e C '(U) and xi...... x,, e C '(U) so that d(df,) = 0 and d(dx,,) = 0. The operator d is called exterior differentiation. The uniqueness of d in Proposition 6.3 implies that its definition is independent of the coordinate system. This makes it possible to define the exterior derivative operator on manifolds.
Let w e fY(M) where M is a manifold. If (U, +:) is a chart on M, then w has the local representation Y f, dx,, n A dx,,, where f, eCx(U), and we define dwI(u.,,) _
Of,
dx, A dx,, n ... A dx,,.
(5)
exi
the sum being taken over all indices i = 1, ... , it and it
0. Two volume elements to, and w2 are said to be equivalent if there is a number c > 0 such that w, = cwt. Note that the orientation for E defined by a volume element depends only on the equivalence class of the volume element. Thus, an equivalence class [co] of volume elements defines an orientation on E, the opposite orientation being [-w]. If v ... , v" is a positively oriented basis with respect to a given orientation A i.", where of E, then that orientation is the one induced by CO = J.' A
;...... i." is the dual basis corresponding to v...... v". Volume elements are determined only up to a positive constant. To specify
a volume element uniquely, we need additional structure such as an inner product on E.
Proposition 6.17 Let E be an oriented vector space over R and let be an inner product on E. Then there exists a unique volume element p e n"(E*) such that p(e,,...,e") = 1
for all positively oriented orthonormal bases et, ... , e" of E. In fact, if A p". More generally, if p', . , µ" is the dual basis then p = µ' A v,, ... , v" is any positively oriented basis of E and %', ... , An is the dual basis
Differential forms
195
p = (det[])"2 A' A ... A AN
(14)
then
Proof Clearly the condition p(e,, . . . , e") = 1 uniquely determines p by multilinearity. If p'. ... . p" is the dual basis in E*, then e...... e" is positively A p" since by virtue of §5.1 (14) we oriented with respect to if = p' A have p(e,, . . . , e") = det[b,j] = 1 , which is > 0. Now let v,, ... , v" be any
other positively oriented basis of E (not necessarily orthonormal) and consider the map qp e GL(E) defined by cp(e,) = v,. We have vj =
a,je, for
some n x t matrix A = [a,j]. Then
a,ja,k' <e1, e,>
a,ja,k
and hence det[(vj, vk>] = det ATA = (detA)2
Therefore, (det[(vj, vk>])"2 = Idet Al, and since det A > 0 we have (det[])ij2 = det A. Now, since is a one-dimensional space, we have p = c A' A for some c e R, and then by §5.1 (14) we get
.
A A"
c = p(v,,... , v") = det[(p`, vj>] = det[a,j]. Thus, c = (det[])"2 which proves (14).
In summary, corresponding to each inner product on E there is a unique n-form it such that p(v,,... , v") = (det[])`12
for each positively oriented basis v,,. .. , v" of E. We call p the canonical volume element determined by the given inner product and orientation on E.
The orientability of a vector bundle can now be expressed in a manner analogous to Proposition 6.26, by using the functor A . Let E be a vector bundle over M of rank n (of course, the rank need not be the same as the dimension of M). If there exists w e C'(M, such that w(x) # 0 everywhere, then for each x e M the nonzero w(x) e determines an orientation p: of E. by Proposition 6.16. It is easy to see that the collection of orientations {p=} satisfies the "compatibility condition" of Definition 6.15, thus { ,u,,) is an orientation of E. It follows that E is orientable whenever the vector bundle A"(E*) has a section which is nowhere 0. The converse also holds, as we show in the following proposition.
Proposition 6.18 Let E be a vector bundle over M of rank n. Then E is orientable if and only if the vector bundle A"(E*) has a section which is nowhere 0.
Proof Suppose there exists co e C '(M, A"(E ')) such that w(x) # 0 everywhere. Then for each x e M the nonzero w(x) e A"(Ez) determines an orientation pX of E,, by Proposition 6.16. We claim that these orientations
196
Boundary value problems for elliptic systems
{lex} satisfy the "compatibility condition" of Definition 6.15'. Let xo e M and choose an open, connected set U a xo on which there exists a trivialization
/3: E I v- U x R'. The sections vi(x) = (fx)-'(e1),
i = 1, ... , n
form a local basis of E on U, and without loss of generality the basis {v1(xo),... , v"(xo)} for Ex0 belongs to the orientation pxo, that is wxo(v 1(x0), ... , v"(xo)) > 0 when x = x0. By continuity we then have wx(v1(x), ... , v"(x)) > 0
for all x e Uo
where Uo c U is an open set containing x0. This means 1'x: Ex - R" is orientation-preserving for all x e UO, and we have thus verified Definition 6.15', i.e. (,ux) is an orientation of E. It remains to prove the "only if" part. Suppose E is orientable, and let U be an open, connected set in M for which there is a trivialization ft: EI v - U x U8" such that the maps Yx are orientation-preserving f o r all x e U. Let e1, ... , e" be the standard basis in R". Then Ps '(e1), . . . , flx '(e") is a positively oriented
basis in Ex, and we let 1.i..... ).x be the corresponding dual basis in E*. It is easy to see that ;Y e C9(U, E*) (see §5.8, Exercise 11) and, if we define wv e C°°(U, n"E) by wv(x) = i.,' A . A ix, x E U, then v1, ... , v" is a positively oriented basis of Ex if and only if wv(vt..... v") > 0. Therefore, we can cover M by a collection of open, connected sets U; on which there exists w1 a C'(U;, n"(E*)) such that, for all x e U1, v") > 0 a v1,. .. , v" is a positively oriented basis of Ex. Let pi be a partition of unity subordinate to { U; }, and set
to = DIM. Because the supports of p1 are locally finite, this sum is finite in some neighbourhood of each point in M, whence w e C'°(M, A"(E*)). (As usual p1w1 is defined to be 0 outside U1, hence it is smoothly defined on all M.) At each point x e M, if v1, ... , v" is a positively oriented basis of Ex, we have (p4w1)(x)(v1, ... , v") >, 0 for all i and p1(x) > 0 for at least one index i. Thus, w(x) # 0. Since the fibres of the vector bundle n"(E*) are one-dimensional we can also write the proposition in the following way: E is orientable if and only if n"(E*) is a trivial vector bundle. Exercises
Let E and F be vector bundles over M. Show that if two of E, F and E ® F are orientable, so is the third. 1.
2.
Let /: N - M be a smooth map, and let E be a vector bundle over M.
Show that if E is orientable, then the pull-back bundle / -'E is also orientable. Find an example showing that the converse is not true.
Differential forms
197
Show that if E is any vector bundle (orientable or not), then E ® E is orientable. Hint: For any vector space E there is a "natural" orientation on 3.
E®E. 65 Orientation of a manifold
A manifold M is said to be orientable if its tangent bundle TM is an orientable vector bundle. It follows from Proposition 6.18 that M is orientable if and only if there is an n-form u e W(M) such that u(x) # 0 for
all xeM. Any two nowhere vanishing n-forms u, p' a f "(M) on an orientable manifold M of dimension n differ by a nowhere vanishing function: u = f p .
If M is connected, then f is either everywhere positive or everywhere negative. We say that p and u' are equivalent if f is positive. Thus on a connected orientable manifold M the nowhere vanishing n-forms fall into
two equivalence classes. Either class defines an orientation on M by Proposition 6.16, and these are the only two orientations on M. Examples
(1) The unit sphere S" c R"+' is orientable. Let n: S" -* R"+' be the unit normal vector field on S", i.e. (x) = x. The orientation on the tangent spaces Tx(S") is defined as follows. A basis v2, ... , v" in Tx(S") is positively oriented if and only if ,,(x), v,,. . ., v" is positively oriented in OB"+' with the standard orientation. We leave it as an exercise to show that this defines an orientation on the tangent bundle T(S") (i.e. satisfies the compatibility condition). (2) The real projective space 18P" (see §5.2, Exercise 9) is orientable if n is
odd, but not orientable if n is even. Let n: S" -, RP" be the canonical projection x i-+ [x]. Since it is a local diffeomorphism, the tangent map Tn: Tx(S") -- T,,J P") is an isomorphism for each x e S". Regarding T(S") as a sub-bundle of the trivial bundle T(R"+') = 68"+' x R"+', we have the map (T_xn)-toTxn given by (x,v)i-+(-x,-v) (15) from Tx(S") to T_x(S"). In view of the orientation defined on S" in Example (1), we see that this map is orientation-preserving if n is odd, but orientationreversing if n is even (since n + 1 is even and odd, respectively). In fact, this holds for any orientation on S" since S" is connected and there are only two
orientations on S", the standard orientation and the opposite orientation. If n is odd, we can define an orientation on each tangent space T,,,I P" by requiring that Txn be orientation-preserving. This definition is independent of the choice of the representative x in the class [x], and it clearly satisfies the compatibility condition of Definition 6.15 since it is a local diffeomorphism.
If n is even, then RP" cannot be endowed with an orientation for if this were possible then we could define an orientation on S" by requiring that Txn be orientation-preserving for each x e S'. But then (T_xn)-' o Txn would be orientation-preserving which contradicts (15).
198
Boundary value problems for elliptic systems
From the point of view of integration in §6.6 we will need to recast the definition of orientability in a more convenient form.
A diffeomorphism gyp: V - W between open sets in 31" is said to be orientation-preserving if
det Drp(x) > 0 for all
x e V,
i.e., the derivative map Drp(x):R" -+ R" is orientation-preserving for all x e V.
Theorem 6.19 Let M be a manifold. The following are equivalent:
(i) The tangent bundle TM is orientable. (ii) There is an atlas {(U,, K j)) on M such that all the overlap maps K, - cj' are orientation-preserving diffeomorphisms, i.e. det D(Ki D h; ') > 0 everywhere on n j(Uj n Ui).
Proof (ii) -+ (i) Suppose M has an oriented atlas {(UU,,c,)}. We define an
orientation µx on TIM, X E U,, as follows: A basis v...... v" of TIM is
"positively oriented" if dK;Is(v,), ... , dreiIx(v") is a positively oriented basis in
11". Since the overlap maps are orientation-preserving, it follows that the definition of µx is well-defined for all x e M. The collection of orientations {µx} obviously satisfies the compatibility condition of Definition 6.15, so TM is orientable. (ii) Conversely, suppose TM is orientable. Choose an atlas {(U,, r;,)} of M where the domains U; are connected. Then the maps dki`x: TIM -+ 18" (i)
are either orientation-preserving or orientation-reversing for all x e U,. In the latter case we can interchange any two component functions of t , to make it orientation-preserving. Thus, we can assume that the dKi's are orientation-preserving everywhere for all i. It follows that the overlap maps, hi 0 is ', are orientation-preserving. We say that an atlas {(U,, tci)} satisfying (ii) of Theorem 6.19 is an oriented atlas. An orientation on M is a maximal oriented atlas e?. Any chart (U, K)
that belongs to G is said to be positively oriented with respect to the orientation. We conclude this section by showing how a Riemannian metric on an oriented manifold determines a unique volume element.
Let M be a Riemannian manifold (see Definition 5.65), and assume that M is oriented. Then the inner product gs = 0 whenever f is real and f > 0. By the Riesz Representation Theorem (see [Ru 1]) there is a unique positive measure nt defined on the Borel sets in X such that
if=ffdin, feQX), and such that m(K) < oo for every compact set K c X. Furthermore, if X is v-compact then n is a regular measure. In particular, a manifold M is a locally compact and o-compact Hausdorff space to which we can apply this theorem. Let dµ a C(M, Ibe Al) a density
on M. It is not hard to see that the map f r j f du is a positive linear
functional on CA(M). By the Riesz Representation Theorem there is a unique regular Borel measure, in,,, such that
Jf dp =
$f dnt
when
feC,(M)
The space L,,(M, in,), p e R, consists of all measurable functions f such that If i P is integrable with respect to in, For p > 1, the norm
IIfII=(JlfIPdm)"
P
Boundary value problems for elliptic systems
206
makes LP(M, mµ) into a Banach space, where functions that differ only on a set of measure zero are identified. For p = 2, the space L2(M, is a Hilbert space with inner product U, 9)L M) =
f
fy dm H
From now on we always write dp rather than dm,,. Exercise 1.
Let d p e C(M, I A I) be a density and f E CC(M ). Verify that if U is an
open set containing the support of f then JM f du = jv f dp. (U is not necessarily a chart domain.) 2.
Let du e C(M, IAI) be a density. Verify that the map f '- f., f du is a
positive linear functional on CA(M). 6.7 Stokes' theorem
Let M be an oriented manifold with boundary OM. Before we can state Stokes' theorem we need to verify that the boundary of an oriented manifold
M is orientable, and then to choose the appropriate orientation on t'M. Recall from Proposition 5.69 that if rp: U -, V is a diffeomorphism between open sets in RR, then 4) restricts to a diffcomorphism 0 for all
x e U,
it follows that det D'(p'(x) > 0 for all x e dU,
i.e. (p' also has positive Jacobian determinant everywhere. We claim that if M is orientable then its boundary OM is also orientable. To see this let {(U1, ic,)) be an oriented atlas on M, and restrict it to get an atlas on 8M
U;=U11- M, +:!=xilc,,,,,M Since the overlap maps for the atlas {(U;, h;)} have positive Jacobian determinant everywhere, the overlap maps for the atlas {(U1, At)} on OM also have positive Jacobian determinant everywhere. Now, let the upper half-space R = 1x: x" 0) in 68" be given the A dx". The boundary orientation on standard orientation dx, A An,. = R"-' = {x: x" = 01j is. by definition, the equivalence class of the volume form
a=(-1)"dx,A...Adxn-1
(20)
The factor (- 1)" is needed to make Stokes' theorem come out sign-free.
Differential forms
207
Let M be an oriented manifold, and let {(U,, r,)} be an oriented atlas on M.
The boundary orientation on am is defined by the following requirement: if : is an orientation-preserving diffeomorphism of some open set U in
M into the upper half-space R+ then the orientation on aU = OM r U is the equivalence class of k*a, that is, if m e aU then the basis v1, ... , v"-, e
Tm,(9M) is positively oriented if the local representatives v'; = dKI,"(vt), j = 1, ... , n - 1, satisfy
a(u1,.... r" ,) _
1)" det[(Y) - ... axi'
n
ax,',"
Proof The sufficiency of this condition is obvious. To prove necessity, we use Taylor's formula. By virtue of (24) we may assume that A C- Diffk(U x C,
U x C), i.e. a scalar operator. Let xo e U and x(xo) = 0. If f e C°°(U, C) then by transporting Taylor's formula for f -K-' by the chart K we obtain
f(x)
a!
a'f(xo)x' + Rk+ 1(x),
xeU
(25)
where
Rk+1(x) =
x'
gk,.(x)
and gk.8 a C°°(U, C).
Note: By x° we mean the function in C`°(U, I8) defined by xi' . x," where x1 e C'°(U, R) are the component functions of the chart K. Since x(xo) = 0 we have x;(xo) = 0 f o r i = 1, ... , n. Then x°, I a I = k + 1, is a product of k + I functions vanishing at xo, so Rk+, vanishes to kth order at xo. Since A e Diffk(U x C, U x C) then by applying A to (25) we obtain that Af(xo) =
JaY
a! aaf(xo) A(x2)(xo) + 0.
This holds for any xo a U; thus A = :ask a, a' where a, = A(x')/a!. Next, we define the principal symbol of a differential operator. We let p: T*M - M denote the cotangent bundle projection and then p-'E and p'F are the pull-backs of E and F to T*M. Recall that the set of vector bundle homomorphisms from p-'E to p-'F is the set of C' sections of
Hom(p-'E, p-'F), that is,
HOM(p-'E, p-'F) = C°°(T*M, Hom(p-'E, p-'F)). Definition 6.32 Let A e Diffk(E, F). The principal symbol of order k is the vector bundle homomorphism nMA a Cm(T*M, Hom(p-'E, p-'F)) defined as
212
Boundary value problems for elliptic systems
follows: for any (x, 1;) e T*M and c e E,, we let
akA(x, ) c = A(
(P
k.
u)(x)
(26)
where rp e C '(M) and u e C'°(M, E) are chosen such that cp(x) = 0, dxcp = S
and u(x) = c. The map irk: Diffk(E, F) -s C°(T*M, Hom(p-'E, p -IF)) is called the principal symbol map of order k.
Of course we must show that the definition of nkA is independent be local coordinates on of the choice of (p and u. Let (U, x,, ... , M such that E and F have local bases on U. The elements c e E. and trkA(x, S)c e FX can then be regarded as column vectors in Cp and CQ, respectively, and we have (26')
nkA(x, )c = Y aa(x) Sa' c I=I =k
where a= = [ajj']q -p is a smooth matrix function on U, S' = Si' Sam" and the ;'s are the coordinates of = El;, dxi. Since (26') does not depend on
the choice of rp and u, then neither does (26). At the same time, we have
shown that the local definition (26') is independent of the choice of coordinates and local bases in E and F. Furthermore, (26') shows that izkA(x, S) is a linear map Ex -+ F, and, since the expression on the right-hand
side depends smoothly on x and , we have defined a vector bundle homomorphism nkA: p-'E - p-'F, i.e. a C'° section from T*M to Hom(p-'E, p-'F). Remark The representation (26') corresponds to the local representation
A = [Ai,] _
(27)
aa(x) da, IalCk
in Proposition 6.31. We say that s e C'(T*M, Hom(p-'E, p- IF)) is a kth order homogeneous symbol if
for all i.>0. The set of all such symbols is denoted Symblk(E, F) and is a linear subspace
of C x(T*M, Hom(p-'E, p''F)). If A e Diffk(E, F) then (26') implies that akA a Symblk(E, F). Not every element of Symblk(E, F), however, is the principal symbol of a differential operator.
Let DSymblk(E, F) denote the set of homomorphisms s C'°(T*M, Hom(p-'E, p-'F)) such that for all x e M, s1: TX M -+ L(E,,, F,) is a homogeneous polynomial map as defined in §5.1, that is, sx a P'k'(T. M, L(EX, F1)). It is clear that nk maps Diffk(E, F) into DSymblk(E, F).
f2.
E. f= is called exact if the image of f is equal to the kernel of f,+, for Recall that a sequence of linear maps E,
i=1,2,...,n-1.
E2
213
Differential forms
Theorem 6.33 The following sequence of complex vector spaces and linear snaps is exact :
0 -+ Diffk _,(E, F) -. Diffk(E, F) - DSymblk(E, F) - 0 Proof The sequence is obviously exact at the first arrow since the inclusion map Diffk_,(E, F) -* Diffk(E, F) is injective. For the second arrow, if A E Diffk_,(E, F) then ttkA = 0; in fact by (26) we have ttkA(x, S)c = 0 since (pktt vanishes to (k - 1)th order at x. Conversely, if A E Diffk(E, F) then locally A has the representation 4
At, =
P
Aijuje;
where
Aij =
a;ji a=.
i=1 j=1
0 when IxI = k, and hence If nkA = 0, then (26') implies that A E Diffk _ 1(E, F) so we have exactness at the second arrow. To complete the proof we must show that the principal symbol map Jtk is surjective. Let s e DSymblk(E, F). We cover M with charts (U,, t:i) such that Eu, and F. are trivial. In local coordinates on U; we have T*(Ui)
Ui x88",
E. = U; x CP and F,,, ^-. U; x C9 and the local representation of s is a kth order homogeneous polynomial of S E 88" with q x p matrix coefficients. It follows that there exists A; E Diffk(Eu,, Fo) such that nkAI = s on the fibres of p-'E over U,. Now let I = Y iyi be a partition of unity subordinate to the cover U; and set A = Y O1 A,. Every point x E M has a neighbourhood that intersects only finitely many of the supports of i/i,, so it follows that A a Diffk(E, /F) and nkA(x, e) _
//
i,(x) . lrkAi(x, S) _
.Ii, Y
/ ,(x) . s(L, S) = s(x, S) //
xx
Definition 6.34 A differential operator A e Diffk(E, F) is elliptic at x E M if
nkA(x, S) is an isomorphism E. -+ Fx for all (x, S) e T*M\0.
The operator A is said to be elliptic if it is elliptic at each x e M. From now on we write it = ztk if no confusion is possible.
Let the tangent bundle TM be given some Riemannian metric g = < . >. This metric induces an "index lowering" operator TTM -* T*X M, defined by v i-4 jp in which case the pth row is 0. In either case, det[6j. j 1 = 0 if ip # jp. If ip = jp then expanding by cofactors about the last column or row we get D P-1 det[bt.jbla.b=1 =det[b,.jja.b=1
so it is evident that (37) follows by induction on p. To show that ,P) we have
it...JP det[g,.j,]°.b= )
11
P! P!
because gij = .
. Denote the induced inner product on each exterior product AP(E*) by the same symbol < , >. If X e E then X # n q>
(43)
for all aeAP(E*)and 1EAP-1(E*)
Proof Let et, ... , e" be an orthonormal basis of E and let p'..... µ" denote the (orthonormal) dual basis in E*. By linearity it suffices to verify (43) when
X = e; and a and q belong to the corresponding orthonormal bases in AP(E*) and nP-'(E*), i.e.
a=/1" n... A u',
and
q=fill A... A U1p
< iP andj1 < .. <j,-,. By definition of the interior product
where it
on AP(E*) is extended by C-linearity to obtain a Hermitian inner product
<x, + ia2' 11 + iq2> = + - M
M[x(x, e)a(x, f)]) dx ds.
It is not hard to see that the derivatives of X(ex, g) are bounded by a constant
independent of e, and for any real number s, each derivative of
is
certainly bounded by a constant times itself, so it follows that we can estimate the integrand by
I<x>-M'%t>-NN[X(ex,es)a(x, )])I 5 const
Ial",+"<S>m-N(x>-M
(and the constant depends only on M and N). If we choose M > n and N > m + n then the integrand belongs to L1(R" x R"). By dominated convergence, I = lim, - 0 I, exists and is equal to (4).
Boundary value problems for elliptic systems
236
Remark When a e L1(R" x R"), the oscillatory integral coincides with the usual value SS
dx dl;.
71 The classes S' Consider a differential operator
A(x, D) = Y a,(x)D',
x e R".
(5)
I3I_< m
Here a = (a ... , an) is a multi-index, D' = D;' D'" where D, = i-' r3/Ox,, and lal = a, + + a" is the order of D'. The Fourier transformation yields A(x, D)u(x) = (21t) a
= (2n)
a,(x) f ei(x.4) S'u( ) dS J
z Jei(z. 41 A(x, FS)u(S)
d
where the polynomial A(x, S) = Ll.,-<m a,(x)S' is called the symbol of A(x, S). Recall that differential operators are local, i.e. A(x, D)u(x) = 0 in every open set in which u = 0. To define pseudo-differential operators we shall use the same formula
Au(x) = (2n)-"
f ei(x.41 A(x,
)d(S) d
with a larger class of symbols A(x, ) suitably chosen so that the operator A still has a pseudo-local property, namely, Au e C'° in every open set in which u e C. Not every function A(x, g) in Om leads to a pseudo-differential
xo fixed, then A is a translation operator, for instance if A(x, S) = operator and hence not pseudo-local. The appropriate class of symbols is stated in the following definition. The pseudo-local property is proved later after we have developed basic properties of the operators A(x, D) (see Corollary 7.14). The symbol A(x, 4) of the differential operator (5) belongs to Sm, m e N, if the coefficients a, are C°° functions with bounded derivatives of all orders, i.e. a2 a Cb (R").
Definition 7.1 If m is a real number, m e 9, then Sm = Sm(R" x R") is the set
of all A e C'(R" x R") such that for all a, fl ID;DxA(x, s)I < C,.a(l +
ISI)m-lal,
for all x, S c R.
(6)
Sm is called the space of symbols of order m. We write S-°" _ 0 Sm.
S"=U Sm.
237
Pseudo-differential operators on R"
It is clear that S' is a Frechet space with semi-norms given by the smallest constants which can be used in (6), and we let JAI") denote the maximum of these constants C,,.Q for Ial + 1#1 0, hence EISI > S on the support of yt - 1. It follows that on the support of y, - I we have S 3111 for all
Pseudo-differential operators on 18"
241
s e R, so we obtain when loci + I#I _< k that IDIDOA(x, 5)I 5 const Ialk"+M+N
JJ m2 - . - > mj -+ -oc as j-* +oo, and let A(x, ) e C'(R" x R"). We will write x
A - Y Aj, j=1
if for any integer r >- 2 we have r- 1
A - Y AjeS
(18)
J=1
From this it follows in particular that A(x, ) a S'".
Proposition 7.7 For any given sequence A e S'' j = 1, 2, ... , with mj_, > mj -+ -oo as j -, +oo, we can always find a function A e S'" (where m = m,) such that 00
A - YAj. j=1
The function A is uniquely determined modulo S - m.
Proof The uniqueness follows immediately from (18). To prove existence,
we use Lemma 7.3. Choose a function X e Co which is equal to I in a neighbourhood of the origin, define X,(i;) = then put W
A = Y (1 - X,,)-Aj,
(19)
j=1
where 0 < ej < I approaches 0 so rapidly as j -+ oo that 2-1(1 +
ID'gD=[(1 -
IUD'",+1 - lal
(20)
for all x, i; and lal + flu m2 -co as j - +oo, > in,
and let A(x, D) e OS". We will write
A(x,D)- Y A.&, D), f=t
if for any integer r >, 2 we have r-'
A(x, D) - Y A,{x, D) e OS',.
(18')
Proposition 7.7' For any given sequence A,{x, D) e OS'"j, j = 1, 2, ... , with oo, we can always find a p.d.o. A(x, D) e OS"" such that z A(x, D) - Y A,{x, D).
m;- I > m; -' -oo as j
i=1
A(x, D) is uniquely determined modulo OS-'.
Proof The uniqueness follows immediately from (18'). To prove existence we use Proposition 7.7. Since the p.d.o.'s A;(x, D) e OS'' have symbols A,{x, l;) a S'j we can find a symbol A(x, l;) a Sm' such that (18) holds. But then (18') is also satisfied for A(x, D). Proposition 7.5 gives the connection between the amplitude a(x, y, l;) and the ordinary symbol A(x, ) of a p.d.o., see (15). We complete this proposition by giving an asymptotic expansion. Proposition 7.8 Let the amplitude a(x, y, l;) e S'". Then the ordinary symbol A(x, i;) given by (15) has the asymptotic expansion il=l
A(x, at!
DeDia(x, y, s)I,=x
Proof Obviously DID,a(x, y, %)1,=s e S' 1:1 and the asymptotic sum is
Pseudo-differential operators on R"
245
therefore meaningful. Since i{"D,'. = c,'., what we have to show is that
A(x, 5) ^ Y
1
(22)
x, I D°O°a(x, Y
x.
Using the notation Op(A(x, )) for the operator A(x, D), we may reformulate this as an asymptotic expansion of operators: A(x, D) = Op(A(x, )) ^
1+ Op(Dk a(x, x, 5)). x.
(22')
By Proposition 7.7, there exists A,(x, S) E S"° with an asymptotic expansion given by the right-hand side of (22), so to complete the proof it suffices to show that
A(x,D) - A,(x,D)EOS
=n OSk. k
Our aim now is to apply Theorem 7.6. To verify the conditions of this theorem, we use Taylor's formula (Remark 5.6) in the y variable in order to
expand a(x. y, 0) in powers of y - x about y = x. This gives, after an integration by parts using the identity (-R.)2 x)' eNx that the operator (13) (which by Proposition 7.5 coincides with A(x, D)) is equal to Au =
(2n)-" JJerIDa(x. x, O)u(r) dy d+ Rnu, IaIEN-1 a.
where
Rnu = jKv(x. y)u(y) dy and
y) _
(2n)-"
N
Jex-r.c,
Ix{ = v x!
x J" (1 -
-'D' a(x. x+t(p - x),{)dtdS
0
Now, the symbol estimates for "a" imply that if N > m + n + k, then Kv E Ck and
{D'D'K.(x, y)1 m + n + k I(x - y)'DPDTKN(x, y)I 2. Then we decompose the original operator (41) as A = A' + A2, where
and we would like to make the change of variable
put the exponential term in the form e'''-='
dx' d
A'u(x) = (2n)-* Jf4(x.)h(r.Ix - x'()u(x')
A`u(x) = (2n)-"JJ
A(x, S)[I - h(e-'Ix - x'I)]u(Y
We claim that A' e OS integral operator
.
)eilz-x.;ldx'd
(43)
(44)
Indeed, the operator A2 has the form of an
A2u(x) =
JK(x. y)u(y) dy,
where
K(x, y) = JA(x. S)[I - h(e - 'Ix - yI)]
dS
Thus A2 is a p.d.o. with amplitude representation of the form (13) where a(x, y, S) = A(x, S)[1 - h(e-'Ix - yi)]. Since a(x, )', ) = 0 for Ix - yI < e,
Pseudo-differential operators on R"
255
then K e C' by Proposition 7.15. By regularizing the oscillatory integral for K as in (37), we obtain
ID'D"K(x,
y)I
< C,B,Ix - )'I
Vx, fi and N -> 0,
and then if we use the inequality I + Ix - yI - 0.
Hence A' a OS - ' by virtue of Theorem 7.6. From (43), (44) we have A. = A,', + AK. Then A- e OS by virtue of the lemma proved above, so it remains to show that A,', a OS" and that (40) holds for A.'. Representations like (41), (42) hold for A' and A,'c, and then by the change of variable y -+'(D(r,-) we have 1
AQ u(y) =
f
where 0(y, z) =
eatr :. ( '(y),
%',:)u(:) d: d;
(46)
z) and
K-'(')I)
D(y,:)
Now. (46) is a familiar representation of the p.d.o. A,'c in the form (13) with the amplitude
a(y,:, S) = A(K-'(y), 0(y,:)S)'D(y,:).
(47)
Note that t(i(y.:) is well-defined and smooth on the support of D(y,:), i.e. when IK-'(y) - K-'(:)I < 2c. Hence a e C. A straightforward application of the chain rule shows that by (38') we have ID,.DCD'a(y.
)I ,.-,(Y,'1) (54) where Am
a S"-'. The conditions (38), (38') imply for all x e R", n e li",
clgl,
c'1,11
0, c' > 0. Now let j(q) be a cut-off function, near q = 0 and l(q) = I for large Iqi. The function
i.e. 1'(n) = 0
'(y, n) = j(q) - X('dK(K-'(y))'n) is C' and satisfies ik(y, q) = 0 for Iql < e and for Inl 3 R (where r > 0 and
R > 0 are independent of y), hence the symbol i(y. q) H"(K -'(y), belongs to S
and we obtain from (54) that
AK(Y. n) = X(n)H"(K '(Y), `dK(K '(1')1 n) + A'"_ i(Y, n)
(55)
where.!.-, e S"-'. Thus A. e OC1-'S" and Proposition 7.21 now gives us the assertion (53) of the theorem.
There is also a version of Theorem 7.17 that holds for (- 1) classical symbols. This is important for Chapter 8. Theorem 7.17' Let X and Y be open subsets of R" and K: X - Y a di/eomorphism. If A E OCI -'S" is a (- 1) classical operator and the kernel of A has compact support in X x X then AK a OCI -'
S",
and the kernel of A" has compact support in Y x Y. Also. nAK(y, q) = nA(K-'(y). 'dK(K-'(Y)) 'n),
y e Y, q e R"
(56)
where A(x, ) = 0 for x outside subset of X, and AK(y, q) = 0 for y outside a compact subset of Y.
7.6 Continuity in Sobolev spaces
In this section we follow Hormander's proof [Ho 3] that differential operator A(x, D) e OS" is a continuous operator A(x, D): W'2(R")
W'2 "(18"),
for every I e R. The Sobolev spaces
fla()l2(l + l2)'d < are endowed with the Fourier norm Ilull,.
any pseudo-
Pseudo-differential operators on R"
261
A summary of definitions and results on Sobolev spaces can be found in the appendix at the end of this chapter.
Proposition 7.24 Let A(x, ) e S'. The commutators of A(x, D) with Dj and multiplication by x j are
[A(x, D), Dj] = i
[A(x, D), x;] =
OA
(x, D).
ax
il aA (x, D). -A i
Proof The first relation states that Dj(A(x, D)u(x)) = A(x, D)Dju(x) - i e (x, Oxj
which follows by differentiation of (2n)-" JeM A(x, S)a(S) dS
A(x, D)u(x) =
is the Fourier transform of Du. The under the integral sign since is -D,6(i ), so an integration by parts gives the Fourier transform of second relation. Proposition 7.25 If A,_., A, c- S° and Fe C'(C') then F(A1, ... , A,t) e S°.
Proof Since Re As., im A, e S° we may assume that A, are real valued and that F e C (Rk ). We use the fact that for A, a S° the values A(x, in) belong
to OR R) for some R and that C'-functions F are bounded on compact sets 8. This implies that F(A) = F(A1, ... , Ak) is bounded, and for the derivatives we have OF OA,
r3F(A) ax,
11F(A)
r=i aA, axj k
AV cA aA,.
is bounded
is 0((1 + ICU-'),
e S°, dA,,leSj e S - ' and c?F(A)/aA is bounded. The proof is finished by induction with respect to the order Jai + I#I of the derivatives since
D'4DXF.
We need also the classical lemma of Schur.
262
Boundary value problems for elliptic systems
Lemma 7.26 If K(x. y) is a continuous function in F2" x R" and
sup J IK(x, y)I dx < C,
sup I IK(x, y)I dy _< C,
(57)
then the integral operator Ku = J K(x, y)u(y) dy has norm < C in L2(W ). Proof The Cauchy-Schwarzfinequality gives IK(x, ),)I dv.
1(Ku)(x)l2 < J IK(x, y)f lu(y)12 dy. J
If the last integral is estimated by C, an integration with respect to x gives f J(K U)(X)12dx < C JJIK(x.
y)I' lu(y)I2 dx dy 2-sup,,., IA(x, )I2. Then
by Proposition 7.25 since M12 _< M - IA(x, S)12 and we can choose
Pseudo-differential operators on R"
263
F e C=(R) with F(t) = 11'2 when t > M/2. Corollary 7.11 and Theorem 7.12 imply that
A(x,D)+T(x,D),
C'(x,D)
where 7(x, D) E OS-' (look at the symbols!). Hence we have
(A(x, D)u, A(x, D)u) _ -(C(x, D)u, C(x, D)u) + M(u, u) + (7(x, D)u, u) M(u, u) + (T(x, D)u, u), or
!IA(x, D)u112 , M I!ull- + II T(x, D) u 11 hull,
which completes the proof since TE OS-' is already known to be L. continuous.
By referring to the Fourier norm of W2, it is obvious that the p.d.o. A' z is an isometry, A"': 10, W2', and we have with symbol (I +
A"' A"'= A'" -12,
r, mz e R.
Using these isometries we can prove the following theorem. Theorem 7.28 If A E S"' then A(x, D) is a continuous operator, A(x, D): W2(I ) -, W2 °(P")
for every I e R.
Proof We write A(x, D) = A(x, D )A-'A', where A(x, D) -, A -'e OS° by Theorem 7.12. In view of Theorem 7.27, the operator A(v, D)' A-m is continuous Wz -. W3, hence A-
A(x, D): WT - W °
AA W'0z
is continuous, where WO = L2. This proves the statement in the theorem if I = in. Now, for any I e P. we use the fact that A' is isometric to obtain JAull,-", = IIA'-tAullo < Cllullt, by virtue of the result just proved, since A'-': A(x, D) E OS'.
Remark The proofs of Theorems 7.27 and 7.28 involve a finite number of adjoints and compositions of operators derived from the symbol A(x, 61.). It follows from Theorems 7.10 and 7.12 that the norm of A(x, D): W2 -+ W; is bounded by a semi-norm of A in S'". The symbol spaces S"'
If the symbol H(x, s) a J", m > 0, is independent of x for large x then it is possible to define operators H(x, D) directly without a cut-off function, and to show the continuity of H(x, D): W'2(RA)
W';-'"(R"),
m > 0.
Boundary value problems for elliptic systems
264
without introducing any S-derivatives. This is important for some applications. For a homogeneous symbol H(x, E J'", differentiation with respect to 5 of H(x, ) would lead to serious singularities near = 0 because for lal > in,
D'H(x, Let in = Jm(R" x P"), m E R, denote the set of functions satisfying (1) and (2) of Definition 7.18, but with condition (3) replaced by the following: (3)- H(x, ) is independent of x for large Ixl -> R.
It is obvious that in c in. The new symbol spaces are defined as follows. Definition 7.29 We say that A(x, S) a S'(R" x R"), m e P. if
(1) all x-derivatives DOA(x, t;) exist, are locally integrable and satisfy the estimate
ID°A(x, )I < CQ(1 + J I)
(58)
.
(2) A(x, y) is independent of x for large Ixl >- R.
Using Remark 7.20, we see that in c 9'
(59)
for m >, 0.
Theorem 7.28' Let A(x, ) E S'", m e R. We define the operator A(x, D) as before (see (7))
A(x, D)u(x):_
(2tr)_" JA(x.
e
u E So
(60)
Then A(x, D) is continuous
A(x, D): W2(P") - W2--(R")
(61)
for every I e R.
Proof The Fourier transform of functions u(x) e C0 '(R") (or 9'(W)) satisfies ICU
for all N > 0.
Putting N = n + 1 + Iml and using estimate (58), we see that the integral in (60) is well-defined.
Let A(ao, ) be the value of A(x, 5) for large Ixl >, R, and set
A(x,S)
Pseudo-differential operators on R"
265
We have C(1 + 141)m
(a)
IA(x, 4)1
(b)
IDA'(x, 4)I x
(c)
A'(x, 4) = 0 for
141)-
(62)
Ix1 >- R.
By a Fourier transformation of the symbol A'(x, 4) in x we obtain from (62) parts (b) and (c) the estimate (1 + I7I)"IA'(7, 4)I < CN(l + 141)"',
for all N > 0.
(63)
To prove the continuity of (61). we decompose the symbol
A(x, 4) = A(x, 4) + A'(x, 4) The action of A(x, D) is multiplication by A(oo, 4) in the Fourier norm,
A(x, D)u = (r-'(A(x, 4)u(4)), and therefore the continuity of A(-c, D): W; -' W'-' follows at once from the estimate (62) part (a). For the operator A'(x, D), we Fourier transform the function e.(x) = A'(x, D)u to obtain, see (60), 0(7) = (2n)-"
f A'(7 - 4, 4)u(4) d4
and then use the estimate (63) to get I£ 0 and H(x, 4) E
H(x, D): Wz(R")
The operator
W2 "(R")
is bounded for all I e R. Also, we have H(x, 4) - X(4) H(x, 4) e S
= n Ss', !eR
where X is an arbitrary cut-offunction.
Proof Since J" c S" for m > 0, the first statement follows from Theorem 7.28'. Now J" c J", hence X H(x, D) is a p.d.o. in the former sense, and the second statement means that the difference between a new and an old p.d.o. is infinitely smoothing. Now for the short proof. If 0 = 1 - X e C0 '(R') is any patch function then from (50) we obtain the estimate 4)I = Iw(4)D°JI(x. 4)I =
(i +141)m
CBx(l + 10-", because cp has compact support. Hence cp(5) H(x, 4) e
bfl and N > 0
as was to be
shown.
Remark 731 If A e Cl -'S" is a (- I) classical symbol with m > 0 and nA = H"(x, ) is independent of x for large Ixl, then by Corollary 7.30 we
can omit the cut-off function in the definition of A(x, D). Indeed, by Definition 7.21 we have A = XH" + A. _ , where A" _ , e S"-', so that A(x, D) = H"(x, D) - 4p(D)H"(x, D) + A"-,(x, D). where QH" e 9- S_ y. xTherefore, the definition of the operator A(x, D) is only if affected modulo we omit the term ep(D)H"(x, D).
We know from equation (31) that the symbol A(x,4) of a p.d.o. is uniquely determined by the operator A(x, D). Further, if A is (-1) classical then the principal part will then be uniquely determined by the operator. The following lemma gives more information on this point. Lemma 7.32 Given x0 e R" and 4o a R", 1401 = 1, there is a sequence (0k) in Co (R") such that
(t) 4k(x) = 0 if Ix - x°1 > Ilk; (ii) 11 Ok ll o = 1 and, for s < 0, we have limk - 11ok II, = 0; (iii) if A e 0C!-'S° then limk - x IIA(x, D)ck - nA(xo, 4°)0,11° = 0. It also follows from (1) and (ii) and dominated convergence that 46k -' 0 weakly in L2, i.e. ! Oki dx -+ O for all e L2.
Pseudo-differential operators on R"
267
Proof Choose 0 E CO' with 110llo = 1 and supp 0 c (x e R"; Ix - xol < 1). Then let Ok(x) = 0120((x - x0)-k) ei(J240),
by a translation we may assume that xo = 0. Property (i) is obvious, and ff0kllo = 1 follows by a change of variables. Further, we have
k(x) = k " 2 ((s - k2So)/k)
(64)
hence for s < 0 the norm II
k Os' =
f (1 + Ik +
k - oo
by the Lebesgue dominated convergence theorem.
Let sk be the isometry of L2 given by f (x) i- k'12 f (kx), and note that [eik'' 4°'0]. Since sk and multiplication by e'k' {O' are isometrics of e-ikt'.{°'sk 'A(x, D)#k = Ok L2, we may establish (iii) by showing that Wk = sk
approaches zrA(xo, o)O in L2. In view of (64) and the definition of A(x, D), we have Wk(x) = k-"(21r)-"
Je1201 A(x/k, S) (( Jeix.si A(x/k, k5 +
(21r)-"
d4
is a Since A is a (-1) classic symbol we have A = X aA + A where cut-off function and A _, a S - '. Then using the homogeneity of nA we obtain by dominated convergence that, for each x e R",
d = nA(0, 0)0(x)
0k(x) -- O(x) 3= (2x)-" JeiM nA(0,
We have shown that 0k 0 pointwise and it remains to show that the convergence holds in L2. We claim that x"Ok(x) is bounded on R'. An integration by parts gives us x°ok(x) =
(2n)-"
Je'.e1 D°[A(x/k, k +
hence by the Leibniz rule Ixa Ok(x)I is bounded by a sum of terms of the form
C.' J 14 +
d4,
y + P = «,
which are bounded by a constant independent of k. Thus, IJfk(x)I C"(l + Ixl)-'"j1)'2 e L2 for some constant C, so the dominated convergence theorem gives us j 144k - 012 dx -' 0.
268
Boundary value problems for elliptic systems
Remark Let OP"(R') denote the set of linear maps A: .9' -4.V of order m. i.e. which satisfy IJAull,
VIE R.
< C,Ilull,,
Theorem 7.28 shows that OS° a OP'. On the other hand, it follows from Lemma 7.32 that if A e OCI-'S' and aA(xo, o) # 0 for some xo, so then
A$ OP' for any ,n'<m. 7.7 Elliptic operators on R'
In this section we prove the existence of a parametrix for an elliptic pseudo-differential operator. We show also that ellipticity is a necessary condition for existence of a parametrix.
Let A = A(x. D) a OS'(R'). A parametrix for
A
is
an operator
B e OS-'"(R"), if it exists, such that (i)
A(x,D)B(x,D)-IEOS
(ii)
B(x, D)A(x, D) - I e OS-
(65)
It is important to note that existence of a (two-sided) parametrix follows from the existence of a right parametrix and of a left parametrix as the following lemma implies. Lemma 7.33 If for some B', B" e OS
we have AN - I e OS-' and
B"A - I E OS-" then B' - B" e OS' ". Proof This follows at once from the equation
B"-B'=B"(1 - AB') + (B"A - I)B' by virtue of Theorem 7.12.
Thus B' is not only a right parametrix but also a left parametrix since
B'A - I = (B'-B")A+(B"A-I)EOSThe same is true of B".
Remark The lemma also proves the uniqueness of a parametrix, modulo OS
Proposition 7.34 Every operator of the form A(x, D) = I - T, where T E OS has a parametrix of the same form
B(x, D) = I - T', Proof We denote T' = T .
T' a OS
- T and define B by the asymptotic series
B-I+T+T2+
(66)
Pseudo-differential operators on if8"
269
By Theorem 7.12 we have TkEOS-k,
k = I,2....,
so B is well defined (see Proposition 7.7') and clearly (66) is satisfied. Setting (67)
E OS-'-', we obtain
where
(I - T) - B - I =(I U-T)_I-,v+,-TN+t
Thus (I - T)- B - I e OS-N-', for N = 1, 2.... , which means that
(I-T)=B-IeOS-`. Similarly we get B = (I - T) - I E OS- '. hence B is a parametrix for I - T. We now consider p.d.o.'s of arbitrary degree nt.
Theorem 7.35 An operator A(x, D) a OS"(I8") has a parametrix if and only if fir some positive constants c and R IA(-x, s)I > c151m
if I I > R
(68)
Proof If A has a parametrix Be OS-" then A(x, D)B(x, D) - I E OS-' and the asymptotic expansion (33) implies that
A(x. )B(x,s)-IES '. Hence IA(x,
11 < i for ISI large, so that
for large ICI, < IA(x, )B(x. s)I < C'IA(x, s)IIsP which proves (68). (The second inequality holds because B(x. ) e S-".) Conversely suppose that (68) holds. Assume first that m = 0. If we choose F E C'°(C) so that F(z) = I/ Z when I:1 > c, then B = F(A) a S° (see Proposition 7.25) and due to (68) with m = 0 we have
A(x, ) B(x, S) = I
when 161 > R
(69)
For arbitrary m we can introduce A(x, y)-(l + Is12)" and B(x. f)
(I + II;I2)"i2 in place of A and B, so that (69) again holds with BeS-'". Therefore,
A(x, D)B(x, D) = I - T(x, D),
T e OS-'
(70)
and we have constructed an operator which inverts A(x, D) modulo OS-'. But our aim is to prove the existence of B(x, D) e OS-" which inverts A(x, D)
modulo OS- ', i.e. such that (65) holds. By Proposition 7.34, 1 - T(x, D) has a parametrix Q(x. D) E OS° and by Theorem 7.12 we can write B(x, D)Q(x, D) = B'(x. D),
where B' e S.
270
Boundary value problems for elliptic systems
Since (I - T(x, D)) Q(x, D) a OS - ', it follows from (70) that B'(x, D) is a right parametrix of A(x, D), that is. 65(i) holds with B replaced by B'. In the same way one can find B" e S`' such that (ii) is fulfilled with B replaced by B". The proof is completed by Lemma 7.33. Definition 7.36 An operator A e OSM(R") is said to be elliptic if there exists B e OS-'"(R") which inverts A modulo OS - ', i.e. such that
A(x,D)B(x.D)-IEOS-' and B(x,D)A(x,D)-IeOS-'. We have shown in Theorem 7.35 that ellipticity is equivalent to (68), and therefore equivalent to existence of a parametrix. If A is (-1) classical, A E OCl-'S'", then (68) is of course equivalent to ellipticity of the principal part nA,
5 #0,xE18".
In Chapter 14 we will need a version of Theorem 7.35 in which the condition (68) holds only for x in some open subset of It".
Proposition 7.37 Let A(x, D) a OSm(R"). Suppose that for some open set U c 18" we have
fI >R,xEU Let W be an open set with W c U. Then there exists B(x. D) E OS-(18") such that, for all cp e C0 '(W), we have (71)
and
Proof By the same method as in the proof of Theorem 7.35 there exists B e S-' such that A(x. S)-B(x, 4) = I
when ICI > R, x e U.
(72)
By choosing p e Co (U) such that p = I on W, we then have
D)B(x, D) - 1) = T(x, D),
Te OS- '(RR),
and in view of the fact that p y, = rp, we obtain rp A(x, D) B(x, D) = N (1 + 7(.r, D))
for all c eCO(W). We can now complete the proof as in Theorem 7.35. Let Q E OS° be a parametrix of I + T, and set B'(x, D) = B(x, D)Q(x, D) a OS'". It follows that
D)B'(x. D) - 1) a OS-'
(73)
for all cp e CO '(W). Similarly there exists B" E OS-' such that
(B"(x,D)A(x,D) -
(74)
Pseudo-differential operators on R"
271
for all q e Co (W). Then as in Lemma 7.33 we have
cp(B"-B')i/i=mB"(1 -AB')-0+q
(B"A-1)B'4VeOS_r
(75)
for all cp, 0 E Co (W), since (I - AB')- aji and p. (B"A - 1) belong to OSby virtue of the lemma below. Now choose ty a Co (W) such that i/i = I on since this expression differs supp cp. It follows that (B" 4A - 1)cp e OS
from (74) by the term B",(1 - t/i)Arp which belongs to OS-" by quasiI on supp ii, then by locality of A. If we choose e C0 '(W) such that quasi-locality of B" we have (4B"i(iA - 1)cp a OS
(76)
and then (75) implies
(CB'r/iA-I)(pEOS Reversing the steps which led from (74) to (76) (this time with B' instead of B") we also find that
(B'(x,D)A(x,D)-1)cge0S-'
(77)
Thus, (73) and (77) demonstrate the existence of an operator B satisfying (71). Lemma Let T E OS "'(R"), and let U be an open set in Ii". If 0 - T E OS
all i/ic C(U) then also Ti/i EOS- x for all 0e
for
(U).
Proof Choose cy' E Co (U) such that I on the support of c/i. Then the supports of 1 - t/i' and of c/i are disjoint, hence the lemma follows from the equation
TO=ty'TO+(1 -c/i')TOi since (1 - i/i') Ti/i E OS- s by quasi-locality and cli' T E OS-' by hypothesis. Remark This lemma implies that we may add the following to the conclusion (71) in Proposition 7.37:
and
(7l')
7.8 Girding's inequality and some results on the relation between the operator norm of a p.d.o. and the norm of its symbol
Girding's inequality for strongly elliptic p.d.o.'s is one of the fundamental results of the p.d.o. theory. It was first proved by Girding for differential
operators and by Calderon and Zygmund [CZ] for singular integral operators. The proofs in this section are based on [KN]. Theorem 7.38 Let A(x, D) E OS°(IF") and suppose that Re A(x, S) 3 0.
(78)
Boundary value problems for elliptic systems
272
Then for any e > 0 there is a constant C = C(E) such that
Re(A(x,D)u.u)+Ellukio> -Cllull?,
(79)
2
for all uECoo .
Proof Since Re A(x, 5) > 0 the symbol B(x, S) :_ [Re A(x. ) + c]"2 belongs to S° by Proposition 7.25. Then by Corollary 7.11 and Theorem 7.12 we get
D)) + e - B*(x. D) - B(x, D) = T(x, D)
2(A(x, D) +
where 71x, D) a OS-'. Thus, Re(Au, u) + E(u, u) _ (Bu, Bu) + Re(Tu, u), so we have Re(Au, u) + E(u, u) > Re(Tu, u). Now, T is an operator of order -1. so I(Tu, u)I < 11 Tull, 2- llull- 112 -< const lull _, 211u11 _, 2 and (79) follows.
Remark There is a sharp form of Girding's inequality which holds with c = 0 and C some constant. For various improvements and for best constants
see [Eg], [Ho 3] and [Tay]. We showed in Theorem 7.27 that a p.d.o. A(x, D) E OS° defines a bounded
operator on L2(R"). For some purposes it is important to have a sharper form which gives some information about its L' norm. The following theorem can be viewed as a corollary to Girding's inequality. Theorem 739 Let A(x, l;) a S° and let
lim sup IA(x. S)I x
Then for every c > 0 there is a constant C = C(c) such that
D)ullo < ('r + e)Ilullo + Cull -, 2
(80)
for all ueCO. Proof Since A e S°. then ,, is finite, and by definition of urn for every E > 0 there exists R > 0 such that sup IA(x, S)l
+ IE
for 15I > R.
x
We take a patch function cp e Q (R"), 0 0
for all .
(see the proof of Theorem 7.39). The new operator differs from the original
one by an operator of order -1, which does not affect the form of (82) as we saw with the operator T in the proof of Theorem 7.38.
Remark When A is a (-1) classical operator the condition (81) is also necessary for G$rding's inequality. We can assume more that m = 0 and c' = 0, then let ¢k be the sequence from Lemma 7.32. By substituting u = 0k in (79) and letting k - oo we find that Re nA(x°, y0) + E 3 0. Since e > 0 is arbitrary, then Re trA(x, 0 for all x, 4, and hence Re A(x, ) ? 0 for large The next theorem is the analogue of Theorem 7.39 for pseudo-differential operators of arbitrary order. Theorem 7.39' Let A(x, 5) E S'" and set
y = lim sup 14,_.
_
IA(x, is)l
(1 + I
I2)m,2
Then for every e > 0 and 1 e R there is a constant C = C(c. 1) such that IIA(x, D)ulll-m < (y + 01ull, + Cliul1,-112
(80')
for all ueCo. Proof The case m = 0 follows from Theorem 7.39 since we can replace u by
A1u in the estimate (80') to reduce it to (80). Now, if A(x, D) E OS' we consider the operator A-` = A(x, D) a OS°, and by using the case just proved
Boundary value problems for elliptic systems
274
we obtain
IIA-"oA(x,D)ullr < (y +e)Ilulll + CIlullr-1/2'
which proves (80'). Note: The definition of y is not affected by this substitution since the asymptotic expansion for the symbol of A-'° o A(x, D)
is equal to (I + I2;12)`12A(x, 4) plus a term of order -1 which does not affect the limit.
The inequality in Theorem 7.39' suggests the following question: how is the norm of A, as a map from W'2 to W2-', related to the symbol? It would seem to be difficult to deal with this question in general. The next theorem, however, does give a positive result in this direction which has important consequences (see §8.10).
In the proof we use the fact that
e-'t 't A(x, D)(u
A(x, D + ts)u,
(83)
for all u e Co , which follows from (31). The theorem is formulated merely for the case I = m = 0. Theorem 7.40 If A E OCI - 'S° is a (-1) classical symbol of order 0, then as a map of L2 into L2 it satisfies inf 11A + TII = sup InA(x, i;)I X.t
T
where II ll denotes the norm of the operator as a map of L2 into L2, and the
infimum is taken over all operators TE OS-'. Here nA is the principal part of A; see §7.5.
Proof It follows by homogeneity of nA that the number y in the statement of Theorem 7.39 is equal to sup,,,, InA(x, S)I. Thus, we must show that
inf!IA+TII=)' T
We start by showing IIA + TII >, y for any operator T e OS-'; since A + T has the same principal part as A we can assume that T = 0. We claim that for all u, v e C0 '(R') and fixed S0 a I8"\0 (A(x, D)(u
as t
eW.'em), v
(nA(', G0)u, v)L2
(84)
oc; once this has been proved it follows that v)I 0, such that IA(x, e) - A(x0, s)!
(I +
\
e
2
for x e B(xo, 6) and all s e R"
(86)
Boundary value problems for elliptic systems
276
Now, we take an open set U a x0 with closure lying inside B(xo, b) and a function 0o e Co (B(xo, b)) with 0 < 0o 0.
Lemma 7.41' Let H(x, ) a J'"`, m > 0. Then for every x0 and e > 0 we can find a_ neighbourhood U of x0 and operators K(x. D) e OS' and T(x, D) e OS
such that for all cp E C0 '(U) we have the decomposition
p(x) H(x, D) = ,p(x) [H(xo. D) + K(x, D) + T(x. D)] with K small: IIKull1-," < ellull, for all u E W2(R").
277
Pseudo-differential operators on I8"
Proof With Corollary 7.30 the proof is very simple. We take a cut-off function X(s) and apply Lemma 7.41 to
p(x) H(x, D) = tp(x)[yH + T, ] = rp(x)[(XH)(xa, D) + T2 + K + T,] = rp(x)[H(xo, D) + T3 + TT + K + T,], where T,, T3 E S - x by Remark 7.31 and T2 E S- T. Clearly we may also take T_ e S' - ' ,
1.
i.e. independent of x for large x. Then T, + T2 + T3 = TES - t.
Exercises The convolution of a pair of functions gyp, ifr on R" is the function ((, * 41)(x) = I
(D(x - v)'(y) dy
L
Show that if rp, Vi a ,' then rp * 4y E .91 and 0 Let f be measurable on R" and I f(x)I _< C(I + IxI)_" and m is a non-negative integer. Show that the Fourier transform Rf has bounded and continuous derivatives up to order <m. 5. The convolution f * (p off e .9' and 4p e Y is defined by 4.
(f * q)(x) = < f(y). Q(x - y)>, where < . > is the duality bracket between .9' and Y. Show that the function
f * 4p is infinitely differentiable and ID'(f * cp)(x)I < C,(1 + Ixi)" for some number N independent of x. 6.
If f e .f ' and 4p e ,9', show that j(f * (p) _ jf jrp.
Let a(5) e S" (independent of x). The operator a(D) a OS' is then defined by a(D)u in particular if m < -n we have 7.
(a(D)u)(x) =
I
h(x - y)u(y) dy.
u e .9
J R"
where the function b = y -'(a(S)) is bounded and continuous by Exercise 3. Differentiability properties of h follow from Exercise 4, for instance, if
ac-S-' show that bECb (R"). Let a(s) e S' where n: is arbitrary. Show that there exists a number I and a bounded. continuous function b such that 8.
b(x - y)( - Ar + I )'u(y')
(a(D)u)(x) = JR "
d y,
u e .f
Boundary value problems for elliptic systems
278
Let K(x, s)e J 'and let jlcl= r K(x, ) dct = 0 (do, denotes the measure on the sphere ISI = 1). Show that for u e 9 the following limit exists 9.
(Au)(x):= lim I K(x, x - y)u(y) dy, c-0 Is-yl>c and defines a p.d.o. of order 0, that is. A e OS°. Hint: Consult the book [Es]. Appendix: summary of definitions and theorems for Sobolev spaces
In this appendix we state the main definitions and theorems of the theory of Sobolev spaces. For the proofs one should consult the books [WI], [Tri], [Ne], [Ad]. We restrict ourselves to the spaces W 2'(S2) where I E R+ (I 3 0) in which the norms are based on L2 norms of the derivatives. Definition 7A.I Let 0 be an open set in R". Suppose first that I is integral, that is, I = 0, 1, 2..... The Sobolev space W2(!) is defined as the set of all functions u e L2(f2) for which the distributional derivatives D'u (or weak derivatives) belong to L2(S2) for Isl 2.) Hence I0g')I2
If we multiply by (1 +
C2
JIQ()I2(l +
z ds" + I;"Iz), /-1
IS,12)'-"2 and integrate, it follows that
I10,-1,2 0, and w = v, for x" < 0, is an element of WZ''(R"). Proof By virtue of a density argument, see Theorem 7A.2, we may suppose
first that u', v' E C0 '(R') and define w' in the same way as w. For any cp e C0 '(R") we have for Ixl < m + I J
dx = (- 01_'lJ
cpD'u' dx + Jx,, 0 there exists a constant c(e) such that for all Ilulli2'< Ellull,, + c(E)Ilullo
Further, (.s) is also true for all u e W2 (f2) if f2 is bounded with smooth boundary, or all u e WZ(M) if M is a compact manifold without boundary. Ehrling's Lemma is also true for the Sobolev spaces on R', as one sees by Fourier transformation.
8
Pseudo-differential operators on a compact manifold
In this chapter we define the class OS-(M) of pseudo-differential operators of order m on a compact manifold M. Essentially, they are linear operators on C(M) which are p.d.o.'s in local coordinates and satisfy a quasi-locality property. The symbol of an operator A e OS'(W) in Euclidean space is uniquely determined by the formula (31) of Chapter 7, that is, A(x,5)=e-i
4 A(x, U)eOx..)
For an operator A e OSm(M) there still exists a symbol - defined locally but it is not unique due to the effect of coordinate transformations. It is possible, however, to define a symbol modulo lower-order terms, which we call a main symbol. The main symbol isomorphism
OS"(M)/OS"(M)
S'"(T*(M))/S'"-'(Ts(M))
is also an algebra isomorphism. It should be noted that other books use the term "principal symbol' where we use "main symbol". We reserve the former term for the special case of classic operators defined in §8.4 which are (-1) classic operators in local coordinates (see §7.5). Such operators A have a
uniquely determined homogeneous principal part nA which is therefore well-defined on the cotangent bundle, and nA e C x(T'(M) ,0). This chapter comprises two parts. In the first part, §§8.1 to 8.6, the p.d.o.
algebra is developed, and in addition to the scalar p.d.o.'s which were indicated above, we also consider p.d.o.'s acting on sections of vector bundles.
In the second part of the chapter. §§8.7 to 8.10, the Fredholm theory of elliptic p.d.o.'s in vector bundles is developed, including the existence of a parametrix. For classic p.d.o.'s, ellipticity means invertibility of the values of the principal symbol. The index of an elliptic symbol is defined and we prove the invariance of the index under homotopies of various kinds, both for general
p.d.o.'s and for classic p.d.o.'s that have a homogeneous principal part. It is beyond the scope of this book to discuss the Atiyah-Singer formula
for the index of elliptic p.d.o.'s (or elliptic symbols) but we develop, essentially, all the analytic properties which are required for its proof. As an illustration, we prove a special case, namely, Noether's formula for the index of elliptic p.d.o.'s of order zero on the unit circle. 287
288
Boundary value problems for elliptic systems
8.1 Background and notation Let M be a compact C' manifold without boundary. A Riemannian metric on the tangent bundle ITM) gives rise to a density dp on M as discussed in §6.6.
If (U, ,) is a chart on M then, locally, this density is given by
du = g,,(x) dx, where g = det[gij] is the Riemannian volume density. We use this density du to define the Hilbert space of square-integrable functions on Af. For functions u, r e C'(M) the scalar product is defined by ('
(u, OM :=
ui du,
I
M
and then L2(M) = L2(M, C) is the completion of C'(M) with respect to this scalar product. If X is an open set in M, we let C0 '(X) = C0 '(X, C) denote the space of
C' functions on X with compact support. Since M is compact then CO '(M) = C'(M). We need also the Sobolev spaces. They are defined here for complex-valued functions and then later on in §8.5 for sections of a vector bundle.
Definition 8.1 Let M be a compact C" manifold. Let I > 0, a non-negative real number. The Soboler space W2(M) is the set of functions u: M -' C such that ((P-u),K-'
E W2(R") V and for every cp e C'o (U). The topology for every coordinate map h: U on W2(M) is defined by the semi-norms
u" II((o'u) c K_' ll,, i.e. the weakest topology for which the semi-norms are continuous.
Let',Uj, nj}j=,...x be an atlas on Af and ;Vj) a subordinate partition of unity. For u e W2(M) we let uj==(q .u), 0. For k > 13 0
Pseudo-differential operators on a compact manifold
289
we therefore have a dense inclusion Wz(M) c W2(M), and by transposing c WZk(M) for -k 0 we consider WZ(M) as embedded in W2(M) by the inclusions WI (M) c W°(M) c W2(M). In this way we regard Wz(M) as embedded in WZ(M) for all k < 1, and we say that 1W2(M)},En forms a scale of Hilbert spaces (see [Pa] for further details). Recall that C"(M) is a Frechet space where convergence of a sequence {u1} is defined to be uniform convergence of all derivatives of u, on compact subsets of chart domains. By Sobolev's Lemma we have
C"(M) = n W2(M). I
with the inverse limit topology, i.e. a sequence (u;} c C-(M) converges to u e C° (M) if and only if it converges in W2(M) for all I E R. The scalar product (u, v)M defined for u, v in L2(M) extends to a pairing between W2(M) and W,-(M) for all 1: if I >, 0 and u is (an anti-linear functional) in WZ'(M) then (u, v)M =
for all v e W2(M );
and if u is in W2(M), I _> 0, then (u, r)M =
for all v e WZ'(M ).
Definition 8,2 A (continuous) linear mapping B: C'(M)
C'(M) is an
operator of order m if for each I E R it extends to a continuous mapping
B: W2(M) - W2-'(M)
(1)
The set of all such operators is denoted by OPm(M). This is a local space of operators, that is, B e OPm(M) if and only if rpB(+y ) E OPm(M) for all functions cp, 0 a C=(M).
If B: C;'(M) -+ C'°(M) is a linear mapping then its formal adjoint B* is defined by means of the scalar product (, )M (Bu, v)M = (u, B*v)M,
U, V E C2(M)
(2)
Note that the left-hand side of (2) is not necessarily a bounded operator of u e L20 f), so the existence of B*v is not assured. However, if B C- OPm(M )
W2-'"(M) denote the maps of Definition 8.2 (1) which extend the operator B, and then the existence of the adjoint B* is proved by we let B1: W2(M)
considering the duality adjoints B!: W2-'(M) -. WZ'(M) of the operators B,. The formal adjoint B* is the restriction of B' to C°°(M). We have B*: C" (M) and B* a OP'(M).
290
Boundary value problems for elliptic systems
The set of operators of order - x. is denoted
op-(M) = n OP"(M), 111ER
that is, B belongs to OP-'(M) if it extends to a continuous map B: W21(M) - W22(M) for all 1 12 a R.
Proposition 83 Let M be a
compact, closed manifold. An operator B: C"(M) -. C'(M) belongs to OP-°°(M) if and only if it is an integral
operator with a C' kernel, that is,
K E Ca(M x M).
(Bf )(x) = J f K(x, y)f(y) dp(y),
(3)
N
Sketch of Proof Let 2'(M) (resp. e'(M)) be the space of all distributions (resp. with compact support) on M. Since M is compact then ([Tri])
r(M) = 2'(M) = U WZ(M), r
with the inductive limit topology. i.e. a sequence [uj} c e'(M) converges if
and only if {uj} c W2(M) converges for some le R. It follows that if B e OP- '(M) then it is also continuous as an operator B: e'(M) - C'(M). Let {(Uj, nj)} be an atlas on M and {rfij} a subordinate partition of unity. Then we can write
tBgjf,
(4)
Bf r. i is continuous from B'(U,) to C'(U;). The points and the operator in Uj and in Ur can be identified with their coordinates in R' (by the maps fi j and hr, respectively), and then by L. Schwartz's "Theoreme de noyaux" is an integral operator with a C' kernel (see [Ho 1] or [Tri]), Krj e C"(UI x Uj), i.e. (*iBi/ij f Xx) = Jv, Krj(x, y)f(y) dy. Now, if we set c&r(x)-Krj(x, y)'cj(y)[gv,(y)]
K(x, y) _
'
1,J
the representation (3) follows.
Conversely, if B has a C' kernel K, we can localize as in (4), and it follows upon differentiating the kernel of IIIBtIij that B acts continuously B: W2" (M) - W22(M) for all l,, 12 e R which means that Be OP- "(M). We write B e OC'(M) if it is an integral operator with C' kernel K(x, .v). i.e.
(Bf)(x) = J I K(x, y)f(y) dy(y).
K e C'(M x M);
M
thus Proposition 8.3 implies that
OP-'(M) = OC'(M).
Pseudo-differential operators on a compact manifold
291
Now, we define the support of an operator and then the transfer (or push-forward) of an operator under a coordinate map.
Definition 8.4 If B: C'°(M) - C'(M) is a linear map we define the support of B to be the complement of the largest open set C c M such that
(1) &p(x)=OifxaC, (2) B(p = O if supp cp c C
(We also make the same definition for a linear map B: Co (R")
C'(R").)
If q e Cs'(M) has its support disjoint from the support of B then clearly
rpB=Bcp=0. If tp e C '(M) is identically I on a neighbourhood of supp B then (I - (p)B = 0, so B = and - ao as d - oc and let A c OS""(M ). We will write
A^-
Ad d=1
Boundary value problems for elliptic systems
298
if for any integer r > 2 we have r- 1
A - E Ad a OS""(M ) d-t
(13)
Proposition 8.16 For any given sequence Ad a OS"°(M ), d = 1, 2.... with and -1 > and -+ - co as d --* oo, there exists A E OS"' (M) such that A - Y The p.d.o. A is uniquely determined modulo OS-'°(M).
I Ad.
Proof The uniqueness follows immediately from (13). To prove existence let 91' = {(W, K,)} be the atlas on M that was chosen just before Theorem 8.9',
and let {co;} be a subordinate partition of unity. Also choose functions li; e Co (Uj) such that 0, = I on supp cpj. The condition (i') in Theorem 8.10 for each operator Ad E OS"°(M) gives us the following: For every j there are operators Ad,; E OS"(R") such that for all q,, 0 e CO'( W j).
q (Ad. j(x, D))"i
By Proposition A'j - Y'= It Ad j e
(14)
p.d.o. A, E OS"'(R") such that r= 2, 3, ... , and it follows from (14) that
7.7, there exists a
\
.-t
(p;(A'j)Ki 'O'; -
(pi
Ad I rj e OS"'(M).
(15)
d
Now, define J
and note that A e OS"'(M) by Lemma 8.11 (see Remark 8.12). Also, ip;(Yda, Ad)(I - 0,) c- OS (M) because p.d.o.'s are quasi-local. Summing over j, using the fact that jj rp; = 1, we see that (15) implies that r-1 r-1 .-t A, = (pj(Z Ad !(l - tfi;) Adal
d=I
j
d=t
/
belongs to OS"r(M) for r = 2, 3.... . 8.3 Main symbols and p,d.o. algebra First we discuss symbols on open sets in R". If X c R" is open we define
x R") as the set of all a e C"(X x R") such that cp(x) a(x, S) e S"(R" x R") for every cp e C0 '(X). This means that for every compact set K c X we have C2
(1 +
x e K. C; E R"
for some constant CQaK. Note that if X = R" then S" c S,a but equality does not hold. We can define an operator a(x, D):.Y' - C t (X) by the formula, a(x D)u = On)` J e'i=.et a(x, (16) )d(s)
Pseudo-differential operators on a compact manifold
299
In fact, if cp e Co (X) then q,(x)a(x, s) E S"' and the corresponding operator cp(x) a(x, D):.9' -+ .9° has already been discussed in Chapter 7. Restricting
a(x, D) to Ca (X) e Y we obtain an operator a(x, D): C0 '(X) - CZ(X) defined by (16). The following lemma will be needed shortly. It says that symbols behave well under a change of variables.
Lemma 8.17 Let X, and X2 be open sets in R" and let 0: X, -+ X2 and 0: X, -+ GL(n, R) be C' maps. Then al(x, S) = a2(4,(x), 1(x) ) is in S;C(X, x R") for every a2 e S,«(X2 x R").
Proof The functions 0 and 0 and all their derivatives are bounded on compact subsets K c X, and cI I < when x e K (where c > 0 and C > 0 depend on K). The lemma now follows easily by repeated use of the chain rule.
For a pseudo-differential operator in OS'(R"), it is clear from (31) in Chapter 7 that the symbol in S' is uniquely determined by the operator. But on open sets X in R" the symbol is only determined modulo S,«'°(X x R"). This is the subject of the next proposition which we take from [Ho 3].
Proposition 8.18 Let X c R" be an open set. If A: C0 '(X) -+ C°"(X) is a linear map such that for all tp, 0 a C0 '(X) the operator .9' - ,9' given by ut--+cpAtiu
is in OS', then one can find a e S;«(X x R") such that
A=a(x,D)+Aa where the kernel of AO is in C'°(X x X), and a is uniquely determined modulo
S1; (X x R"). Remark We call the equivalence class [a] e of the operator A.
the complete symbol
Proof Let Y j = 1, Oj E C(X ), be a locally finite partition of unity in X. Then 4, Ati/ u = Ajk(x, D)u, u e J, where Ajk e S' and Ajk(x, t;) = 0 when x 0 supp 4,. Set
a(x, ) = Y'Ajk(x, ) where we sum over all j and k for which supp Oj n supp 4,k # 0. The sum is locally finite since any compact subset of X meets only finitely many supp t/ij and they in turn meet only finitely many supp 4,,,. Hence
Boundary value problems.for elliptic systems
300
a e S' ,(X x R"). If K(x, y) is the Schwartz kernel of A then the kernel of .4 - a(x, D) is the sum, Y",;(x)K(x, y)O,,(y), taken over the indices for which supp qi; n supp fir = 0. It is in C"(X x X) since the sum is locally finite and the terms are in CZ by Corollary 7.14. To prove the uniqueness, we must show that if a r= S;C(X x R") and the kernel of a(x. D) is in CZ(X x X) then a E Si c'(X x R"). Taking cp, Vi E C(X) with o = ip = 1 near any given point in X. the operator B(x, D)u = ipa(x, D)(Jiu belongs to O$'" and its Schwartz kernel is in C0 '(11" x R"). Hence the symbol B(x, S) = e-i(x.4) B(x D) e,(x.,)
is rapidly decreasing when r; -+ x (see the proof of Theorem 7.6), so in fact BB(x, D) e OSChoosing >V equal to 1 in a neighbourhood of supp ip, it follows that <pa(x, D) = 4pa(x, D)a(i + rpa(x, D)(1
is in OS
by virtue of Corollary 7.14.
We now define the symbol space S,a(T*(X )) when X is an open set in the manifold M. Let (U, K) be a chart on M with U c X. For each x e U, consider the differential dK(X): TA(M) - R" (see §5.3) and the transpose map `dK(x): R"
T*(M).
that T*(U) = K(U) x R". The push-forward a e C°"(T*(X)) is the function in C°`(T*(U)) defined by Recall
a(K-'(y), `dK(K-'(y)) n),
of a
y E K(U), n E R".
More formally, we define the map t(-, K): CZ(T*(X))
function (17)
Cs(T*(U)) by
t(a, K)(y, tl) =
Then t(a, K) E CZ(T*(U)) is called the transfer or push forward of a function a e CZ(T*(X)) by the coordinate map K. Definition &19 Let X be an open set in M. We define S" (T*(X)) to be the
set of all a e Cz(T*(X)) such that the pushforward function (17) belongs to S
that is,
t(a, K) E Sl«(K(U) x R"),
for every chart (U, K) on X. If X = M, we write STM(T*(M)) instead of
S,(T*(M)). By virtue of Lemma 8.17, it is enough to require that this condition hold
for all charts (U, K) in an atlas on X. In particular, the definition of So(T*(X)) agrees with the earlier definition when X is an open set in R", on which there is the atlas consisting of the single chart (X, identity). If A E OSTM(M) then the push-forward operator A": Co (K(U))
C D(K(U ))
in R" satisfies the conditions of Proposition 8.18 with X = K(U) (see (8)
Pseudo-differential operators on a compact manifold
301
and Definition 8.7), so it defines a complete symbol in
Letting
a"ESI'(ic(U) x R") denote a representative (see (20) below), and then pulling it back to U, at(x. b) = a"(h(x), dK(.Y)-' S),
x c: U,
e TsM
(18)
we get a function a" E S,'(T*(U)) which is called a complete symbol of A on the coordinate chart (U, K), and it is unique modulo S,« (T*(U)). It is important to take note of an inconsistency in our use of notation: A. is the push-forward to R" of the operator A, but a" = t(a,,, K - ') is the pull-back to U c M of the complete symbol a" of A", and therefore involves a "double transfer". Lemma 8.20 Let A e OS'"(M). Let (U, K) and (U', K') he charts on M and let a e S' (T*(U)) and a., e S'(T*(U')) be the corresponding complete symbols
of A on each chart, as defined above. If U n U' # 0 then a" - a". E S o. '(T*(U n U')).
(19)
Proof By definition, we have
a
A,; =
D) +
the kernel of
A0
is in
C z(K'(U') x K'(U')). Let ip E Cp (K(U n U')) and choose t, E Co (K(U n U'))
such that aG = I on supp Co. The operator cpA,,(y, D)u a OS'
has a kernel which is compactly supported in K(U n U') X K(U n U'). By virtue of Theorem 7.17. with the coordinate transformation co = K' K-', it follows that (ibA,, ) a OS"' and (SPA"W )(w
n) - (NA" ), (}'. n) E S"'-'.
for all)' E K'(U n U'), n e R". Then,by Corollary 7.13 (i.e. by commuting iGA,
and 0) and the fact that (OA"i ) = cp o
w-' and using (20),
we obtain
N(co '(Y))'a"(w '(}'), `duHw '(}'))'n) - cD(w 'U'))'a. ly, n) e S"Substituting K'-'(y) = x.'dK'(x)'n = S and cp = ip = K, and then using Lemma 8.17. we find that (P( x)'(a,(K(x).'dK(x)-'s) -a., (K'(X),'dK'(X)_'s))E
S'"-
xEUnU',SETzM (21) for every (p a Co (U n U'). By definition, a. and a,,. are the pull-backs of a. and a,,. by the maps K and K', respectively, and therefore (19) is a consequence of (21).
Our aim now is to define the symbol of a p.d.o. on a manifold, and to investigate its properties. Lemma 8.20 indicates that it is possible to define such symbols only modulo S,'-'(T*(M)).
Boundary value problems for elliptic systems
302
Definition 8.21 Let A E OSm(M). Then a E Sm(T*(M)) is said to be a main symbol of A if
a - .tK E S'j '(T*(U))
for all charts (U. K) on M.
(22)
where a is the complete symbol of A on the chart K (see (18)). Note: By Definition 8.19, the condition (22) just means that the push forward of rp (a - a,,) belongs to S' ' for every cp c- Co (U ), that is,
w(y)'(a(K-'(y),'dK(K-'(y))-q) - aK(y, q)) E S"_'(R" X R") for every (p E Cp (K(U)).
When checking that a is a main symbol of A, it suffices to verify (22) for all charts in an atlas on M. Indeed, let {(pj) be a partition of unity subordinate to a given atlas 91 = {(Uj, Kj)), i.e. supp cpj c U,, and let (U. K) be any chart for M. Then, for every rp c- Co (U), the push-forward of
(P(a - aK) _ Y_ tP(.t(a - a,) + I "i(aK - a.) by the map K is an element of Sm'-'((R" x R"). The terms in the first sum are in S'` by hypothesis, while the terms in the second sum are in S'"-' by Lemma 8.20. It remains to be shown in Theorem 8.23 that every p.d.o. A E OSm(M) has a main symbol. But three facts are clear at this point: (1) If a and a' are both
main symbols of A then a - a' e S'-'(T*(M)): (2) If A E OS'"-'(M). then the 0 function is a main symbol of A (when it is regarded as an operator in
OS'"(M)). and (3) If A has main symbol (., then any other function a' c CI(T*(M)) such that a'(x, 4) = a(x, S) when (x, 1;) E T*(M)\,K, K a compact set in T*(M), is also a main symbol for A. Lemma 8.22 Let (U, K) be a chart on M, and let A(y, D) E OSm(I2") have compact support in K(U). Also, let 2(y, q) be the symbol of A. Then the pull-back operator A = T(A, K-') = (A(y, D)),, E OSM(M) (see Lemma 8.11) has a main symbol a e C'(T*(M)) given by
a(x, ) =
if X E U,
a(x, S) = 0,
if X E M \U, (23)
i.e. the main symbol of A is the pull-back, t(A, K-'), of the symbol 2 to T*(U), extended by 0 off T*(U).
Proof Formula 7.31 for the symbol of a p.d.o. in R" shows that A(', n) = 0 when y E K(U) \supp A (see condition (1) of Definition 8.4). Hence a(x, S) = 0
when x e U \supp A,
(24)
and it extends by zero to a C' function on T*(M). As mentioned above, to prove that a is a main symbol of A it suffices to verify (22) of Definition 8.21 for all charts in an atlas on M. Let 21 = {(Uj, Kj))
be the atlas that was used in the proof of Lemma 8.11. For j
jo we can
Pseudo-differential operators on a compact manifold
303
=0 choose d", = 0 since A., = 0 (supp A does not meet U;); therefore on T*(UU). and by (24) we also have a = 0 on T*(UU). On the other hand, when j = j0, we choose a" = A and then it is evident by definition, see (18). that a = a" on T*(U). Hence a = a", for all j, and a satisfies (22) for all charts in this atlas. Suppose that we are given an operator A E OS'(M) with supp A c U. Then the push-forward operator T(A, K) = A: Co (R") -. C0 '(R") belongs to OSM(R"). In fact, this is immediate by Definition 8.7 if we observe that A = cpArp when cp c- Co (U) is equal to I on supp.4. Therefore, Lemma 8.22 can be phrased in a different form:
Lemma 8.22' Let (U. K) be a chart on M, and let A E OS'"(M) have support in U. A has a main symbol a e Cx(T*(M)) given by
a(X,) = A(K(X),'dK(X)-'S)
i f X E U,
a(X. S) = 0,
if x E W, U,
where A(}, n) is thesymbol of the push forward operator 71A, K) = A: C0 '(R") Co (R").
Lemmas 8.22 and 8.22' are the basis for computing the main symbol of a pseudo-differential operator which is defined via a partition of unity. Theorem 8.23 Every A E OSm(M) has a main symbol a c- Sm(T*(M)) (i.e. satisfying Definition 8.21). We have a main symbol isomorphism
,#: OS-(M)/OSm-'(M) - S'"(T*(M))/Swi-1(T*(M)) which is the linear map d e fi n e d by [A] :
[a]. We write also fr.4 = a, to avoid
being too meticulous.
Proof Choose an atlas '(Uj, Kj)}"_ 1 on M, and a subordinate partition of unity ;(pj}. Let yj E Co (Uj) be equal to I on supp rpj, then write A=
pjAyj +
gpA(1 - y,).
Because of quasi-locality the second sum is in OS-'(M) and thus has a main symbol of 0. By Lemma 8.22', the operator c Ayj, with support in Uj, has a main symbol aj a S°(T*(M)). Then it is easily verified that aj e S"(T*(M)) is a main symbol of Y pAyj, and hence it is a main
(5
symbol for A. The main symbol map [A] i-+ [a] is well-defined because a E S°'-'(T*(M)) when A E OS'"-'(M). To prove injectivity. we must show that if A E OSt(M )
and has a main symbol a e S'(T*(M)), then A e OS'- I(M). But from (20) we have for every gyp, yi e Co (U),
1,,++p"Ao/K / ,I, '"a"(r. D)7"
(25)
second term belongs to OS-"(R") while the first term belongs to OS'-'(R") if a E S'"-(T*(M)). Hence A E OS'"-'(M) by Definition 8.7.
Finally, to prove surjectivity let a e St(T*(M)). Then choose lJ/j e Co (Uj)
Boundary value problems for elliptic systems
304
such that Y o; = I and set A,u= fi;(aJ(x,U))K,-,IG,u,
uc-
C'(M),
where a, is the push-forward of a to x,(U,) x a", see (17). By Lemma 8.11 it follows that A, a OS'(M) and then Lemma 8.22 shows that A; has a main symbol O/ a, whence A = Y A; has a main symbol Y 0j 'a = a. Our next aim is to develop the p.d.o. algebra on M, i.e. the composition and adjoints of operators. If A E OSm(M), the adjoint operator A is taken with respect to some inner product (, )M on L2(M), see (2) in §8.1. The definition of As depends on the choice of inner product. but (b) of the following theorem shows that its main symbol does not depend on this choice.
Let ;L;, n,},-,. .,, be an atlas on M and ;cp,} a subordinate partition of unity. Also let dp = gr,(x) dx be the density on U,. We write down for later use the formulas
5 ut dp =
J
L
lp,(x)u(x)r(x)gi (x) dx,
(26)
and when u and r have their support in U,, uc dp = SM
u(x)v(x)yu,(x) dx,
(27)
G,
where dx is of course Lebesgue measure. (To simplify the notation we have identified points in U, with their coordinates in I".) Theorem 8.24
(a) If A e OS""(M) and B e OS"(M) then A B e OS'"I*m'(M ). Further. if a and f are main symbols of A and B, respectirely. then a( is a main symbol of AB:
, (A=B) = (b) If A e OSm(M) then the adjoint operator, A', is once more a p.d.o. belonging to OS'(M). If a is a main symbol of A, then .i is a main symbol of A*:
,*A = pA Proof (a) Let W. 0 e C, (U) where (U, K) is any coordinate chart on M, and choose X e CJ (U) equal to I in a neighbourhood of supp w. Then (pABi/i = ((pAX)(XBO) + (pA(l - X2)BI/.
The second term on the right-hand side has a C' kernel.
In fact
(l - y2)BO a OC'(M) = OP-'(M) since the supports of I - X2 and ip are disjoint. It follows that rpA(I - XZ)B'i a OP- `(M), so it has a C' kernel by Proposition 8.3, and therefore belongs to OS-'(M) by Proposition 8.15. As for the first term on the right-hand side, the push-forward operator ((PAX)" V,B')"
Pseudo-differential operators on a compact manifold
305
belongs to OS"I"(12") by Theorem 7.12. The condition of Definition 8.7 therefore holds with m = ml + m2, and we have proved that A o B E OS---2(M). of those of A Next we prove that the main symbol of AB is the and of B. It suffices to assume that A is the operator Aj with main symbol a that was constructed in the proof of Theorem 8.23, and B is an analogous operator B,, becauseA -YAjEOS"'-`(M)and B-YBJa and then
A -YA., )B+FAJ(B-YBk)eOS"'`"2-'(M).
AB Thus, we assume that A=
B=
ii(aj(x, D))",-,OJ.
Wk(bk(x, D))kk k.
where aJ e S" and i a S". Then we have AB = Y
tfil(ai(x, D))K, Y J +yk(bk(x. D))K Wk,
J. J;
and the (j. k) term in this sum has main symbol iG i/ii a e by Lemma 8.22. By linearity of the main symbol map, we obtain that AB has main symbol
(since=1).
!=a G
Now to the proof of (b). Note first that the adjoint A* exists and belongs to OP"(M); see §8.1. We will show that A* satisfies the conditions (i') and (ii) of Theorem 8.10. For condition (ii), observe that if B e with kernel K(x, y), then B* e OC"(M) with C" kernel K(y, x). Therefore, if cp and 0 have disjoint supports, it follows that pA*ty = (t/,Aip)* e OC'(M). To verify (i') we use the atlas {(W, Kj)) 1 introduced just before Theorem
8.10. Let Aj(x, D) a OS"(R"), j = 1, ... , N, be any p.d.o.'s such that the representation (pAO = tp(Aj(x, D)),,,-.0
(28)
holds for all tp, s e Co (W ). j = 1, ... , N. We must obtain a similar representation for A*. To simplify the notation, let us identify points in U, with their coordinates in KJ(U,) c R". Now, if we recall 13 c UJ, we obtain for 'p, li e C"(W) that 0u A(t/v)gw, dx
((pA*iku, v)M = (iku, Atpv)M =
(see (27))
J
+/ru-A,(x D ttgjdx H"
=
J
q,
'A;(x,D)(9JOu)iigJdx
306
Boundary value problems for elliptic systems
where gwf and (gw)-' are extended to functions gj and 4--' in Ce (R'), respectively, such that l on the support of the operator Aij(x, D) (see Definition 8.4 for the definition of support). Hence (pA*hG = (p(9; 'A;`(x, D)9j)"; ,i
(29)
for every (p, i' e Co (W). In other words, AS satisfies condition (i') of Theorem
8.9' with the role of the p.d.o.'s Aj being played by 4, ' Aj (x, D)9, a OS'(I8"),
j = 1, ... , N,
where we used Theorems 7.10 and 7.12. (Theorem 7.10 also shows us how to get A*(x,1;) from A j(x, c).) It remains to show that a is a main symbol of A*. Let {(pj} be a partition of unity subordinate to the cover { Wj}, and choose functions lij a Co(W ) which are equal to I on supp (pj. By Lemma 8.22, the operator (p1Ahrj has a main symbol aje S"(T*(M)) given by
aj(x, ) = (pj(x)Aj(Kj(x),'dKj(x)-'1;)1fj(x)
(=0 when x# W)
and by virtue of (29) and Theorem 7.10, and Lemma 8.22 once again, aj is a main symbol of q, A*Lij. Hence at= Y aj is a main symbol of A = F(pjAOj + Y A(1 - A,), while Y aj = a is a main symbol of A* =
(pjA*f[j +
(pjA*(1 - ayj).
Corollary 8.25 If A E OS'"'(M) and Be OS'"(M) then
AB - BA E OS""+""-'(M).
Proof The operator AB - BA e OS"" +'"'(M) has a main symbol at ea = 0, and therefore by the main symbol isomorphism, i.e. Theorem 8.23, we get AB - BA e OS"'+`2 - l(M). The next corollary follows at once from Theorem 8.24.
Corollary 8.26 If A E OS'"(M) has main symbol a, then for every
(p,
0 a C'°(M), we have (p o A o ip e OSm(M) with main symbol (p(x)Ji(x)a(x, s), and ((p o A 0)* = A*ip e OSM(M) with main symbol (p(x)O(x)a(x, s).
It follows that OS-(M) is a local space of operators, i.e. A E OS"'(M) if and only if (pAiI E OS"'(M) for every pair of C'° functions (o, ii on M. The definitions and results of this section extend to matrix p.d.o.'s with the obvious modifications. A matrix pseudo-differential operator is a p x q matrix, A= E OSm(M, p x q) with entries, A;j e OS'"(M), that are scalar p.d.o.'s of order m. For any open
set X in M, let S (T*X, p x q) be the set of p x q matrix functions whose entries belong to S,,c(T*X ). Then Definition 8.21 of the main symbol extends
to matrix operators. It is easy to see that the matrix operator A has a main
Pseudo-differential operators on a compact manifold
307
symbol a = [a,j] E Sm(T *M, p x q),
where a,; E S'"(T*M) is a main symbol of A,j. Once again, we have a main symbol isomorphism 4 and Theorem 8.24 also holds with obvious modifications. For instance, if A E OSm(M, p x q) has a main symbol a then the adjoint operator A* E OSm(M, q x p) has main symbol, ka, the Hermitian adjoint (conjugate transpose) of a. 8.4 Classic operators on M
An operator A E OS"(M) is said to be (-1) classic, and we write A E OCIS'(M), if for any chart (U, K) on M and for all cp, ' e C0 '(U) the push-forward (cpAfi)" E OSm(1B") is a (-1) classic operator
(Definition 7.21). Naturally it suffices to verify this condition for all charts in an atlas on M (see Theorem 7.17'). Let 91 = {(Up Kj)}N 1 be an atlas on M, and shrink it to obtain an atlas 2i' = ;(W, Kj)} , with W c Uj. Then we have (-1) classic operators
j = 1, ... , N, such that condition (i') in Theorem 8.10 holds. The operators Aj have a Aj(x, D) E OS"(l?),
principal symbol, homogeneous of degree m, defined by
nAj(y,q) = lim Aj(ytn) to" ,_M
(see Proposition 7.22)
for y E K1(Wj) and q # 0. and if W n W # 0 then
nAk(Y,q)=nAj(co-'(y),'dw(w-I(Y)).q),
w=Kk°Ki',
for all y E Kk(W n W) and q e R"\0. Consequently, we can define
irA e C"(T*(M)\0) by letting
irA(x, ) = iAj(Kj(x),'dKj- '(x) when x e W, s E TxM\0. (30) This definition does not depend on the choice of atlas because the union of two atlases is once again an atlas.
Theorem 8.27 Let A E OCISm(M) be a (-1) classic operator. Then the function trA e C'°(T*(M)\0) defined by (30) satisfies the following properties: (a) nA is positively homogeneous of degree m in l;,
iA(x, cy) = cm trA(x, ),
c > 0,
(b) for any X E C'°(T*(M)) which vanishes near the 0 section of T*(M) and equals I outside a compact subset of T *(M ), the function X nA is a main symbol of A (in the sense of Definition 8.21).
308
Boundary value problems for elliptic systems
Moreover, nA is the unique function in C'(T*(M)\0) having properties (a) and (b).
Proof The first property (a) is obvious. We prove that a = X nA is a main symbol of A and leave the proof of uniqueness as an exercise. From the definition of A j in the proof of Theorem 8.9' it is evident that we can choose a.,(Y,'1) = Aj(y,'7),
Y C- j(J3)
as the representative in the symbol class S /S,a' of the operator
A,,: Co (Kj(W )) -. C'(kj(W)) on the open set nj(W) c R. Then are is defined by (18), and we must show that a - a K a S, ' (T *(W )) for all j, that is, (see Definition 8.21)
ip(y)
for all rp e Co (rj(Wj)), or. since a = X irA, what we must show is that N(Y)'(X(q)'nAj(Y, q) - Aj(y, q)) a Sm-" where X is a cut-off function (see the definition of nA in (30)). But this follows
at once from the definition of (-1) classic symbols in §7.5.
We call nA the principal part (or principal symbol) of A. Clearly OS' __'(M) c OCISM(M ), i.e. every p.d.o. of order m - I is a (-1) classic operator of order m (with principal part 0). To be precise then we should denote the principal part of A by n,"A but normally we omit the subscript m. If A E OCIS"(.M) then 0 if and only if A e OS'-'(M). Let Symbl'M(M) denote the class of all functions A(x, 5) E C" (T*(M)\0)
which are homogeneous of order m in l:. The next theorem shows that Symbl"'(M) is the class of all principal parts of operators in OCIS'(M). Theorem 8.28 If A e Symblm(M) then there exists a (-1) classic operator A e OCISm(M) with principal part irA = A.
Proof If we multiply A by a function X e C"(T*(M)) which vanishes near the 0 section of T*(M) and equals I outside a compact subset of T*(M) then we obtain a function
a= By Theorem 8.23 there exists a p.d.o. A e OSm(M) with main symbol y A
(in the sense of Definition 8.21). It is not hard to verify that A is a (-1) classic operator and nA = A. Corollary 8.29 The following sequence is exact:
0 -.OSm`(M) `- OCIS"(M) = Symblm(M) - 0
Pseudo-differential operators on a compact manifold
309
Remarks 8.30
(i) If A', A E OCIS'"(M) are p.d.o.'s with the same principal part, rrA' = irA,
then A' - A e OS°'''(M) by Theorem 8.23. (ii) OS- "(M) = OCIS-'°(M ), i.e. operators of order - cc are always (-1) classic.
(iii) If A - Ya Ad is an asymptotic expansion (see Proposition 8.16) and m2 -< m, - 1, then .4 is (-1) classic if and only if A, is (-1) classic. and we have rrA=irA1.
To prove this, we need only observe that relation (13) gives A - A, e OSm'(M) C OS'"' -'(M).
One easily verifies that if A and B are (-1) classic operators then so are the operators A - B and A*. Thus, Theorem 8.24 implies the following: Theorem 8.31
(a) If A E OCIS'"(M) and B e OCISm=(M) then .4 , B e OCIS'"' i m=(M) and we have
rr(A B) = irA n6 (b) The adjoint of A e OCISm(M) is an operator.4* e OCIS'(M) and we have rr(A*) = rzA 8.5 Definitions for operators in vector bundles
The purpose of this section is to generalize the concepts in the introductory section §8.1 to operators acting between sections of vector bundles. Then we define pseudo-differential operators in bundles in §8.6. Two vector bundles (El, a,, M,) and (E2, it2, M2) are said to be isomorphic
if there is a diffeomorphism x: Mr -, M2 and a smooth map x: E, -, E2 which is linear and invertible on each fibre and such that tr2 such that the diagram
K o n1, i.e.
x
E,
R,1
E2
l.,
M, - M2
commutes. We say that X is an isomorphism over K. There is then a pushforward operation which maps sections f: M, -+ E1 to sections ic* f: M, defined by
E2. (31)
(the sections are not assumed to be smooth). Of course, there is also a pull-back operation Z* defined by y*f = ;t - ' - f = n which maps sections of
E2 to sections of E,. In the case of trivial bundles E, = M1 x C' and E, = M2 x C' and a smooth map ;: M, -+ M2, the push-forward and
310
Boundary value problems for elliptic systems
pull-back operations become simply
K*f = f--K-' and Of = f K. The Sobolev space 11'2(M, C") for the trivial bundle M x C" is defined as v
W2(M, C") = (D LV (M). j=1
To define the Sobolev spaces for a general vector bundle E, we will use the push-forward operation when X is a local trivialization of E over a chart K on M. The basic idea is that u c- W2(M, E) provided that whenever x...... .r
are local coordinates on U c M and e ... , e" is a basis for Et then for all tp e Co (U) we have
tp'u = I ui(x)ei, where ui E W' (in the local coordinates). It is then clear that all the definitions given above can be generalized to vector bundles with no essential difficulty.
To formalize this definition (see (32) below), let K: U - V be a chart for M on which there is a trivialization /i: Et; -, U x C". Then the vector bundle isomorphism
y:Et,-+VxC" over K is defined by K = (K, lo) = fJ. The push-forward (3l) of a section f e C 7-(U, Et;) by the trivialization X is a section of the trivial bundle V x C ", that is, Z* f = X f - K -' E C' (V, C"). There is also a push-forward l* f = fi f e C-(U, C"). but in this case we omit the asterisk * and denote it simply by fif. We call (U, K, X) a trivialization-chart and (U, fi) a trivializing patch for E. The letter f is meant to suggest a "basis" for the vector bundle, while K and X denote "coordinates".
Definition 8.32 Let l > 0, a non-negative real number. The Sobolev space W;(M, E) is the set of sections u: M - E such that X°
((D. u) , K
E W;(R , C")
for every coordinate map K: U -A, V and trivialization X: Et.
(32)
V x C" over K,
and for every tp e Co (U). The topology on W2(M, E) is defined by the semi-norms u H IIX
(tp u) K
' llr,
i.e. the weakest topology for which the semi-norms are continuous.
Vj x C" trivializations Let {UJ, Kj;j-,...... be an atlas on M and Xj: Et,r over Kj, and choose a subordinate partition of unity {cpj}. For U E W2(M, E) we let u j = X j (qpj u) = Ki ', and we make W2(M, E) into a Hilbert space by defining the scalar product (u, V)W"(M.E) = Y_ (uj, ri)h
j=I
Pseudo-differential operators on a compact manifold It
311
is possible to show that the topology defined by the norm pull` =
(u. U)W3(M.E) coincides with the topology in Definition 8.32, in other words. two equivalent atlases and two corresponding subordinate partitions of unity yield equivalent norms. For I < 0 we define 1412(M, E) as the anti-dual space
Wi(M, E) = (W,(M. E))*, or, alternatively, we could use Definition 8.32 with distributional sections. Now we endow E with a C" inner product (, )E.,, X E M, making it into a Hermitian bundle, see §5.10. For sections u, r e C' (Af, E) we then define a scalar product by (U, r)E'= f (u(x), v(x))E= dp(x),
(33)
N
and L2(M, E) is defined to be the completion of C'(Af. E) with respect to this scalar product (dµ is the invariant volume element on M introduced in §8.1). We have WW2(M, E) L2(M, E) as before, and the general comments about the scale of Sobolev spaces following Definition 8.1 hold once again. Let E and F be complex vector bundles over M. Just as in Definition 8.2, we say that a linear mapping B: CT(M, E) -+ CX(Af, F) is an operator of order m if for each I e R it extends to a continuous mapping B: W2(M, E) -- W2-'"(M, F). The set of all such operators is denoted by OPm(E, F). and it is a local space of operators. If B: Cx(M, E) -+ CX(M, F) is a linear mapping then its formal adjoint B* is defined by means of the scalar products (, )E and (, )F,
UECx(Af,F),reC"(M.F) If B e OPm(E, F) then we have B* E OP"'(F, E). (The existence of B' is proved as before.)
The set of operators of order - r_ is denoted
OP- x(E, F) _ () OPm(E, F), mER
that is, B belongs to OP- x(E, F) if it extends to a continuous map B: WZ'(M, E)
W2"(M, F),
for all l,,12 E R. We say that B e OC"(E, F) if it is an integral operator with a kernel K which is a C" section of the bundle Hom(n ' E, n2 ' F) over
M x M, i.eL .Bf(x)
KeC"(M x M,Hom(nI 'E,n2'F)), whe re n,: M x M - M and n2: M x M -+ M are the projections on the first
and second components, so that K(y, x) is a linear map from Er to F,,. Once
312
Boundary value problems for elliptic systems
again the classes OP-'(E, F) and OC"(E, F) coincide; using local trivializations the generalization of Proposition 8.3 to vector bundles is straightforward.
U x C" V on M and local trivializations f': Er, Given a chart h: U and fl': Ft, -+ U x Cq of the bundles E and F. there is a notion of the transfer of an operator B: C'(M, E) -+ C"(M, F) similar to that for scalar operators. As in the discussion preceding Definition 8.32 we let XE = (k, lo) fl' and XF = (K. lq) fl" and we call (U, K, XE, XF) a trivialization-chart for the pair of bundles E, F. The push-forward operators B. and B, are defined by the diagram e
Co(U,EU.) -' C'(U,Fi,) qf)
Ip
C'(U,Cq)
Co(U,C°)
(34)
Co(V,C°) that is, Bp = 13F B- (f3E)-1. Since X. = h
fl", and similarly for X,, we
have
B,=Xs-B°(%;) = K*
Be K*.
The push-forward by fl (resp. the push-forward by X) takes an operator B acting in the bundles Et, - FL to a matrix operator Bp over U c M (resp. to a matrix operator B, over V c R").
To be precise, the operator B in the first row of (34)
is really
eF B 1E, where ,E is the natural embedding :E: Ct; (U. E'.) -+ C'(M, E) C'(M, F) -+ C'(U. F,,,) is the restriction operator. and The support of a linear map B: C'(M, E) -, C'(M, F) is defined as in Definition 8.4. If the support of B lies in a chart domain U, we have an extended definition of the push-forwards Bp and B,. Definition 833 Let (U, A. /E, XF) and (U, /3E, /3F) he trivialization-charts for
E, F, and let B be a linear operator
B: C"(M, E) -+ C"(M. F), with supp B e U. Also let y e Ca (U) be such that s = I on a neiyhhourhood n( supp B. We define the following transfers (or push forwards): (i) Bp = T(B. /3): CZ(M, C°) C'(M, C") is a matrix operator, T(B. fi)f:= f31 s-B-(#E)-1;..f = and we hare supp T(B, j3)
= supp B c U;
(ii) B, = T(B, y): Co (R", C°)
Co (08", Cq) is a matrix operator,
T(B, %) = T(T(B, f3), h) = [nB,.,, K)],,. p.
Pseudo-differential operators on a compact manifold
313
where the transfer K) is from Definition 8.5, and we have supp T(B, X) _ h(supp B) c V. Obviously, we have
T(B,X)g = where
= 5 h ', and we see the complete analogy with (7).
The remarks mentioned after Definition 8.5 hold in this case as well; in particular, we point out once again that if B = (pAIi where 4p, y e C0 '(U),), there is no need to introduce another function Theorem 8.6 remains true with obvious changes in the proof. Theorem 8.34 Let the assumptions be as in Definition 8.33. Then the following are equivalent:
(i) B e OP'(E. F) with supp B c U, (ii) 71B, T) a OP'(M, q x p) and supp 77B,1) = supp B c U, (iii) T(B, X) a OPm(l2", q x p) and supp T(B, X) is a compact subset of V.
8.6 Pseudo-differential operators in vector bundles The definition of the transfer of an operator introduced in §8.5 is crucial to
the development of p.d.o.'s in vector bundles. Now that we have this definition, we can take over all theorems and all proofs from §§8.2 to 8.4 almost without change. As usual, let OS'(M, q x p) denote the set of q x p matrix operators with entries in OS^(M). The next definition stipulates that pseudo-differential operators in bundles be locally of this form.
Definition 835 A linear operator A: C" (M, E) - C"(M, F) belongs to OS-(E, F), A e OS'"(E, F),
if and only if for every trivializing patch (U, Pt, the push forward operator
#') and for all tp,
T((pAl, f) = #F((pA4,)PE-' belongs to OSM(M, q x p).
' e C0 '(U),), (35)
Since T(VAgi, X) is the K-push-forward of 779AO, /1), then using Definition
8.7 we have the equivalent definition:
X-Definition 8,35 A linear operator A: C'(M, E) - C'(M, F) is called a pseudo-differential operator of order m e R, and we write A e OS'(E, F), if for
every coordinate patch (U, K,)(E, XF) trivializing E and F, and for all rp, 41 e Co (U). the push forward operator T((pA#P, X) = K* -fF((pAo) fE- K* belongs to OSm(Q2", q x p).
(36)
Boundary value problems for elliptic systems
314
If we were studying p.d.o.'s in vector bundles ab initio, it would be necessary to work with the X-definition, which involves not just a change of basis fi in the fibre but also a coordinate transformation tc on the manifold. We can rely, however, on the work already done in §8.2, so we need only consider S-transfers. We leave it as an exercise for the reader to verify that
the results stated below in terms of #f-transfers have their counterparts in terms of X-transfers (and vice versa). As in Proposition 8.8, one proves that from X-Definition 8.35 it follows that A is quasi-local: if rp, 1i e CD(M) have
disjoint support, supp tp r supp 0 = 0, then QAt/i a OP- z(E, F).
Then we obtain the following theorem, the proof of which follows by substituting h by X in the proofs of Theorems 8.9 and 8.10 and then pulling back to the manifold. QF)} j
Theorem 8,36 Let 9.1 = {(Ui,
E and F. A linear operator A: C if and only if
1 be an "atlas" on M which trivializes
E) -i C"(M, F) belongs to OS'"(E. F)
(i) For any trivializing patch (U, fl f, /1J') a 91 and for all p, iy e C0 '(U,) we have
T(gpAtii, /f) a OS"(M, q x p).
(ii) A is quasi-local.
Let the "atlas" `It' = {(Wj, /1j, /15')}.. t be obtained by shrinking W e Ui; see Lemma 5.44. We can replace (i) by the following condition, obtaining thus another equivalent definition:
(i') There exist matrix p.d.o.'s Al a OSm(M, q x p), j = 1, .... N, such that the following representation holds for all (p, 0 e C( W ): 71VA'', flt) = ioAto or using pull-backs by #,-: We can also choose the operators A; so that they have compact support in U;.
Note that in the statement of Theorem 8.36, we do not assume that the sets Ui are chart domains, although in any application of it we will assume tacitly that the U's have been chosen this way. The following lemma which is the counterpart of Lemma 8.11 shows how to construct p.d.o.'s in bundles. X-Lemma 8,37 Let (U, K, XE, XF) be a trivialization-chart on M, and let the matrix operator A: C0 '(11", C°) Co (R", C9) have compact support in
ic(U) = V e R". Then the pull-back operator A: C" (M, E) -b C'(M. F) defined by A
71A,;(-
is in OSm(E, F) if and only if 1 e OS' (R", q x p).
Pseudo-differential operators on a compact manifold
315
It is obvious that the corresponding #-Lemma 8.37 is also true: If A has compact support in U c M, then
A = flA,l-')eOS'"(E,F) if and only if A e OS'"(M, q x p). The proof of continuity in Sobolev spaces is analogous to Theorem 8.13. Theorem 8.38 Let A e OS'O(E, F). Then for each l e R the mapping
W,- -(M, F)
A: W,' (M, E)
is continuous.
Proof Let {cpj} be a partition of unity subordinate to the cover {Uj} and let Orj a Co (Uj) be functions which are equal to I on supp ppj. We have
AcppAOj+q,A(I -0j), j
j
where the second sum belongs to OP (E, F) by quasi-locality, so it remains to prove that the first sum belongs to OPm(E, F). By Definition 8.35(i) we have T(cpAOj, Xj) e OSm(R". q x p). Then using Theorem 7.28 we get n(pjAgfrj, yj) a OPm(R", q x p),
whence Theorem 8.34 implies that cp1A*j a OP'(E, F),
which finishes the proof. (The support property in Theorem 8.34 is obtained from (5), i.e. supp cp1AJrj c supp (pj u supp 0j c Up) Now we define the main symbol of an operator A a OSm(E, F). Once again, since we can rely on the work done in §8.3 we need only investigate the effect
of a change of basis in the fibre, i.e. we look at fl-transfers instead of y,-transfers.
Let tr, be the projection s,: T*M -+ M, and as usual let Hom(zrs'E, n;'F)
denote the vector bundle over T'M whose fibre over (x, ) e T*M is the vector space of linear transformations E. F.. To obtain a main symbol for A E OS'"(E, F) the general idea is as follows. We push A forward by a to get a matrix operator AU a OS'(M, q x p), trivializing patch (U, flf, for which we have already defined the main symbol j4Ao E C'(T*M, Hom(C°, Ce))
(see the last paragraph of §8.3), and it can now be pulled back by (fF)-' _
= fl' to obtain the section ae
as a main symbol for A.
Hom(7r='E. n* 'F))
Boundary value problems for elliptic systems
316
To make this idea more precise. some definitions are required. Let (U. fit, fir) be a trivializing patch on M. The transfer of
o e C"(T*M, Hom(zr:'E. 7r* 'F)) is defined by
t(,, #)(x, 5)'= Qx a(x, S) (fl:) ', (x, E T*Afl u. = T*U. (37) Then t(.., l) E C'(T*Mlc, Hom(C°, CQ)), i.e. it is a q x p matrix function on T*Afli,..
For short we often use the abbreviation Horn
Hom(n*'E, n* 'F).
Definition 839 Let X he an open set in M. We define So,(T*X, Hom) to he the set of all a E C '(T*X, Hom) such that the transfer matrix (37) belongs to S,1, that is, t(a.fl)ESi x p) for every trivialising patch (U, p l, fir) on X. If X = M, we write S°(T*.M, Hom) instead of S oC(T*M. Hom).
It is sufficient to require that this condition hold for a collection of trivializing patches covering M (an "atlas") since the transformation from one basis (U, fE, fl') to another, (U', N'E, fl"), just involves multiplying by smooth matrix functions on T*(U n U'), that is, t(a, /i') =gr,r(a, p)-(g')-', (#E)- I and 9F = (RF)- l are the transition matrices. In where g' = l3-E
particular, the definition of S a(T*(X ), Hom) agrees with the earlier definition
when the bundles are trivial, E = M x C' and F = M x C. on which there is a trivialization given by /tE = identity on E and far = identity on F. Let
U. fE, fir) be a trivializing patch, and let W be an open set
with W c U. Let A e OSM(E, F). Then for all cp, ' e C0 '(W), ), the transfer T(cpAiI', J3) _ ((pAfi)p is given by
((pAiji)p = 4 Ac(/.
(38)
for some At. E OSm(M. q x p) with supp A,; c U. If (38) holds for another a OS-' for all (P E C0 '(W), as follows if representative At then rp(At we take qi e C0 '(W) such that 41 = 1 on supp gyp. Passing to main symbols E S « '(T* AI, q x p), and we obtain by Theorem 8.23 that therefore
x p).
/tAt - k;iveStoc For further use we introduce the notation
ap:= t(/,Av, fl-').
(39)
and we call a p a symbol of A in the basis P. This local symbol is therefore unique modulo S;- '(T*Ml w, Hom(a-'E, it-'F)I w ).
Definition 8.40 Let A eOSm(E. F). Then a e S'(T*M, Hom(n'E,,t* 'F)) is said to be a main symbol of A if ape Sla
Hom(a,'E, 1t*'F)IW)
(40)
Pseudo-differential operators on a compact manifold
317
for all trivializing patches (U, P', #') where IT' e U. (By Definition 8.39 this is equivalent to t(a, f) - 14AU E Soc q x p).) It suffices to verify (40) for a collection of trivializing patches covering M (an "atlas"), as follows from the next lemma; see the text after Definition 8.21.
Lemma 8,41 Let A E OS '(E, F). Then let (U, f u, flu') and (U', f' , f'.) he two trivializing patches, and Au and At,. the local representatives of A. see (38). If W c U, W' e U' and W n W' 0 0, we have ap
(41)
Proof If we denote by
flu, fu' the transition matrix for a vector
bundle, then we have for different transfers, (gpAJi), = pF!= D) = CA' a.,, (x. D)]a. P Thus we have established that
(pA*l = W(Ai )pi ,
(46)
for every rp, 0 e CO NW), where A; E OS"(M, q x p). In other words, A" satisfies condition (i') of Theorem 8.36.
Pseudo-differential operators on a compact manifold
321
It remains to show that "a is a main symbol of A. By means of a partition of unity - see the proof of Theorem 8.24(b) - this reduces to the problem of
finding a main symbol of (46) given the symbol a of A. Now, we have I,Aiq = /r(Aj)B, ,cp for q, 0 e Co (W), and by Lemma 8.42 the main symbol of tiA,O is equal to the matrix fi(flfa/tE ')rp. Using Theorem 8.24(b) and matrix algebra we see that the symbol of ipA /i is equal to rp(f 'a Applying once more Lemma 8.42 we get the symbol of (46):
, ( A'O) =
ipsa4i.
Finally, we define classic p.d.o.'s in vector bundles. Definition 8.46 An operator A E OS°(E, F) is said to be (-1) classic and we write
A e OCIS'(E, F) if for any trivialization (U, P) and for all rp, 0 a Ca (U) the push forward ((pAifi)8 a OS'"(M, q x p) is a (- 1) classic matrix operator on M, see §8.4.
Let `II = ;(Uj, fJ, jLL'__)} be an atlas of trivializing patches on M, and shrink
it to obtain an atlas `A' _ ;(W, fJ , #j")j with W C Uj. Then we have classic matrix operators supp A, C Uj, Aj(x, D) a OCISm(M, q x p), such that condition (i') in Theorem 8.36 holds. The operators A; have a principal symbol, homogeneous of degree m, defined by where
e
xp
irA j(x, S)'= xp \0). Since supp A; e Uj, then we have zrA;(x, S) = 0
for x outside U;, and from (42) it follows that on R; r Rk # 0 we have nAj, (x, ,, - gU,,u,(x)trAj(x, x9i,u,(Y)) Consequently, by pulling itA; back to the manifold. i.e. trA(x,
when x e W,
a TFM\\0
(47)
we have that aA e C'(T*M'\\0). Hom(ir;'E, n* 'F)) is well-defined, and, further, its definition does not depend on the choice of atlas or of trivializing
patches. We call aA the principal part or principal symbol of A, and the proof of the following theorem is now essentially the same as Theorem 8.27.
Theorem 8.47 Let A e OCISM(E, F) be a (-1) classic operator. Then the section nA a C'(T*M \0, Hom(n;'E, n;'F)) defined by (47) satisfies the following properties: (a) rrA is positively homogeneous of degree m in
(b) for any X e Cz(T*M) which vanishes near the 0 section of T*M and equals I outside a compact subset of T*M, the section is a main symbol of A.
Boundary value problems for elliptic systems
322
We denote by Symbl'"(E. F)
the vector space of all C" sections d E C'(T*M\0), Hom(n 'E, n;'F)) which are homogeneous of degree m in
By homogeneous of degree m in
y, we mean that the maps l(x, S): Ex - F, (x, ) E T *M \0, satisfy the condition h(x,
c' - A(x, )
for all c > 0.
Clearly OS"-'(E, F) c OCIS'(E, R). As in Corollary 8.29 we have the following result.
Theorem 8.48 The following sequence is exact:
0-.OS"-'(E, F)= OCIS'"(E, F)' Symbl'(E, F) ---+0 (Exactness at the middle two arrows means that n",A = 0 if and only if A E OS'"-'(E, F), while exactness at the last two arrows means that n", is surjeetive.)
Further, the mapping n is a homomorphism: Theorem 8.49
(a) If A E OCIS"(F, G) and B e OCIS'"'(E, F) then A B E OCIS"" +"I(E, G) and we have
n(AoB)=iA'nB (b) If the bundles E, F are Hermitian then the adjoint of A e OC1S'"(E, F) is an operator A* e OCIS'(F, E) and we have
nA* =''(nA) The proofs of Theorems 8.48 and 8.49 follow from Theorems 8.44, 8.45 and 8.47(b). Theorem 8.50
(a) For any given sequence A,, e OS'"°(E, F), d = 1, 2, ... , with me_, > Ae and - - x- as d -» +x, there exists A E OS'"(E, F) such that A (see (13)). The p.d.o. is uniquely determined modulo OS-''(E, F).
(b) If m1 S m, - I then A is (-1) classic if and only if A, is (-1) classic, and we have nA = nA,. The proof of (a) is obtained by localizing as in Proposition 8.16, while (b) is
obvious since we have A - A, a OS'"=(E, F) a OS"" -'(E, F). We end this section with a lemma which gives some information on the relation between an operator and its principal symbol. By virtue of Lemma 8.67 (see §8.7) there is no loss of generality if we consider just p.d.o.'s of order 0.
Pseudo-differential operators on a compact manifold
323
Lemma 8.51 Let E and F be Hermitian bundles. Given x0 e M. there exists a sequence {fk} in C°°(M) with the following properties: for all sections e e C' (M, E) with e(x) of length I for x in a neighbourhood of x0, and 1 forms E C"(M, T*M) with i;(x°) # 0,
(i) the support of fk converges to x0, (ii) IlfkeI L, = I,
(iii) for each A E OCIS°(E, F), we have
IIA(fke)(x) - irA(x, (x))'fk(x)e(x)IjL,
0
as k -+ oe.
Proof First suppose that E = F = M x C are trivial bundles, then we may take e(x) - I. Consequently, it suffices to prove existence of { fk } satisfying (i), (ii) and (iii') IiAfk(x) - 7rA(x°, 5o)'fk(x)11L, - 0
since it would follow from (i) and (ii) that
IInA(x, S(x))'fk(x) - nA(xo, W'fk(x)IIL, - 0 where s0 = S(x0), and (iii) would then follow.
Let c: U -+ V be a coordinate map in a neighbourhood U of x0. Let 0 e C"(M) equal I in a neighbourhood of x0 with supp i/i c U; we have OAP = P(A(x, D)),;- 0 for some p.d.o. A(x, D) E OS°(R"). Letting z0 = x(x0) and Zo
='dx-'(x0)(So),
and choosing k a C0 .(R-) as in Lemma 7.32 for the point (z0, 0) a R" x R", we have II A(x, D)0, - nA(xo, S)' k 11, 0. Since the support of $k converges to x0, then for k sufficiently large, we may let Ok = 10k o K E C°`(M) be the pull-back to the manifold. For k sufficiently large we also have W 2ck = 0k, so that Ask = /At/r4k + A - , Ok, where A _ 1 E OS- 1. Now A _ l k - 0 since Ilk II, 0 when s < 0, and it follows that IIAOk - nA(Y0, So)'OkDIL, -. 0.
In view of (30) we have nA(xo, So) = nA'(z0, 50), and since II0kOi, _ f.uu) I$k(x)IZgu(x) dx 3 C > 0, we can set fk = thereby establishing (ii) and (iii').
In the case where E and F are general bundles, choose trivializations /3F: Fu -+ U x RQ and /3E: Eu - U x R9 with ft' -e = e, (the first coordinate vector in R"). Then we have
OAO =,(#F)-'(A(x. D))x- ipEo for some matrix operator A(x, D) = [A1J(x, D)] a OSm(R", q x p). Now choose Ok as above. Since i,/t" , e = e, it follows that OAi,li0ke = dr(#F)- 1([Ai1(x.
D)]9=1)g_.04,p.
Then by Lemma 7.32 we have as before IIA11(x, D)Ok - 7rA;l(.xo, So)' Y'k IIo -. 0,
and the sequence fk = 04/II40IL, again has the required properties, in view of (47).
324
Boundary value problems for elliptic systems
Corollary 8.52 It follows that II A(fke)II LZ - I7rA(xo,o)e(xo)IFF0, and hence, for any 0 0 l;o e T oM, the section e in Corollary 8.51 can be chosen so that as k -+ oo.
IIA(fke)IILZ --' IImA(xo, o)II
(48)
Proof Let fE and aF be the trivializations introduced above, and in addition choose them so that they map orthonormal bases to orthonormal bases. It follows that IIiA(x, (x))-fk(x)e(x)IIL'Z =
-
Ifk(x)I2 du(x) SM
InA(xo, to)e(xo)I
xo
since JM Ifk12 du = 1 and the support of fk converges to xo. Then (48) follows
from Corollary 8.51 if we choose the section e e C'(M, E) so that
Ind(xo, o)e(xo)IF. = Ilnd(xo, o)II and l e(x)I E,
=1 in a neighbourhood of xo.
8.7 Elliptic operators An operator A e OS'(E, F), with main symbol
a e S'(T*M, Hom(n*'E, n* 'F)), is said to be elliptic of order m if there exists some
e e S-'(T*M, Hom(zc* 'F, 7r* 'E))
such that aG - IF E S-1(T*M, Hom(7r* 1F, st* 1F)) l
(49)
ea - lE e S-1(T*M, Hom(n* 1E, 'n*-'E)) J where lE and IF denote the identity mappings E -* E and F - F. We will also refer to a as an elliptic symbol. Let Ell'"(E, F)
denote the set of all elliptic operators A e OS'"(E, F) of order m and Ell = U. Ell' the set of elliptic operators of arbitrary order. Note that if A e Ell then A e Ell' for a unique m (see Exercise 8). Similarly, the set of all elliptic symbols a e Sn(T*(M), Hom(n* 1E, n* 1F)) of order m is denoted
ettt(E, F), and ett = UT dtt is the set of elliptic symbols of arbitrary order. It can be shown that the existence of G satisfying both conditions in (49) implies that E and F have the same fibre dimension (see Exercise 19). If we had assumed this at the start, the two conditions would have been equivalent; see the addendum at the end of the chapter.
Pseudo-differential operators on a compact manifold
325
We begin by showing that elliptic p.d.o.'s are Fredholm operators. Theorem 853 Let A e OSm(E, F). If A is elliptic then A: W2(M, E) -,
W2-",(M, F)
(50)
is a Fredholm operator (for all I e 68). The kernel, ker A, is contained in C°°(M, E), and the image, im A, is the orthogonal space of the kernel e C`°(M, F) of the adjoint A* a OSm(F, E), i.e.
im A = f f e W2-m(M, F); (f, g)F = 0 for all g e ker A*)
(51)
Consequently, the index is independent of I since ind A = dim ker A -
dim ker A*. If A' e OSP"(E, F) is another elliptic operator such that
A - A' c- OS'"-'(E, F) then ind A = ind A'; hence if two elliptic operators in OSm(E, F) have the same main symbol then they have the same index.
Proof By Theorems 8.43 and 8.44, the condition (49) is equivalent to the existence of an operator B e OS-"'(F, E) such that
AB-1=:T,aOS-' and BA-1=TeOS-'.
(52)
T, and T are operators of order -1 in the scale of Sobolev spaces by Theorem 8.13 and by the compact embedding theorem for Sobolev spaces, Theorem 7A.13, it follows that
T:
W2-'"(M, F) -, W2-"'(M, F) c W2 "(M, F)
and
T,: W2(M, E)
W2+'(M, E) c WW (M, E)
are compact operators. Hence B is a right and a left regularizer, and by Theorem 9.6 the operator A is Fredholm. If u e ker A c WW then u = - T u e W2+', and by iterating this result we get
u e n WZ(M, E) = C°°(M, E) Similarly, the adjoint A* e OS'n of A is an elliptic p.d.o. that maps W.1-'(M, F) to WZ '(M, E), and ker A* c C°°(M, F). Since the image of A as an operator
from W2(M, E) to W2-"'(M, F) is closed, it is the orthogonal space of the kernel of the anti-dual operator which can be identified with the adjoint A*: WZ -'(M, F) - WZ '(M, E) by means of the duality brackets, (,)E and (1)F1 see §8.5, and this proves (51). The index is independent of I because ind A = dim ker A - dim ker A*, and both ker A e C°° and ker A* c C°° are independent of 1. The last statement in the theorem is clear by virtue of Theorem 9.11 because the operator A - A': W2(M, E) -+ W2-'"+'(M, F) c' WW-'"(M, F) is compact.
Substituting I = m we obtain the following.
326
Boundary value problems for elliptic systems
Corollary 854 If .4 E OS'(E, F) is elliptic, we have the orthogonal decomposition
L2(M, F) = im A Q ker As.
(53)
where both subspaces, im A and ker A*, are closed. By im A we mean the image of the operator A: WZ (M, E) -+ L2(M, F).
From (52) we obtain the following regularity theorem. The discussion of smoothable operators at the end of §9.1 is relevant here. .4ueW21-m+'(M,F) Corollary 835 (Regularity Theorem) If ue W21(M,E) and then u E W21+'(M, E). Hence if u E U W21(M, E) and Au E W'(M, F) then u c- W2" '(M, E).
Proof Since Au a 14'21 -m ' I(M, F) and B has order -m then BAu e W21+'(M, E). Now, by (52) we have u = BAu - T,u E W21+ '(M, E) which
proves the first statement. The second statement follows easily, for if u E W21(M, E) with I - m + I < r then u e W21 +'(M, E), and we can iterate
this result until I - m + I = r.
There is also the following result which shows that A: C'(M, E) -+ C'(M, F) is Fredholm and has the same index as the operator (50). The topology on C"(M. E) is the Frechet space topology in which a sequence ,u,} c C'(M, E) converges if all derivatives of u, converge uniformly on compact subsets of chart domains. Corollary 836 If u e W21(M, E) for some I and Au a C(M, F). then u E C'(M, E). (This is known as Weyl's Lemma.) Hence
C'(M, F) = A(CZ(M, E)) $ ker A*, (53') where both subspaces A(C'(M, E)) and ker A* are closed in C'(M, F). Moreover, A(C'(M, E)) consists of all f e C'(M, F) for which (f, g)r = 0 for all g e ker A*. Proof If u e W21(M, E) for some 1, and Au a C'(M, F), then u e n WW(M, E) _
C''(M, E) by virtue of Corollary 8.55. Now, since ker A* c C'(M, F) it follows from (53) that
C"(M, F) = (C-(M, F) n im A) ® ker A* which implies (53') by Weyl's Lemma just proved. Once again it follows from Weyl's Lemma that A(C"(M, E)) is closed in C"(M, F) since im A is closed in L2(M, F) and the embedding C'(M, F) c' L2(M, F) is continuous. Obviously ker A* is closed in C'(M, F) because it is a finite dimensional subspace. Finally, the last statement in the corollary follows from (51) and Weyl's Lemma.
We now turn to the problem of determining a parametrix for an operator A e OS'(E, F).
Pseudo-differential operators on a compact manifold
327
Definition 837 An operator B e OS - '(F, E) is a parametrix for A if
A,-B-IeOS-"(F,F) and B,A-IEOS-r(E,E)
(54)
We show in the next theorem that ellipticity is the necessary and sufficient condition for existence of a parametrix. To prove this result two lemmas are required.
Lemma 838 If TE OS-'(E, E) then I - T has a parametrix.
Proof We let 7- T. .
= T and define Q by the asymptotic series
Q-I+T+T2+ Since by Theorem 8.44 we have T" e OS-'(E, E), then Q is well-defined (see Theorem 8.50). Setting Q = 1 + T + + T" + T"+, where T'"+, c OS-"-'(F, E) we obtain
(I -T) Q - I=(I - T)(I+T+
+T"+TY,,)-I
=I-T"+'+(I-T) TN,, -I =(1-T),T;,.,-T"+'
which belongs to OS-"-'(E, E) for N = 1, 2,.... whence
(I -T)-Q-IEOS-c(E,E). Similarly, we get Q = (I - T) - 1 E OS-;(E, E). Hence Q is a parametrix for
I - T. The following lemma shows that if A has a left parametrix and a right parametrix then any right or left parametrix of A is in fact a parametrix. Lemma 8.59 Let A E OS'"(E, F). If for some B e OS `'"(F, E) and B' e OP-"(F, F) (i.e. an operator of order -in) we have
A=B-IeOS-Z and B'-A-IEOS ', then B - B' c- OS- ". (In particular, it follows that B' c- OS-"(F, E).)
Proof This follows at once from the equation
B'-B=B'(I -AsB)+(B'^A-I)B by virtue of Theorems 8.44 and 8.38 and the fact that OP- '(E, F) _ OS- -(E, F).
Theorem 8.60 Let A e OS"(E, F). Then A has a parametrix if and if it is elliptic.
Boundary value problems for elliptic systems
328
Proof If A has a parametrix B satisfying (54) then certainly AB - I E OS"' and BA - I E OS"'. If we let a denote a main symbol of B, it follows from Theorems 8.43 and 8.44 that (49) holds, i.e. A is elliptic. Conversely, if A is elliptic then as in the proof of Theorem 8.53 there is an operator B, e OS- '(F, E) that inverts A modulo OS"', i.e.
where T e OS -'(F, F). A o B, = I- T By Lemma 8.58, 1 - T has a parametrix Q e OS°(F, F) and if we let B:= B, o Q e OS-'"(F, E) it follows that
AoB - I = AoB,oQ - I = (I - T)oQ - IEOS so B is a right parametrix. Similarly, there exists a left parametrix B', and by Lemma 8.59, we have B - B' c- OS- °°. Thus B is a two-sided parametrix of A.
We turn now to the case of classic pseudo-differential operators A E OCISm(E, F). Classic operators have a principal part nA a Symblm(E, F) which exists by Theorem 8.47. Theorem &61 Let A E OCISm(E, F). Then A is elliptic if and only if nA(x,1;): E -+ Fx is invertible
(55)
for all (x, l;) e T*M\0.
Proof The inverse [xA(x,is taken pointwise for each (x, l;) e T*M\O. Then [aA]"' E C°°(T*M\0, Hom(n#'F, n-'E)) follows from Proposition 5.2 in local coordinates because nA is C. And since nA(x, c > 0, it follows that
[nA(x, ct)]-` _ [c'"nA(x, t)]"' = c"m[nA(x, and therefore [nA(x, l;)] a Symbl- '(F, E). By Theorem 8.48 there exists B e OCIS-'"(F, E) such that nB = [nA]"', whence n(A o B - IE) = 0 and n(B o A - IF) = 0 (by Theorem 8.49), i.e.
A o B -1F E OS"'(F, F) and B o A- IE a OS-'(E, E). Applying the main symbol isomorphism, Theorem 8.45, we get a o e - IF E S
and G o a - 1 E E S', which is ellipticity in the sense of (49). To prove the necessity of (55), choose some Riemannian metric on M, i.e. a smoothly defined set of inner products < , > on each tangent space 7 M. Let ICI = < , i>zI: where 0. In view of Theorem 8.28 there exists an operator Be OC!Sm, 2(M) with principal part i.12. We set A = I + B*B which belongs to OCISm(M) and has principal part ;.111. i 12 = A by Theorem 8.31 because m > 0. Now, (Au. u)M >, (u, u)M, so the operator A: W2(M) -+ W2-'(M) is one-to-one, and being self-adjoint its image is dense in W,-'(M). On the other hand, A is elliptic because i. > 0, so the image of A is closed. Thus A is bijective, hence an isomorphism by virtue of the Open Mapping Theorem.
Further, by Corollary 8.62, A has a (-1) classic parametrix. Since A' e OP - m is certainly a parametrix (it is an inverse!) and then by Lemma
8.59 it follows that A-' c-OCIS-m(M). By Theorem 8.31, the equation A = A - ' = I implies that the principal part of A-' is i,-'. Thus, the case m < 0 may be handled by applying the result just proved to the function i.-' a Symbl-m(M) since -m > 0.
Remark The same proof works even in bundles if A e Symblm(E, E) and ,.(x,1;) is positive definite for all (x, s) e T*M \0. But for most purposes the following weaker version is sufficient.
Lemma 8.68 Let A e Symbolm(M) be real-valued and suppose that i. # 0. If A e OCISm(M) is any p.d.o. with principal symbol A then ind A = 0. Proof Since A is real-valued, then A and A* have the same principal symbol.
The difference between A and A* is therefore in OSt-'(M), and by virtue of Theorem 8.53 we have ind A = ind A* _ -ind A, whence A has zero index.
We now consider briefly the case of p x p systems which are elliptic in the sense of Douglis-Nirenberg. Let the real numbers s...... sp, t,, ... , tp be given (we can take any real numbers without restriction). We denote by OCIS"'(M, p x p) the set of all p x p matrix operators
A=[Aij]pxp,
A1jeOCISm"(M)
334
Boundary value problems for elliptic systems
such that each AijeOCISm,j(M) is (-1) classic of order m,j < s; + tj. The DN principal part is the p x p matrix zrDA(x, ) __ [atj(x, )]
where the entries are a,j = rrs,+,,Atj if the order of Aij is si + tj, and 0 otherwise. Its elements a,j are homogeneous functions in of order s, + tj, so we have the homogeneity property c > 0,
nDA(x,
(62)
where S(c), T(c) are the diagonal matrices
S(c) = [btjc'']pxp, T(c) = [S,jc',]yxp. (63) In general itD is not an algebra homomorphism since the DN numbers of two operators A and B do not necessarily match up to give DN numbers for AB.
The matrix p.d.o. A is said to be DN elliptic if there exist real numbers s,, ... , sp, t1,.. . , tp such that A e OCIS' (M, p x p) and det 2rDA(x, ) # 0,
for all (x, l;) e T*(M)\0.
(64)
Theorem 8.69 Let A = [A,j] a OCISS+'(M, p x p) be a DN matrix p.d.o. with DN numbers sl, ... , sp, t1,.. . , tp. Then the following are equivalent: (i) The operator A: ®J?- t WZ+'l(M) -* Q°=1 W2- `(M) is Fredholm (for any I e R) (ii) A is DN elliptic, see (64). (iii) A has a right and left regularizer B = [B,j] e OCIS-'-s(M, p x p) with DN numbers -t,, ... , -tp, -s,, ... , -sp.
(iv) A has a right and left parametrix B = [B,j] e OCIS-'-'(M, p x p) with DN numbers -ti, ... , -tp, -s1,. .. , -sp. Note: It follows from the equivalence of these conditions that if (i) holds for some 1 e R then it holds for all I e R. Proof Choose any elliptic operators Ak e OC1Sk(M), k e R, say with principal symbol 1 1k for some Riemannian metric. (By Lemma 8.68 these operators
have zero index but we do not need this fact here.) Let P = [SijA-S-] and Q = [EtjA-'7] and note that P and Q are also elliptic. (i) . (ii) If A is a Fredholm operator, then A' = PAQ e OC1S°(M, p x p) is a Fredholm operator in L2. Hence det A'(x, ) A 0 for all (x, g) a T*M \0 by Theorem 8.66. Now, 7rA' =
so det nDA(x, l;) 0 0 for all (x, g) e T*M\0, i.e. A is DN elliptic. (ii) (iv) If det i1A(x, l;) 0 for all (x, g) a T*M\0 then the same is true for det rrA'(x, ). By Theorems 8.60 and 8.61, A' has a right and left classic regularizer B' c- OCIS°(M, p x p), i.e. B'(PAQ) - I e OS and (PAQ)B' - I e OS
Pseudo-differential operators on a compact manifold
335
Now, if we multiply the first expression by Q on the left and by a parametrix of Q on the right, we find that QB'P is a right parametrix for A. Similarly, if we multiply the second expression by P on the right and by a parametrix of P on the left, we find that QB'P is also a left parametrix for A. (iii) - (i) This follows as in Theorem 8.53. (iii) (iv) This follows as in Theorem 8.60.
Remark The analogues of Theorems 8.64 and 8.65 also hold for rectangular q x p DN systems. This is accomplished by the simple device of left and
right multiplications by diagonal matrices P and Q as in the proof of Theorem 8.69. Any questions concerning the index of A can be reduced to the index of the operator A' = PAQ since ind P = ind Q = 0 and therefore ind A' = ind A. For the sake of completeness we indicate a more explicit way to construct the left parametrix of A e OC1S' "(M, q x p) in the case when the principal part of irA is injective, i.e. c e C9, nA(x, )c = 0 => c = 0. Let a = irA and (x°, 50) e ST*(M). The injectivity of the p x q matrix a(xo, o) implies that
it has a non-vanishing q x q minor. By continuity, the minor of a(x, formed from the same rows and columns will not vanish for all (x, S) in some neighbourhood VO a (x0, ii0). Thus, there exists a q x p matrix function b0(x, i;) with entries in C" (V0) such that
bo(x, )' a(x, ) = 1, (x, S) C- V0. By compactness ST*(M) can be covered with a finite number of such neighbourhoods, and piecing together these local constructions using a
partition of unity we obtain a q x p matrix function with entries in C"(ST*(M)) such that
(x, ) e ST*(M).
b(x, )'a(x, S) = 1,
(65)
Extending b by homogeneity we obtain
b = [b;,],
b, j homogeneous of degree - t; - sj
and (65) holding for all (x, ) e T*(M)\O. Now let B = [B,,] e OC1S where Bit is (- I) classic with principal part b,;. In view of (65) we have
BA = I + T, where each entry of T is in OCIS-'(M). Since I + T has a parametrix Q e OCIS°, then A has left parametrix QB. 8.8 An illustration: the Hodge decomposition theorem Let M be a compact, oriented manifold. As usual we assume that M has no boundary. We consider the following vector bundles over M
l P(T*M),p=0,...,n:
A(T*M) = +$N(T*M), P=O
336
Boundary value problems for elliptic systems
and the spaces of sections Q"(M) = CX(M. Ar), C2(M) = CY(M, n). Recall
from §6.9 that the Laplace-de Rham operator, A = db + bd, is an elliptic second-order differential operator. A: C'c(M, Ap) Cz(M, n°). A form x for which Aa = 0 is called harmonic. Let
JrP = kerpA = ;a e C''(M, Ar): Ax = 01; denote the vector space of harmonic p-forms. In §6.9 we introduced the L2
inner product in C'(M, A) defined by
(a, n)=J 1A
?I
M
and showed in Theorem 6.43 that d and d are adjoints. i.e. (dx, y) = (x, 6q). Therefore, for cq,
e C" (M, A) we get
(AO. q,) = (0d1', (p) + (dal, (p) = (dc, d(p) + (60, aw) = (0, A(p),
and it follows that A = M. i.e. A is seif-adjoint.
Proposition 8.70 If to e C" (M, N) then Aw = 0 if and only if dw = 0 and bw = 0. Thus harmonic forms are closed and co-closed.
Proof If dw = 0 and bw = 0 then by definition Aw = dbw + bdw = 0. Conversely, if dw = 0 and bw = 0 then the previous computation shows that 0 = (Aw, w) = (dw, dw) + (be), bw), hence dw = 0 and bw = 0. Since A is elliptic then by Theorem 8.53 we have
ker A c C" (M, A),
dim ker A < x.
Theorem 8,71 (Hodge Decomposition Theorem) Let w e C"(M, np), p = 0, ... , n. Then there exist forms x e C' (M, A `), R e C" (M, A ') and
y e C`(M, N) such that
w=dot +bfi+y and Ay=0.
(66)
Further, this decomposition is orthogonal, hence unique, so we have the direct sum
C'(M, N) = kerpA ® imp d ® imp 6
(67)
where imp d is the image of d: C"(M, Ap- `) - CX(M, A) and imp b is the image of b: C'(M, np. `) C"(M, np). Proof Since A is elliptic and A = A' then by Theorem 8.53 we have
kerp A,= C'(M, A"),
dim kerpA < x,
(68)
and by Corollary 8.56 we have the orthogonal decomposition
C''(M, IV) = A(C %(M, M)) e kerpA,
p = 0, ... , n
(69)
Thus we can write a p-form w e C' (M, np) as w = Aq + y for some y e kerp A and p e C' (M, /\p). Therefore, w = dbq + bdry + y, and to get (66)
Pseudo-differential operators on a compact manifold
337
we can choose a = Oil and /3 = dil. Hence we have established the sum C°°(M, M) = kerp A + imd d + imd b.
If we show that the factors in this sum are orthogonal to each other, we are done because uniqueness of (66) follows from orthogonality. (1) im, d 1 im b. Let u = dot a imp d and v = 6/3 a imp d. Then, since d is the adjoint of 9, we get (u, v) = (da, 0$) = (d2a, /3) = 0. (2) imp d 1 kerp A. Let Aw = 0 and u = da a imp d. Then bw = 0 (see Proposition 8.70) and (w, u) = (co, da) _ (bw, a) = 0.
(3) imp 61 kerp A. Now, from Am = 0 it follows that dw = 0, thus composing with v = bfl we get (w, v) = (w, 69) = (dw, /1) = 0.
Definition 8.72 A p -form a on a manifold M is called closed if da = 0; it is called exact if there is a (p - 1) form /3 such that a = dfl. Since d2 = 0, every exact form is closed. We define the vector spaces
Z"(M):= (w a CW(M, N); dw = 0} B"(M)
(co e Zp(M); w = da, a e C°(M, Ap-')}
and the quotient vector space
Hp(M) = Zp(M)/Bp(M),
p = 0, ... , n
(70)
is called the pth de Rham cohomology group of M. The points in Hp(M), i.e. the classes in Zp(M) modulo B"(M), are called cohomology classes.
Theorem 8.73 Each cohomology class in Zp(M) contains a unique harmonic representative from kerp A. Proof Let co e Zp(M ). From the Hodge decomposition theorem 8.71 we have
w = da + 6/3 + y, where y e ker A. Since y is harmonic then dy = 0 and, further, since w e Zp(M) then Iw =' 0, so that dbfi = 0 too. It follows that (b$, 6/3) = (/3, d6/3) = 0, so b/3 = 0. Thus co = da + y, and the cohomology class [w] contains the harmonic representative y, i.e. [w] = [y]. If two harmonic forms a, and a2 differ by an exact form dli, then we have
0 = dfl + (al - a2).
(71)
But since a, and a2 are harmonic then ba, = Sae = 0, and it follows that dli and (a, - a2) are orthogonal because (d/3, a, - a2) = (/3, ba, - bat) _ (/3, 0) = 0. Hence by uniqueness of orthogonal decompositions we have
dfl=0 and a,-a2=0. Thus, there is a unique harmonic form a, = a2 in each de Rham cohomology class.
Using Theorem 8.73 we immediately obtain that the de Rham cohomology groups are finite dimensional.
338
Boundary value problems for elliptic systems
Theorem 8.74 The vector spaces
HP(M) _- ker,A,
p = 0'...,n
are isomorphic; and since the vector spaces kerp A are finite dimensional, we have established the finite dimensionality of H"(M), dim HP(M) < oc,.
Proof We map HP(M) - kerp A by [co] r y where y is the unique harmonic form such that to - y is exact. It is easily verified that the map is linear, and it is evident that it is surjective (since harmonic forms are closed). It is also injective because if y = 0 then to is exact, whence [w] = 0. This establishes an isomorphism HP(M) kerp A. 8.9 Limits of pseudo-differential operators
In this section we look at limits of pseudo-differential operators in the operator norm. Much of the Fredholm theory of elliptic p.d.o.'s can be carried over to limits of such operators. The primary reason why we introduce these limits is the following. If
A: C'(M) - C'(M) is a pseudo-differential operator on M then we can define
A': C'°(M x N) - C'(M x N) by letting A act on just the first variable. But A' need not be a p.d.o. because
in local coordinates the estimates for a function a'(x, y; i;, q) = a(x,
),
namely, IDi{.,,)D =.r)a(x, )I 0, however, it can be shown that A' = A ®1 is a limit in OP'" of pseudo-differential operators of order m (see Lemma 8.78) and then we may use Theorem 8.76. The basic idea here is that when m > 0 the principal i;/ICI), is continuous for kI + I'iI > 0 part, (nA (9 1)(x, y, , q) = (including S = 0). We begin with the following lemma.
Lemma 8.75 Let Aj e OCISm(E, F) be (-1) classic operators, j = 1, 2, ... , and assume that Aj converges in the operator norm in £°(W2, Wv") for some I e R. Then the principal parts nAj converge uniformly on compact subsets of
TsM\0.
Pseudo-differential operators on a compact manifold
339
Proof If A e S'(R" x R") has main symbol X nA then t-M(A(x, D)(u
v ei(-.'4))L2 = (t-mA(x, D + t )u, v)L2 - (mA(-, S)u, v)
for u, v e Co (I8"); see the proof of Theorem 7.40. It follows that I (mA(-, S)u, v)L211< lim
t-"'I(A(x, D)(u
v ei( Jt')I
and, by making use of (58), V)L21 max(2, I) is homogeneous of degree 0 when III > 2. Then X Rn+"'), A.(x, y, , 1) = A(x, y, )X(+;, eI) e CISm(R"+"'
the principal symbol nAt converges uniformly to ttA on Iii' + III2 = 1 as a - 0, and the norm of A,(x, y, D, D,) - A(x, y, Dx) as operator from WW(R"+"') to WZ-1"(R"+"') tends to 0 ass - O for every I e R. (In particular, it follows that A(x, y, Dx) e
Proof Since X is homogeneous of degree 0 when ICI' + III' > 8 (in fact >4 for the function constructed above) it follows that X e
OCIS0(R"+"
Rn+n')
XX
Also it is clear that Ae e S' in both variables l;, I, that is, ID' 'DY DfD,',1'Ae(x, y, 4, I)I < const(l + ItI + III)'"-1P1
-ia
q):0 0 only when III < and then it is not hard to see that A, a OCISt with principal part nA(x, y, en). The magnitude of the difference between the principal parts nAe and nA is bounded by a constant because
times eI) - II which tends to 0 uniformly on 141' + III' = I as 8 -+ 0 because m > 0, rrX is bounded, and nX(1, eI) 0) = 1 uniformly when ICI >, const > 0. Now we must prove convergence of the corresponding p.d.o.'s in Sobolev ICI"'.
I
norm. For fixed y, let C(y) the norm of the operator A(x, y, Dx): WZ(R") L2(R") and note that C(y) is bounded by a semi-norm of A, thus C(y) is bounded by a constant C independent of y. If v e .°(R" x R") we then have for fixed y J
IA(x,y,Dx)(x,Y)I2dx
Clly( ,Y)IIM,
Boundary value problems for elliptic systems
342
and then integrating withh respect to y,
IIA(x, y. D)olC IIv(, y)IId= C J
ff
+
Now, since
A,(x, v, D, D,)u - A(x, y, D,)u = A(x, y, DD)r,
where r, = y,(D. cD,)u - u, it follows that 1)AE(x. y, Dx. D,)u - A(x, Y, DD)ullo
InAI. Then A0/ - A is
an elliptic p.d.o., so it has a finite dimensional kernel; the orthogonal projection PO on this kernel is therefore compact. Since A0 is real, the operator
A0/ - A - PO is a self-adjoint Fredholm operator, and it has trivial kernel so it is invertible; and further there is a d > 0 such that IAOI > lnAl + b and Al - A - PO is invertible for IA - A01 < S. We claim that if A. is a real number
such that 0 < 1). - A01 < d then A # sp(A). In fact, if ;.f - Af = 0 then (A -- A0).f = -Aof + Af belongs to the image of A0/ - A and is hence orthogonal to the image of PO. Since A. - A0 # 0 it follows that f is orthogonal to the image of P0, so that PO f = 0. Hence Af - A f - P, )f = 0, which implies
f = 0 since li. - A01 co. Therefore, Al - A has trivial kernel, so it
is
invertible.
Thus we have shown that the eigenvalue of A satisfying IAI > InAI are isolated points of finite multiplicity. Let E > 0 be given. There are only finitely
many eigenvalues A of A such that IAl > InAI + e; we denote them by A.1, ... , A.N and let P1.... , PN be the projections on the corresponding eigenspaces. Then i.jPf is compact, and Al - (A - i.;P;) is invertible for IAI > InAI + E, that is,
spI A - YAPjl
ji.;lAl
InAI+E},
hence
-y_AP,
InAI + E,
which proves the required inequality. In the case where A is not self-adjoint, we can consider the operator on L,(M, E e F) = L2(M, E) ® L2(M, F) defined by A(e, f) = (A *f. Ae). Here
E 8 F denotes the orthogonal direct sum of the bundles E and F, and L2(M, E) (BL2(M, F) is the orthogonal direct sum of the Hilbert spaces L2(M, E) and L2(M, F). It is not hard to verify that 112 11 = 11 All and InAI = InAI, and A is a self-adjoint p.d.o. Further, A can be represented in block operator form by the matrix
0/ (0A A' and in view of what we proved above there is a matrix K of compact operators such that the operator
K,l
(A + K2,
A*+K,2 K22
has norm < InAl + E. Since the norm of each entry in such a matrix is dominated by the norm of the operator represented by the matrix, we have 11 A + K2111 < I nAl + e, and the lemma is proved.
Boundary value problems for elliptic systems
346
We denote the closure of OCIS°(E. F) in the L2 operator norm by
Y=Y(E,F). Clearly 5'z) K, the set of compact operators L2(M, E) -i L2(M, F). In fact, operators of finite rank Pf = Y_ (f, gj)Ehj, where gj e C'(M, E), h, e C '(M, F), are dense in K, and such operators belong to OCIS°(E, F) because they are operators of order -oo. Since we can identify Symbl°(E, F) with C'°(ST*M, Hom(n*'E, n*'F)) by restriction to ST*M, we have a principal symbol map it =
OCIS°(E, F) - C°°(ST*M, Hom(n*'E, n* IF))
(78)
which is surjective by Theorem 8.48. By virtue of Lemma 8.80, the map it induces an isometry n: C(ST*M, Hom(n* IE, 7r* IF)), which necessarily has closed image because ?/.7Y is a Banach space (i.e. a complete space). Further, the image of n contains C'(ST*M, Hom(n*'E, n* IF)) and these sections are dense in the continuous sections by Lemma 8.63, so it is in fact surjective. In other words, the principal symbol map (78) extends to a map
n: 9(E, F) - C(ST*M, Hom(n*'E, n* IF))
(79)
which is surjective, mapping onto the space of continuous sections of Hom(n*'E, n* IF). Theorem 8.81 Let M be a compact manifold, and let 9 = 3(E, F) denote the closure of OCIS°(E, F) in the L2 operator norm. The principal symbol map
(78) extends to a surjective map (79) which has the set JY = K(E, F) of compact operators L2(M, E) -+ L2(M, F) as kernel. The following properties hold:
(i) The induced map n: i/jY - C(ST*M, Hom(n*'E, 7r* IF)) is bijective and an isometry; (ii) Y is closed under composition of operators and taking of adjoints, and the map it is a * homomorphism. Moreover, an operator A e Y(E, F) is Fredholm if and only if nA(x, i;): Ex -+ F,
is invertible for all (x, ) e ST*M. If A e ?(E, F) is Fredholm then it has a regularizer B e _P(F, E), and we have ind A = ind, nA.
Proof The property (i) follows because (77) in Lemma 8.80 carries over to operators A e by taking limits, and it is immediate that nA = 0 if and only if A e Y. Property (ii) is obtained by taking limits in Theorems 8.44 and 8.45. We have already proved in Theorem 8.76 that invertibility of nA on ST*M
is sufficient for A to be Fredholm. In that case there is an operator (F, E) such that aB = (iA)-1 and it follows that n(AB - 1) = 0 and n(BA -1) = 0, hence AB - I e K and BA - 1 e .A, i.e. B is a regularizer Be.
for A.
To prove necessity, let A e Y(E, F) be a Fredholm operator. Then A*A has index zero (since it is self-adjoint), so there is a compact K such that
Pseudo-differential operators on a compact manifold
347
A*A + K is invertible, thus, II(A*A + K)u JJ % cliull,
c > 0.
(80)
The conclusion (48) in Corollary 8.52 carries over to operators in _60, so by applying it to the operator A*A and using (ii) we obtain II A*A(fke)ll
I nA*(xo, So) o nA(xo, so)e(xo)I
as k -- oo.
Also fke -+ 0 weakly in L2(M, E), thus K(fke) - 0 in L2(M, F), and by substituting u = fke in (80) and taking the limit as k -+ oo we find that JiA*(xo, o) o irA(xo, o)e(xo)I > c > 0 for all e(xo) a
of length 1. Therefore, irA(xo, So) is injective and an identical
argument shows that tr(A*) = knA is injective at any (xo, l;o) E T*M\O, so nA has invertible values.
Let Y,"(E, F) denote the closure of OCIS"(E, F) in the operator norm W2'(M, E) -+ W2-"'(M, F). In Theorem 8.82 we will show that the index of A E OCIS'" is independent of the homotopy class of irAlsr.M. By the usual device (see the proof of Theorem 8.76) it suffices to consider the class
Y='q o, i.e. the closure of OCIS°(E, F) in the L2 operator norm. Let a, e Symbl°(E, F), 0 0 it is evident that A has zero kernel and its image consists of all f e L2(T) with kth Fourier
coefficient equal to zero for k = 0, ... , in - 1. Hence ind A = 0 - in = - in. On the other hand, if in < 0, then the image of A is all of L2(T) and its kernel consists of all it ck ell a L2(T) with Fourier coefficients satisfying the conditions
ck-m+ck=0fork=m,...,-1.
ck=0fork>, -mork<m, Hence ker A has dimension
Imp
and im A has codimension
0, so
ind A = jm( - 0 = -in. In any case we have ind A = -m as was to be shown. The next theorem is the matrix version of Theorem 8.83. Theorem 8.84 Let A e OCIS°(T, p x p) with principal symbol iA(z, i;) equal to a+(z), S > 0. and equal to a_(z), < 0, and suppose that det at(_) 54 0 for all z. Then A: L2(T, CD) L2(T, C') is a Fredholm operator with index
ind A= - n [arg det a+(e'x)]T + 2n [arg det a Proof We can assume that
since this operator has the same principal symbol as the given A. In view of Proposition 15.5 there exist homotopies a'+': T - GL,(C) and a+': T -+ GLp(C), 0 < t < 1, such that
at' = a= and at' _
Pseudo-differential operators on a compact manifold Then A
351
a() - P + at) - (I - P) is a homotopy of Fredhoim operators with B 1
A(0) = A and A(') _ 1/
where B denotes the 1 x I operator det a+ - P + det a _ (1 - P). Now by Theorem 8.82
ind A = ind A") = ind B and then Theorem 8.83 finishes the proof.
Remark Theorem 8.82 is an important tool for calculating the index, and it was proved using the closure, 9, of OC1S ° in the L2 operator norm. However, this theorem can be proved by other means as we show below in Remark 8.88.
Our aim now is to extend the definition of the symbol index, finds a, to the case where a is any continuous section T*M - Hom(s*'E, 7r* 'F) (not necessarily homogeneous) that is invertible outside a compact subset of T *M. To do so, however, it is necessary to allow a wider class of homotopies of elliptic operators than was considered in Theorem 8.82.
But first we need a few preliminary definitions. By definition of the manifold structure on T*M, a e C°°(T*M)
if and only if for every coordinate chart (U, K) on M, the push-forward function ac = t(a, K), see (17), belongs to C°°(K(U) x R"). Then for each compact set K C K(U) x U8", consider the semi-norms sup (Y..fl
IDgD,,.aK(y, n)I,
d = 0, 1, 2, .. .
IC
lal+IBlsd
(derivatives in local coordinates of the push-forward function). If we let (U, K)
vary over all charts, and K over all compact sets in K(U) x R", we get a system of semi-norms defining a topology on C°°(T*M). A sequence a" a C°°(T*M) is convergent if, for each chart (U, K) on M, all derivatives of the push-forward functions converge on compact subsets
of K(U) x R". The limit function a = lim".,. a" will then also be in C°°(T*M). By using a locally finite partition of unity on T*M, one could write down a countable family of separating semi-norms on CI(T*M), so the topology is metrizable and hence a Frechet space topology. For the case of C00(ST*M), it would suffice to use a partition of unity on M due to the compactness of ST*M.
Boundary value problems for elliptic systems
352
The semi-norms in S' = SM(R" x R") can also be transported to the manifold: for an arbitrary chart (U, K) and compact set K C K(U) we have the semi-norm Ila Ilx a =
sup
g)I'(1 + Igl)l
,"
d = 0, 1, 2,...
(83)
ial+Idl"
where aK = t(a, K), and this system of semi-norms defines a topology on Sr(T*M), with continuous inclusion S'"(T*M) c' C°°(T*M). Once again, using a partition of unity on M we obtain a countable collection of separating semi-norms for Sn(T*M), and the topology on Sn(T*M) is a Frechet space topology.
A subset C. c S' is said to be bounded if each semi-norm in Sr has bounded values on It is not hard to see by Ascoli's theorem that, on bounded sets in S', the following topologies are identical: the C'° topology, the topology of pointwise convergence, and the S" topology for any m' > in. Definition 8.85 A sequence a" a Sm(T*M), n = 1, 2, ... is said to converge weakly as n - oo if (i) {a")M'= I is a bounded set in SM(T*M), i.e. each semi-norm is bounded on this set;
(ii) {a"} 1 converges in C"(T*M). (As remarked above, condition (1) implies that (ii) is equivalent to pointwise convergence.)
It
is clear that if a" is weakly convergent then the limit function
a(x,1;) = lim" - " a" is also in S(T*M). If Z is any topological space, then we say that a map S' -+ Z is weakly
continuous if the restriction to any bounded set in S' is continuous in the C" topology. In the proof of Theorem 8.23 we defined a right inverse to the main symbol isomorphism
Sn(T*M) a a t--+ A =
if;(ai(x, D))K, ,+/i; a OSM(M),
(84)
where aj is the push-forward of a to Kj(Uj) x 18", see (17), and it defines a continuous map SM(T*M) &(W2, W2-'") when S' has the semi-norm topology and £°(W2, W2-') is given the operator-norm topology; see (86) below.
Lemma 8.86 Let at - a weakly in S' = SM(R" x 18") as j -+ oc. Then we have a,(x, D)u -+ a(x, D)u
in W2-'"(P8")
(85)
for every u e W2(R").
Proof The operator norm of a(x, D): W2 W;-" is bounded by a seminorm of a in S' (see the remark following Theorem 7.28), i.e. Ila(x,
constlalk hIlullt,
for some k,
(86)
Pseudo-differential operators on a compact manifold
353
where the constant depends only on 1. Consequently, the operator norm of aj(x, D) is bounded by some constant, independently of j, so it suffices to verify (85) on the dense subset u c -Y. In fact, we will show that if u e .5" then aj(x, D)u -+ a(x, D)u also in .5o as j -+ oo. Choose a function X E Co (l ") equal to I in a neighbourhood of the origin and, for b e S'", set b'(x, ) = X(x/v)X(S/v)b(x, S),
v >, 1.
By Lemma 7.3 and the estimates for the semi-norms in Theorem 7.12, it follows that the symbols {b`},,, form a bounded set in S', and b' - b in
S"' as v
co. In fact, more is true: since any semi-norm of b" - b in Sm+' is bounded by a semi-norm of b in Sm and a semi-norm of X(xJv)X(l;/v) - 1
in S', it follows that if v is a bounded set in S' then b' -+ b in S` as v - m, uniform in b e G. Hence b'(x, D)u - b(x, D)u in J', uniform in b e C (see the estimates in the proof of Proposition 7.2). To be precise, this means that if V is any neighbourhood of 0 in .9' then there for some vo it follows that b'(x, D)u - b(x, D)u e V for all v >, vo.
Now, if aj - a weakly in S" then we write aj(x, D)u - a(x, D)u = (aj(x, D)u - of (x, D)u) + (a' (x, D) - a'(x, D))u
+ (a'(x, D)u - a(x, D)u).
(87)
Since {aj} is a bounded set in S', then, for any neighbourhood V of 0 in ,9', there is some v such that aj(x, D)u - a,(x, D)u E ;V for all j and a(x, D)u - a'(x, D)u e V. It remains to show that with this (fixed) v, the middle term on the right-hand side of (87) lies in IV for sufficiently large j. Now, since aj -+ a on compact subsets of R2" as j -+ oe and aj - a' is equal to aj - a multiplied by X(.r/v)X(S/v), a function with compact support, it
follows that a -+ a' in S"'. Hence aj(x, D)u -+ a'(x, D)u in .9' as j -+ oo, that is, aj(x, D)u - a"(x, D)u a jV for all j 3 jo. We conclude that aj(x, D)u - a(x, D)u e V for all j >,jo. Hence aj(x, D)u -+ a(x, D)u in .51 as 1--'cc.
Lemma 8.86 implies that the map (84) is weakly continuous when .(W2, W2-') is given the strong operator topology. For this reason we can use Theorem 9.13 to prove the following theorem, which we take from [Ho 3].
Theorem 8.87 Let I be the interval CO, 1] in R. Let a: I -+ C" (T*M) and 4: I -, Cx(T*M) be continuous maps such that, for t e 1, a(t) is bounded in S'", G(t) is bounded in S-1", a(t)G(t) - 1 is bounded in S', and G(t)a(t) - 1 is bounded in S. If Ao, A, e OS"(M) have main symbols a(0) and a(l), respectively, then
ind Ao = ind A,. Proof By composing t u-+ a(t) with the map (84) we obtain a continuous map I a t f--+ A(t) e.(WW(M), W2'-'"(M))
Boundary value problems for elliptic systems
354
with the strong operator topology, and similarly for t F- B(t). This verifies the first part of the hypotheses in Theorem 9.13. Choose a covering of M by coordinate patches (Uj, Kj) which is so fine that Uj u Uk is contained in another coordinate patch (Ujk, Kjk) whenever Uj n Uk # 0 (see Exercise 16). By virtue of (84), the operators A(t) _ Y,Oj(aj(t, x, D))Kj-,t&j and
B(t) _ E Oj(bj(t, x, D))K,_,Oj,
have principal symbols a(t) and e(t), respectively. We know that A(t)B(t) - I is of order -1 (because it has main symbol a(t)6(t) - 1 e S-'), and we wish to show that its norm as an operator WZ -+ WW+' is uniformly bounded in t (for any 1). Analogous to (84), let P(t) = 1 Pj+*/k((ajtbjk)(t, x,
an operator with principal symbol a(t)G(t), where a jk is the push-forward of a to Kjk(Ujk) x 18", see (17), and similarly for bjk. Then P(t) - I = Y_ Ij4,k((ajkbjk - 1)(t, x, D))K;kcjOk
(88)
is uniformly bounded in t as an operator WZ - W2"' by the hypothesis on a(t)G(t) - 1, so it remains to verify the same for the operator A(t)B(t) - P(t). We have
A(t)B(t) _
_
O (aj(t, x, D))K, ' ,' k(bk(t, x,
Ot
,jik(aj(t, x, D))K,-,(bk(t, x, D)). ,z YjOk + T1(t)
_ Y Oj'Yk(ajk(t, x, D) °bjk(t, x, D))K;kOjIk + T2(t) + T1(t),
where the operator T, (t) is of order -1 and contains terms involving the commutators ['k , (aj(t, x, D))K,-,] a OS'"-'
and
(bk(t, x, D)), . ] e OS-'-1,
multiplied by (bk(x, t, D)),,,,, e OS-' and (dj(x, t, D)),,,-, e OS'", respectively,
while T2(t) is also of order -1, and contains terms that arise from the asymptotic expansion in Theorem 7.17. If one looks carefully at these terms and the definition of the Sobolev norm on M it is clear that the boundedness
assumption on the main symbols a(t) and 1(t) implies that the norms of T1(t) and T2(t) as operators WZ -+ W2+1 are uniformly bounded in t. Finally, by applying Theorem 7.12 to the composite operator a jk(t, x, D) o bjk(t, x, D) we get A(t)B(t) = P(t) + T3(t) + T2(t) + T1(t) (89)
where T3(t) is also of order -I with uniformly bounded Sobolev norm. In view of (88) and (89) we conclude that the norm of A(t)B(t) - I as an operator WW -+ Wz*' is uniformly bounded in t. Similarly for B(t)A(t) -1. All the hypotheses of Theorem 9.13 have been verified, and the proof is complete since by Theorem 8.53 we have ind AO = ind A(0) = ind A(1) = ind A,.
Pseudo-differential operators on a compact manifold
355
The preceding definitions of the semi-norm topologies on Cz>(T*M) and Sn(T*M) extend to matrix functions, C'°(T*M, p x p) and SM(T*M, p x p) in the obvious way, and then we define weak convergence of symbols just as in Definition 8.85. Moreover, since the definitions are stated locally we
can even extend them to Sr(T*M, Hom(nc'E, n;'F)) if we first take push-forwards $Fa(IJE)-' of the symbols under local trivializations of the bundles E and F in order to get matrix methods. As before, we say that a set S c Srn(T*M, Hom(n*'E, n*'F)) is bounded if each semi-norm is bounded on G. Theorem 8.87 may now be generalized as follows.
Theorem 8.87' Let a: I 6:1
C'((T*M, Hom(nc'E, ir* 'F)) and
C'(T*M, Hom(n*'F, n,'E))
be continuous maps such that, for t e I, a(t) is bounded in S', 6(t) is bounded
in S-", a(t)6(t) - I is bounded in S', and 6(t).a(t) - 1 is bounded in S'. If A0, A, a OSM(E, F) have main symbols a(0) and a(1), respectively, then ind AD = ind A,
Proof Let {(U1, xj)) be an atlas on M for which there exist trivializations
#j:FU, -Uj xCP and as before choose functions j e Co (Uj) with OJ = 1. If a a /3j:Ev,
Sm(T*M, Hom) then let a-j be the push-forward of the matrix function #j afl! -' to Kj(Uj) x R", i.e. see (17) and (37), '(Y))-1) Pj -,, which is a p x p matrix function with entries in S a(Kj(U1) x R"). We may then define a right inverse to the main symbol isomorphism, 4, by aj(Y, h) = Qj a(Ii
'(uj(y.D))",.Pj"ojaOSm(E,F) (90) The rest of the proof is essentially the same as for Theorem 8.87.
Remark 8.88 In view of (86) and the semi-norms (83), the map (90) is continuous when S' has the semi-norm topology and 2'(W2, W2-') has the operator-norm topology. This leads to an alternative proof of Theorem 8.82 as follows: If nA and nB are homotopic then by Exercise 14 it can be shown that they are C' homotopic (cf. proof of Lemma 10.22), i.e. there exists a
C' map
h: ST*M x I - Hom(n*'E, n* 'F) with do = nAlsr.M and A, = xBIST.M Now extend d by homogeneity so that
it becomes a smooth map T*M\0 x I - Hom(irs'E, n; 'F), and let y be a cut-off function as in Theorem 8.47. Then a, = y f, defines a continuous map a: I - Sn(T*M, Hom) with the semi-norm topology. By composition of this map with (90) we obtain a continuous map I -+ £°(W2, WI-"') with the operator norm topology; denoting its values by t i - A(t), we obtain by
Boundary value problems for elliptic systems
356
Theorem 9.12 that ind A = ind A(0) = ind A(1) = ind B, as required.
The next theorem shows that in the context of determining the index of elliptic operators, it suffices to consider operators which are (-1) classic. This is where we need the full strength of Theorem 8.87'.
Theorem 8.89 Let A e OS'(E, F) have main symbol a such that a(x, S) is invertible when ICI > C. Then any operator with main symbol a(x, elliptic and has the same index as A when r > C.
is
Proof It suffices to prove the statement when C is an arbitrarily small positive number C, by virtue of the homotopy a(x, re), 1 C. For 0 E 2, whence
Pseudo-differential operators on a compact manifold
357
A, has main symbol a(x, S/ICI) (see the comments following Definition 8.21).
The proof is now complete when m = 0. For the case m :A 0, we choose a p.d.o. A-` c- OS-'"(M) with principal symbol I, by Lemma 8.68, the index of A- m is zero, so A -A-' has the same index as A. In view of Theorem 8.24(a) the operator A o A-` e OS° has main symbol a(x, l) Ill -'", so it has the same index as any p.d.o. with main symbol a(x,
r5 -'" ISI
- a(x, rS/I5I)r-"'
for some sufficiently large r. This completes the proof.
This result enables us to make the final extension of the symbol index function.
Theorem 8.90 There is a unique function, ind defined on all
a e C(T*M, Hom(ir*'E, it* 'F)) such that a(x,1;) is invertible outside a compact set in T*M, and such that
(i) ind, a = ind A if A e OCIS°(E, F) is a (-1) classic elliptic operator; (ii) ind, a, is independent of t if a, is a continuous function oft e 1 with values
in C(T*M, Hom(ir*'E, n,'F)) such that a,(x, S) is invertible except when (x, S) is in a compact set.
Proof We have already defined ind, a when a e C(T*M, Hom) is homogeneous of order 0. Now, even if a is not homogeneous, there is a constant C such that det a(x, ) # 0 when ICI >, C, so we define
ind, a = ind, a', where a'(x, l;) = a(x, rr;/IfI) is a homogeneous symbol of order 0. The definition is independent of the choice of r > C. When a is homogeneous of order 0 we have a' = a, so the property (i) holds. The property (ii) follows by Theorem 8.87'. The uniqueness of the index function, ind is a consequence of the homotopy used in the proof of Theorem 8.89.
Remark In particular, the definition of ind, does not depend on the choice of Riemannian metric on T*M. We review some basic facts about tensor products of finite dimensional vector spaces, then conclude with a multiplicative property of the index. Let
E and F be finite dimensional vector spaces over C. The tensor product E ® F is characterized up to unique isomorphism by the following universal
property: There is a bilinear map q : E x F -+ E 0 F such that for any bilinear map 4i: E x F - G there exists a unique linear map ' : E 0 F -, G
Boundary value problems for elliptic systems
358
such that the diagram
E®F
ExF - G commutes. We write x ® y = ip(x, y) for x e E, y e F. It follows from the universal property that if {ei} and 11f j) are bases of E and of F, respectively, then (e1 O f} is a basis of E ® F. If E and F have inner products then there is a unique inner product on E ® F such that (x ® y, x' O y') = (x, r )E' (Y, Y'')F-
Then {e, ®fj} is an orthonormal basis of E OF if {e,} and {f} are orthonormal bases of E and F. For A e 2'(E,, F,) and B e 2'(E2, F2) we define A 0 B to be the unique linear map in .Y'(E, ® E2, F, 0 F2) such that (A 0 B)(x ® y) = Ax ® By. Lemma 8.91 Let E,, E2. F, and F2 be finite dimensional vector spaces endowed with Hermitian inner products, and let S e 2'(E,, F,) and T e ..P(E2, F2), and define
S#T:S®I
:1:*)e .(E10E2®F1OF2,F1®E2Ei®F2)
in block operator form. We have
ker(S # T) = ker S ® ker T@ ker S* ® ker T*,
(92)
ker(S # T)* = ker S * 0 ker T ® ker S ® ker T*.
(93)
Hence if S and T are isomorphisms then so is S # T.
Proof First of all, note that the adjoint (S # T)* is taken relative to the natural Hermitian inner products on E, ® E2 ® F, 0 F2 and F, 0 E2 ® E, 0 F. in which the summands are orthogonal, and is of the same form with S replaced by S* and T by -T. Let Z = S # T, then it follows that *
ZZ
_
S*®1 1®T* S®1 -I®T* 1®T S®I I®T S*01
_ S*S®1+1®T*T 0
0
1®TT* + SS* ®1
(94)
Let {ej} be an orthonormal basis of E, of eigenvectors of S*S with eigenvalues ii >, 0. Let { f }, fee), {f} and y j, ij, pj be the corresponding objects for SS*, T *T and TT*, respectively. Let e,j = (e, ® e;, 0) and f j = (0, f 0 f;); then the set (e,j, fki) is an orthonormal basis of E, ® E2 ® F, 0 F2 and Z*Zeij = ()i + ).;)eij,
Z*Zfij = (pi + f;j)f.,.
Pseudo-differential operators on a compact manifold
359
Since all the eigenvalues are non-negative it follows that (A; + A )ei1= 0 if and only if A, = A j' = 0 and (p + µ;)f,1= 0 if and only if pf = u = 0, and then it is easy to see that ker Z = ker S ® ker T ® ker S* ® ker T*. Replacing S
by S* and T by -T we also get the equation for ker Z*. Finally, if S and T are invertible then we obtain ker Z = 0 and ker Z* = 0, so Z is also invertible.
Theorem 8.92 Let M and N be compact manifolds, and let E,, F, be vector bundles over M and E2, F2 vector bundles over N. Let A e OC1SM(E,, F,) and B e OCISm(E2, F2) be elliptic operators of order m > 0, and define the operator
A#B=
A®I -1(DB* C`°(MxN,El I®B A * } El 0
E26) F1 21 F2)
F2)
Then A # B is a Fredholm operator, and
ind(A # B) = ind A ind B.
Proof Applying Theorem 8.79 to each of the four operators in A # B, it follows that E2(D
x
x
with n(A # B) = nA # iB. Now, by Lemma 8.91, the linear map (nA # nB)(x, y, ia, r) = irA(x, ) # nB(y, rl) is an isomorphism for each (x, y, , q) a T*(M x N)\0, thus, ker(A # B) c C°° and ker(A # B)* c C°° are finite dimensional spaces by Corollary 8.77. The same calculation which gave (94) also gives
(A # B)*(A # B) =
CA*A ®1 + 1®B*B
0
0
10 BB*+AA*®I
and a similar formula for (A # B)(A # B)* with A*A replaced by AA* and so on. It follows that (92), (93) remain true, so ind(A # B) = ind A -ind B. Corollary &93 Suppose that
a e C(T*M, Hom(n*'E,, n* 'F,)),
e c- C(T*M, Hom(n*'E2, x* 'F2))
are isomorphisms outside compact subsets K, c M and K2 c N. Then
a# d e C(T*(M x N), Hom(n*'E,
7r* 'E2
Q) n 'F1 Ox n* 1F2, n*'F, Ox n*'E2 ®n* 2E, 0 * 1Fni))
is an isomorphism outside K, x K2, and
ind3(a # t) = ind, a finds G.
(95)
Boundary value problems for elliptic systems
360
Proof The fact that a # 6 is an isomorphism outside K, x K2 follows from Lemma 8.91. To prove (95) we may assume that a and 6 are homogeneous of degree I and C' outside the zero section, for a and 6 are homotopic to such functions. Then a # G is also homogeneous of degree I and continuous outside the zero section, but will not in general be C'° when S = 0 or p = 0. If we choose classic pseudo-differential operators A and B with principal symbols a and 6 then it follows that ind,(a # 6) = ind(A # B) = ind A ind B = ind, a - ind, X. Exercises 1.
The Cauchy principal value of 1/x is the distribution defined by Cpv
1,
X
q) = lim c-o
fi(x) dx f1X1_>r
X
Show that pv(1/x) a ,f'(8R). 2.
Show that 1(x ± i0) a 9'(UR) where (
I
XfiO'
V
) := lim
£-o
//
f- ox) _x,X±iW
q, a Y(R)
dx,
Also show that I/(x ± iO) = pv(l/x) + ia8, where S is the Dirac distribution.
Show that the Fourier transform of i/2n(x + iO) is equal to the Heaviside function 0 (where 0(e) = I when S > 0, and = 0 when < 0). Hint: Apply the residue theorem to evaluate the limit 3.
1
lim
R-m
a- 'x4
-R
X+IE
dx.
Show that the Fourier transform of pv(l/x) is equal to - ni sgn S, where sgn i; =l if > O and sgn = -1 if < 0. 4.
5.
(a) If Q e C'(T, p x p) show that I
d(det Q) = tr(Q - ' dQ)
det Q
denotes the trace where the operator d is exterior differentiation and of a matrix. (b) Show that the index formula of Theorem 8.84 can be written in the form
ind A = -2n J tr(a T
da+) + 2n J tr(a-` da-) T
Pseudo-differential operators on a compact manifold
361
Demonstrate that the one-dimensional singular integral operator on a smooth closed (compact) curve r a R2, defined by 6.
K(t, T)
Au(t) - a(t)u(t) + lim
e..,0
1$-s)>,
t-T
u(T), dT,
t, ,r E r,
for a(t) e C°°(r) and K(t, T) a C°°(r x r), belongs to the class OS°(r). Also show that the operator is (-1) classic and compute the principal part. 7. Prove that f o r any sequence ad E S'""(T *M), d = 1, 2, ... with md_ 1 > and - - oo as d oo, there exists a e S"(T*M) such that a - Y-a 1 ad, that is,
r-1
a - E ad a S"''(T*M)
for any integer r > 2.
d=1
The function a is uniquely determined modulo S-°°(T*M). Hint: See Proposition 8.16 and recall that every function in S(T*M) is the main symbol of an operator in OSm(M). 8. Show that A e OSm(M) cannot be elliptic when it is regarded as an element of OS"(M) for m1 > m.
Let A e OSm(E, F) be an elliptic operator. Suppose that ind A = 0 and ker A = 0. Show that A: C'(M, E) -+ C°°(M, F) is bijective and 9.
A-1 a OS-'(F, E). Further, if A e OCISm(E, F) then A
a OC1S-'(F, E).
Hint: See Lemma 8.59. 10. Prove Girding's inequality: Let A e OSm(M), m e R, and suppose that Re A(x, ) > 0 for large ICI. Then for any e > 0, there is a constant C = C(e) such that
Re(Au, u)1,2(m) + elluII 12 >
Cllul1(2m-1)i2
for all u e C`°(M). Hint: Choose AA' e OCIS'k(M), k e R, with principal part such that Ak: WW(M) -+ WW-'(M) is an isomorphism for all I e R. Then replacing A by A-n,/' o A o A-1°/2, we may assume without loss of generality that m = 0. The rest of the proof is just as in the proof of Girding's inequality in R", Theorem 7.38. 11. Formulate, and prove, Girding's inequality for A e OSm(E, F), where E and F are Hermitian bundles.
Let P e OSm(M), m > 0. Suppose P is self-adjoint and that P has a main symbol A with 12.
Re,A(x, 4) > C141'",
141 large.
Show that there exists a (formally) self-adjoint P.. a OSm(M) such that
P+-PeOS-oD and P+>cI>0. 13.
Let A e OSm(M) be elliptic and in > 0. If A is formally self-adjoint, i.e. (Au, v)M = (u, Av)M,
u, v e C(M),
show that ±ii + A: W? (M) -+ L2(M) are isomorphisms.
Boundary value problems for elliptic systems
362
14. Let M be a compact manifold and E a Hermitian bundle over M. Show that C'°(M, E) is dense in the Banach space, C(M, E), of continuous sections
f: M -+ E with the supremum norm, IfI = sup:EMllf(x)II Hint: Using a partition of unity one can reduce the proof to the case where f e C0 '(R", CP). 15. Let (X, d) be a compact metric space and (U,) an open covering of X. Show that there exists a positive number ). (called a Lebesgue number of the covering) such that each ball B(x, A) is contained in at least one U. 16. Let M be a compact manifold. Show that it is possible to choose a covering of M by coordinate patches {Uj} which is so fine that U, U Uk is also contained in some coordinate patch whenever Uj n Uk # 0. Hint: The topology on M is metrizable, hence one can use Exercise 15. 17.
Let M and N be compact manifolds, and E, F vector bundles over M
and G a vector bundle over N. Prove that finite sums Y fj ® gj with fj e C'°(M, E), gj e C'°(N, G) are dense in C'(M x N, E El G). Hint: If a function has compact support in {x; Ix;l < n, i = 1, ... , n} then it can be viewed as a function on the torus T" for which a multiple Fourier series argument can be applied. 18. If A is a continuous linear map C'(M, E) - C°°(M, F) prove that there is a unique continuous linear map
A':C'(M x N,El G)-.C`°(M x N,Fpx G) such that
A'(f-g) = Af -g,
f e C'(M, E), g e C'O(N, G).
Let rn > 0 be an integer. Show that if A has a continuous extension A,: WZ WI'"' for all integers I then A' has a continuous extension A;: WZ -i Wv-' too, and (96)
IIA;II < const max IIASII
for any integer l (the constant depends on 1, of course). Hint: To verify (96), divide the proof into three cases: I > m, I 0, and 0 < I < m. If I > m, then in local coordinates we have if Ixl + I J3I I - In, u e C0 "(R"+"), ), and if we integrate first with respect to x with fixed y:
ffR" x R^
DxDy A'ulz dx dy = J I
dy fR. IDA'DeuI2 dx
C,,II dy
f
Y
IDxDY
dx
1,14121+m
C, $j.
>
ID=D,uIZ dx dy,
where the constant C, depends only on the norms that is, IIA'uIIt-, < of A: Wz(R") -+ WW-"'(R") for 0 < s < 1. Passing to adjoints gives the
Pseudo-differential operators on a compact manifold
363
corresponding estimate for l < 0. The estimates for 0 < I < m can be handled easily by using the Fourier norm in the Sobolev spaces. 19.
Prove that if both conditions hold in (49), then dim E = dim F (i.e. E
and F have the same fibre dimension). Hint: see Theorem 8.94 in the addendum below. Formulate, and prove, versions of Theorems 8.64, 8.65 and 8.66 for any operator A E OSm(E, F), i.e. when A is not assumed to be a classic operator. 20.
Addendum The proof of (iii) ca (ii) in Theorem 8.64 can be generalized to the case where
the operator A is not assumed to be a classic operator. Theorem 8.94 Let A E OSm(E, F). The following are equivalent:
(1) If a E Sm(T*M, Hom(tr*'E, n*'F)) is a main symbol of A then a has a left inverse G e S-'(T*M, Hom(ir*'F, n* 'E)), i.e.
IE E S'. (2) For each point x0 E M, let (U, K) be a coordinate patch on which there exist trivializations Xu: Ev - U x C' and XU: Fu - U x C9. Then lim
inf
inf Ia.(y, I)vl/(l + Jill)' > 0
IM -. X IVI m I Ye K(W)
where dK E S;C(K(U) x R", q x p) is a complete symbol of the matrix operator A. on K(U) (see (20)) and W is an open set such that x0 e W e U.
Proof To simplify the notation, we assume that E and F are the trivial bundles M x C° and M x CQ, respectively. This does not alter the proof in any significant way; we leave the details for the reader.
(1) = (2) We have a - aK E
'(T*U, Hom) by Definition 8.21, and
multiplying by L gives G(a - aK) E S- (T*U, Hom), whence I - e aK E S10 (T*U, Hom).
Choosing tp a C, '(U) such that 0 5 (p } and a(A2) - s is the dimension of the kernel of this fl(A1) x 8(A2) matrix. If one proceeds similarly with the equation A*z = ArA2*z = 0, one deduces that it has a(A') = a(A2*) + (a(A!) - s) linearly independent solutions (where s is the rank of the transpose of the same matrix as before). Therefore we have ind A = a(A) - a(A*)
=a(A1)+a(A2)-s-a(At)-a(AZ)+s = ind A, + ind A2 Finally, to prove part (b) of the theorem we factor A, ®A2 = (Ir1®A2) o (A, a IX,),
370
Boundary value problems for elliptic systems
and observe that ind(I1, (D A2) = ind A. and ind(A1 ® IX) = ind A1, and then apply part (a).
Corollary 9.9 If A e QI(X. Y) is a Fredholm operator and S1 a L(Y, Z), S. e L(V, X) are isomorphisms, then S1 o A o S2: V - Z is again a Fredholm operator with the same index.
In fact, because an isomorphism is Fredholm with index 0, the corollary follows from part (a) of Theorem 9.8.
Theorem 9.10 If A e ID(X, Y) is a Fredholm operator and if R is a left regularizer for A, then R is also a Fredholm operator, R e (b( Y, X), and we have
indR= -indA By duality the same holds for right regularizers.
Proof We have dim ker R o A = dim ker(IX + f) < oo, and from linear algebra dim ker R (X, Y) is again a Fredholm operator and
ind(A + S) = ind A.
'(X, Y) is an open set in T(X, Y), and homotopically stable in the operator-norm topology. It follows that
that the index is
Proof We know from Theorem 9.6(c) that for any A e 1(X, Y) there exists a regularizer R which is simultaneously a right and left regularizer. Assuming IISII'IIRII < 1, the Neumann series for (lx+RoS) and (I> + S c R) converge, giving bounded linear operators. It is easily shown that R,=(Ix+RoS)-'cR
is a left regularizer and
R,=Rc(Ir+SoR)-'
is a right regularizer for A + S, so that A + S c- 4)(X, Y). Since the equations
R,y = 0 and Ry = 0 are equivalent we have a(RI) = a(R), and passing to duals we get in the same manner /3(R,) = a(R!) = a(R') = fl(R). We apply Theorem 9.10 and obtain ind(A + S) = -ind R, = -ind R = ind A. To prove the last assertion of our theorem, consider a continuous path {A,; 0 < t < 1) in V(X, Y). Since a path, being a continuous image of [0, 1], is connected and compact, we may apply the first part of our theorem finitely often, obtaining thus ind AO = ind A, = ind A 1.
The index has much stronger stability properties than stated in Theorem 9.12; this is the content of the following theorem of Hormander [Ho 3]. Recall that an operator function A, e 2(XI, X2), t e I, is said to be strongly continuous if, for t" - t, the sequence A,. f - A,e (converges in X. for every f e X1. Theorem 9.13 Let X1 and X2 be two Banach spaces and let I be a compact
space (i.e. I = CO, 1]). If A, e.(X1, X2) and R, e.(X2, X1) are strongly continuous as functions of t e I, and if
,f1t=R,A,-11,+(2:=A,R,-12 are uniformly compact in the sense that for j = 1, 2
Mj= {,(j,f;te1, f eXj,llfII,
1)
372
Boundary value problems for elliptic systems
is precompact, in XX, then A, and R, are Fredholm operators, dim ker A, and
dim ker R, are upper semi-continuous and ind A, = -ind R, is locally constant, hence constant if I is connected.
Remark 9.14 The hypotheses of Theorem 9.13 are very natural in the study of elliptic pseudo-differential operators A, on a compact manifold M; the operators R, are their parametrices (see §8.6). For the proof we need the classical lemma of F. Riesz, which we state in a form appropriate to our purposes. Lemma 9.15 Let L be a closed subspace of the Banach space X. We have
dim L>k if and only if there exist fj a L, j =
1,
... , k such that
11411 = 1 < llfj
+ E a`f III
for all Proof If the last condition holds the f's must be linearly independent; hence
dim L > k. To prove the necessity, let us suppose that f1, ... , f _ 1 have already been chosen, and denote by L;_ 1 the linear space spanned by these elements. If j - I < k we choose g e L outside Lj_ 1, then there is a point h e Lj_ 1 which minimizes 11g - h11, because Lj_ 1 is finite dimensional (hence
locally compact) and the norm -- oo if h - oo. Then
Ilg - h - E aifll I<j
IIg - hIl
for all a1eC
and fj = (g - h)/ II g - h 1l has the required properties.
Now we prove Theorem 9.13. First of all, we claim that for any finite dimensional space W c XZ the set
Nw= {(t, f)eI x X1; 11f111 = 1,A,fe W} is compact in I x X1. In particular if we let W = (0) it follows that
N={(t,f)eIxX1;11f11,=1,A,f=0}. is compact in I x X1. To show first that N is closed let (t,,, f,) e N and let tp - to and J. - fo By the principle of uniform boundedness of Banach-Steinhaus (here we use strong continuity, see [TL]), we have 11 A,. - A,. 11 < M < oo, whence 11 (A,
- A,0)(ff. - fo)11 < M 11 fe - .fo 11 - 0.
373
Elliptic systems on bounded domains in 68"
Since A,,, J" = 0 and
(A, - Am)(f, - fo) = A,"fn - A,of,, - A,"fo + A,ofo, we obtain by letting n
oo,
0=0-A,ofo-A,ofo+A,ofo Therefore A,o fo = 0, and (to, fo) e N, so N is closed. In the same way if we have (t", f") e Nw then and Nw is also closed.
E Wand it follows that A,o fo e W, so (to, fo) E Nw
The compactness of N now follows immediately from our assumptions.
In fact N is a subset of 1 x M1 because A, f =_P implies f = -1,J e M,; therefore N is closed in the compact space I x M, and is therefore compact
itself. To prove compactness of Nw for any finite dimensional space W requires only a little more proof. When (t, f) e Nµ. we have f = R, - A, f - A 1, J where ( f e M1 and IIA,fll2I< C11f11, = C since A, is strongly continuous in t (by Banach-Steinhaus). The compactness
of Nµ, follows from the fact that the map (t, g) -. R,g is continuous and the set {(t, g) e 1 x W; 1191121< C) is compact.
The hypotheses of Theorem 9.13 imply that A, and R, are Fredholm operators by Theorems 9.6(c) and 9.10. Consider now the set
Ik= {teI.dimkerA,> k}. By applying Lemma 9.15 we get
Ik={te1;3fl,...,fksuch that l
icI Im,
VXEs2,Se I8",
(11)
where c is independent of x and 1;, and m is the degree of the characteristic polynomial X = det nsl. Ellipticity means that the inverse matrix (nd)-'(x, i;) exists for x E S2, 0 # S E QB"; in the proof of Theorems 9.29, 9.32 we shall need an estimate for the entries li.,{x, ) of this matrix. From (9) we obtain S) =
=
ICI'
14/ S-I(151)-
In view of (11) the entries of the matrix nsl-'(x, /Ii;I) are bounded on the compact set Q x S" a (x, /l) by some M < cc, whence the entries l,,(X, s) =
of nsal(x, S) are bounded by MIDI-u,+°j' for all 0 # E R". Hence for = r) with 0 # i;' E I8"-' we have the estimate (12)
which shows that for S' 0 0 there is no singularity at r = 0. In the proof of Theorem 9.32, we will need to work in local coordinates.
For a point lying inside a compact subset of D we use the standard coordinates in R". For points near the boundary, a12, we use admissible coordinates defined as follows. By Theorem 5.73, a tubular neighbourhood of ail in I8" exists and can be identified with ail x (-1, 1). A chart (U', x') on ail gives rise to a chart
(U' x (-1, 1); x,x"),
x'c- R'-', x. c- (- 1, 1),
on the tubular neighbourhood ail x (- 1, 1). Then (x', x") are coordinates on ail x (- 1, 1) such that the vector fields a/ax,,... , a/ax" _ , are tangential to ail, and we can assume that a/ax" is normal to aft (see text preceding Theorem 7A.9). We refer to (x', x") as admissible coordinates; note that x"
is a global coordinate on aft x (- 1, 1), while x,.... , x.-, are local coordinates.
Elliptic systems on bounded domains in l8"
381
Remark For future use, let us also note that the set it =Q U (OQ x (- 1, 0)) is an open set in R" containing D. Let v: Oil - T(il8") be the inward-pointing unit normal along ail. For each x e 00 there is the direct sum decomposition T. R" = Tx(0i2) ®span(vX), and by passing to the dual, we have T,*, R" = [span(vs)]° ® [T=(ail)]°. By with the cotangent space identifying the annihilator subspace T*(c(l) by means of the restriction map S' rxieni we have T*R" = TX(OD) ® [T,,(ail)]°
where S' e Ts(O(l) is cotangent to cQ and Zi" is "conormal" to Oil, i.e. v> = 0 for all v e T,(e). Now let n: Oil -. T*(W!) be the image of v by the index-lowering operator
T(R") - T*(R") that maps O/Oxi to dxi. Because the space of conormal vectors at a point x e Oil is one-dimensional and n(x) # 0 for each x, then every e T*(R") can be written uniquely in the form l;".n(x),
where t' c- T*(CQ), S. E R.
This defines a vector bundle isomorphism T*(R") we are justified in writing for each 1; a T:'R",
(13)
T*(Oil) ® (OQ x R) and
where t' c- T*(Oil), S" a R.
(13')
The definition of proper ellipticity at a point x e Oil on the boundary is
as follows. When we substitute t = (c', i.) = ' + 7, n(x) E T* R" in the characteristic polynomial y(5) = det nsad(x, ), it is to be understood that the canonical isomorphism T*(R") = 11" is taken into account. We may also permit A to be a complex number and then S e TX P" ® C c C". Definition 9.24 Let n 2. The elliptic operator a/(x, D) is proper at x e Oil if for each 0 0 S' e T*(Oil) the polynomial in 1, e C
P(i) =
i.) = det nd(x, (c', i.))
(14)
has as many roots, r, in the upper half-plane Im 1. > 0 as in the lower half plane Im i. < 0, counting multiplicities.
Here y = det tr,sol is the characteristic polynomial (7). Since it polynomial of degree m in (5', i.) then P(i.) = det n %f(x,
is a
a)) = am(x)im + ..,
with a,"(x) independent of S'. By ellipticity with ' = 0, i. = 1 it follows that a",(x) 96 0, x e On, whence m is the degree of P(i.). Also because of ellipticity
there are no roots on the real axis, so if a is properly elliptic we must have m = 2r. Hence m is even and the number r in Definition 9.24 is independent of x e an and i;' # 0. Using the fact that any pair of linearly independent vectors, S,, 2 a R", n > 2, is homotopic to the pair (1, ... , 0), (0, ... , 1) through pairs of linearly
Boundary value problems for elliptic systems
382
independent vectors, we obtain the equivalent definitions of proper ellipticity:
(i) det ird(x, (c', A)) is proper (Definition 9.24);
(ii) If l; S2 E R" are linearly independent, the polynomial in the complex variable A, det nsal(x, , + y2), has as many roots in Im A < 0 as in
ImA>0; (iii) The polynomial det nsif(x, (1, ... , 0) + 2(0,..., 1)) has r roots in Im A < 0 and r in Im A > 0.
For n = 2, either 1;,,1:2 or ,, -b2 is homotopic to the pair (1, 0), (0, 1) and then using (8) with c = -1, we again obtain the equivalence of conditions (i), (ii), and (iii). It is also well known that proper ellipticity differs from ellipticity only in the case n = 2. Remark One could try to generalize proper ellipticity and propose that the number r of roots of P(A) = 0 in the upper half plane Im A > 0 be independent of ' e T,' (BSI) \0. But by homogeneity of det n.sjf(x,1;) with c = -1 we have det nsal(x,
-A)) = (-1)' det nd(x, (c', A)),
x e BSI,
from which it follows that the number, m - r, of roots in the lower half-plane Im A < 0 is equal to
m - r = r. Thus we get proper ellipticity._
Let .4(x, D) be elliptic on Q. We say that s/ is properly elliptic on f2 if condition (ii) holds for all x e S2. Since the domain SI is connected, it suffices
that this condition hold for one point x0 e SI and one pair of linearly independent vectors ,, b2 a R". For instance, we can take x0 e BSI, a single cotangent vector 1;, = 1;'e T(B() and a conormal vector 1;2 = n(xo) as in Definition 9.24. Definition 9.25 If in the definition of DN ellipticity (Definition 9.21') we can
. = sp and t, _ = t p, then the operator of is said to be homogeneously elliptic. Because s, and tj can be replaced by s + constant, = tp = 0. (Another possible choice tj - same constant, we can choose t, _ 0.) is of course s,
put s, =
If t, =
= t = 0 and s,
sp = 1, then writing the homogeneous
operator sad in tie form s+d (x, D) _ Y Aa(x) D" IaI
where the AQ(x) are p x p matrices, we have
irsl(x,1;) = Y Aa(x) I2I=1
and the characteristic polynomial order I p. Often we have I = 2s.
det nd(x, l;) has the (maximal)
Elliptic systems on bounded domains in R"
383
Definition 9.26 If in the definition of DN ellipticity we can put either = s,, or t, = . = t,,, then d is said to be elliptic in the sense of S, = Petrovskii. In the former case one can put s1 = = sp = 0 and have tj >' 0, = tp = 0 and have s; 3 0. while in the latter case one can put t, _ If s1 =
= sp = 0, then formula (9) reduces to c5) = ir.d (.r, S) T(c). Examples (see [Vo 1])
(a) The system Au1 = 0, a'"u1/axi + Au2 = 0, with the 2 x 2 matrix r
e
of
Lam/axi
It
has characteristic polynomial x(S) = 114 It is DN elliptic with
s1 =2,S2=m,t1 =0,t2=2-m and it is homogeneously elliptic if n = 2. (b) The system Gpu1
a9n2
a'u1
o axi
axl
+
apu2
= o
ax2
ax1
with the 2 x 2 matrix Cap/ax;
a'/ax2
-a9/axZ
as/ax,
has the characteristic polynomial X(S) = i;s + Si+9 It is DN elliptic if p + s = r + q = even, with the DN numbers
s1 =P,s2=r,t1 =0,t2=q-p. The system is elliptic in the sense of Petrovskii if either
p=r,q=sandp+siseven
orp=q,r=sandp+riseven.
It is homogeneously elliptic if p = q = r = s. (c) The linearized system of Navier-Stokes
eu;-au4=0,i= 1,2,3, ax;
ant
ax,
+
au2
ax2
+
au3
= 0 axi
is DN elliptic with S1 = S2 = S3 = 1, S4 = 0, t1 = t2 = t3 = 11 t4 = 0,
but not Petrovskii elliptic.
384
Boundary value problems for elliptic systems
Addendum: the marriage lemma
A fundamental question in combinatorics is the "marriage problem". Let
X = [xj] v X r be a matrix of zeros and ones. Suppose that the p rows
correspond to girls g1,.. . , gp and p columns correspond to boys b...... bp. Then xi; = I if gi and b; are willing to marry, and x1 = 0 if not. A full set of p marriages (if it is possible) is called a complete matching. A complete matching means of course that x;,;, = 1 , i = 1, ... , p, for some permutation
j...... jp of the integers 1, ... , p. For example, consider the matrix 1
X=
1
1
1
0 0 0
1
0
1
0
1
0
1
0
0
It is evident that a complete matching does not exist for this matrix because the last three girls like only two boys. In other words, the lack of a complete matching is due to the existence of a 3 x 2 submatrix of zeros, namely, the submatrix consisting of rows 2, 3 and 4 and columns I and 3. One necessary
condition for a complete matching is therefore evident, known as Hall's condition:
There is no r x s submatrix of zeros for which r + s > p (otherwise, there would be r girls who like < p - s < r boys). It is remarkable that this condition is also sufficient for the existence of a complete matching.
Lemma 9.27 Let X = [x,1]c be a matrix of zeros and ones. Hall's condition is necessary and sufficient for existence of a complete matching.
Proof If we have an r x s submatrix of zeros then the r girls corresponding to the rows of this submatrix like no more than p - s boys, and vice versa. Consequently, Hall's condition can be stated another way: Every set of r girls. 1 5 r < p, likes at least r boys
We will prove the sufficiency of this condition by induction on the order p of the matrix. For p = 1, it is obviously sufficient. Now suppose that Hall's condition holds for some p, and that it is sufficient for existence of a complete matching for any matrix of order < p. There are two cases to consider. Case 1. Every set of r girls, r < p, likes more than r boys. Then the first girl can marry anyone she likes, and Hall's condition holds for the remaining p - 1 girls and p - I boys. By the induction hypothesis the remaining p - 1 girls and boys can be completely matched. Case 2. Some set of r girls, r < p, likes exactly r boys. This is the more difficult case. After reordering the rows and columns, the matrix X takes
Elliptic systems on bounded domains in R"
385
the form
X= [A C 0
B
where 0 denotes an r x (p - r) matrix of zeros. The r x r matrix B corresponds to the r girls who like exactly r boys. Within this smaller matrix,
Hall's condition still holds (it applied to every set of girls). Hence, by induction, r marriages can be arranged between the r girls and r boys in B.
To complete the proof we show that Hall's condition holds for the remaining p - r girls and p - r boys, i.e. for the matrix A. Consider any R girls in this remaining subset; we must show that they like at least R of the remaining (unmarried) boys. The trick is to consider these R girls together with the original r. By Hall's condition, this group likes at least r + R boys, and since the r girls liked only the r boys they married, then the R girls necessarily like at least R of the unmarried boys. Hall's condition holds therefore for the matrix A and, by induction, p - r marriages can be arranged between the girls and boys in A. In this way we have obtained a complete matching.
The proof of Lemma 9.27 was taken from the book [Str]. This book contains many interesting insights into the topics of applied mathematics, including a chapter on linear programming. For instance, the following lemma is just the strong duality principle for the "assignment problem". Lemma 9.22 For any p x p matrix a = [au] whose elements are integers, or equal to -oc, there exist integers s,, ... , sP, t,, ... , tP such that a-i
(si+t;)=m si+tj - a;j
where, as before, m denotes the maximum of a,, j, + permutations (j,_., jP) of the numbers (1, ... , p).
+ aP,,, taken over all
Proof Let s, ... , s,,, t1, ... , t,, be integers such that s; + t j >, ai j and Efs, (s1 + t!) = m' (the minimum of such sums). Define a p x p matrix [xrj] of zeros and ones as follows: x1j=1
=0
if s;+tj=a;j if s;+tj>ajj.
We claim that this matrix has a complete matching, i.e. there exists a permutation (j,, ... , jP) such that x,, j, = 1. From this fact the lemma follows at once. Now, to prove the claim, we argue by contradiction. If such a permutation does not exist then by Lemma 9.27 there must be an r x s submatrix of zeros with r + s > p that prevented a matching. Define new integers s;, . . . , sa, t'1,. .. , tP by subtracting 1 from the tj's corresponding to the s columns of this submatrix and adding I to the s; corresponding to the
386
Boundary value problems for elliptic systems
p - r rows not in this submatrix. Then the inequalities si' + tf >, aj are still true, but v
v
I=t
f=1
Y (s,+t;)=
(s;+t;)+(p-r-s) (Si + [j)
a;j and now the matrix [x;j] has the form
[x;j] =
0
1
0
1
1
0
0 0
1
which has a complete matching x, 2 = x21 = x33 = 1. The sum 3
(s;+t;)=18
r=1
is minimized because it is equal to m = x12 + a21 + 233
Elliptic systems on bounded domains in R"
387
9.3 Boundary operators and the L-condition We turn now to the formulation of boundary value problems for a (properly) elliptic operator d, with DN numbers s1,. .. , sP, t,, ... , ti,. As usual, we let
D. = F' a/an where n is the inward pointing unit (co)normal vector field
on 852. Let D.' be the normal trace operator 0120) - W2 K
> 1, from Theorem 7A.10.
Points on the boundary will be denoted by y e M. Let r be the number of roots of the polynomial P(A) = det n.rd(y, (', A)) in the upper half-plane Im A > 0, i.e. half the order of the characteristic polynomial X. In addition
to the p equations, d(x, D)u(x) = f(x), in 52, we consider r boundary conditions P
1=1
k = 1, ... , r,
BJ.(y, D)u/Y) = gk(Y),
(15)
that is, R(y, D) u(y) = g(y), where 9(y, D) is the matrix operator [Bk1(y, D)], x ,. The boundary operators Bk, are taken in the form r
b,- 0,
'a
(22)
It should be noted that there is a slight difference in our treatment here from that in Part I. In Chapters 1 to 4 we studied the connection between the spectrum of L(A) and the equation L(d/dt)w = 0, whereas here - because of the Fourier transformation - we are concerned with L(
d
Jw = 0,
where L(A) = rc.d(Yo, (c', A)).
The solutions of this equation are p-columns of exponential polynomials of
the form Y pj(t) a"j' where the pj's are polynomials in t and the ;.;s are eigenvalues of L(A). The solution space IM = directly into
of (22) decomposes
M=tAt-(Dg+, where ¶W+ consists of all solutions w(t) with lim,-
w(t) = 0. We have
dim9J2=m, dimT?+=r and dimtW=m-r, and any weM+ is a p-column of exponential polynomials such that Im Aj > 0 for all j. It is
evident that if w e TV then w and all its derivatives belong to L2(R+) since le"''I < e-" for some constant c > 0.
Elliptic systems on bounded domains in R°
389
Before continuing on, we make some observations about the weights s ... , s,,, t...... tP in the definition of ellipticity. Note that
max s; > - t;
for all j
(24)
i
Otherwise, there would exist jo such that s; + tjo < 0 for all i, so that A,;o(x, D) = 0, in contradiction to ellipticity. Replacing the weights s, and ti with s, - const and t; + const we may assume that the weights are normalized as follows:
for all i, j,
si 5 0, tj '> 0,
(25)
maxs,=0. Definition 9.28 The pair of operators d(Y, D), -V(y, D).
y e os>!,
is said to fulfil the L-condition f one of the following three statements holds for all y e oft. 0 # ' a T, *(00). 1. The zero initial value problem tts>T
y,
d S',
w(t) = 0,
dt
r !d aR Y, I " i dt
t > 0,
W(t)l,=o = 0,
has in ¶W+ the unique solution w(t) = 0.
II. The initial value problem
xd
Id y'
sMy,
dt))w(t) = 0,
i
t > 0,
dt))w(t)It=o = g,
has for every choice of g e C' a unique solution w in Dl
III. Let
l>, maxs,=0 and I>, maxm,t+q+l(for gsee(21))
(26)
Then the operator
(nd(Y.
('. I i
d )),
n_4(y,
(c',
Ii dt) d) t=o)
.W-l-s(R+) X Cr (2)
is an algebraic and topological isomorphism.
390
Boundary value problems for elliptic systems
In the statement of the condition III, we have used the spaces
7f-"(l+)== W"",(f$+) x ... x WZ+' (R+) Yi-'-'(R+):= WZ-s,(R+)11.fx ... x Wi-sP(l+) IIuIl; , and Ili-. Ilfj11i respectively. It is important to recall that we defined the Sobolev spaces W' (f8+) only for I >, 0. In view of (24), the condition I >, max si ensures that I + tj '> 0 for all j and hence all spaces Wz+'J(R+) and WZ s'(l8+) are well-defined. With with norms 11ulii+1=
the normalization (25), the first condition in (26) becomes I -> 0. The second condition is necessary when taking boundary values. Theorem 9.29 Suppose that the operator .W(y, D), Y E i Q, is properly elliptic
(Definition 9.24). Then the statements 1, II, 111 from Definition 9.28 are equivalent. If III holds we have in particular the estimate
II4II'+t s c(s) [I It d(y, (c',
i dt)}
7r-4 Y, '-'
(s', at)) w(0) I
} (28)
where on compact sets K c T*(cQ)\0, the constant Cff) can be chosen independently of ' e K. Proof We have tAI+ c Y/'^'+t(IB+), and in fact it is not hard to show that ' . 1 : 1 I + = Dl n '# '+'(I8+),
so the implications III - II -' I are obvious. To prove the implication I - II, we consider the map r 0
(44)
and w(t) -+ 0 rapidly as t --+ oo. Let VA(x, C.) = Al/2-, e,A(x'. 4u) T-'(A)wo -x.)
Let p(x") e Ca ((-b, 8)) be equal to
I
near x" = 0 and ¢(x') a C0 '(W)
and set O(x) = O(x')'P(X.) where x = (x', x"). Then we have v e C0 '(R,') and uA e Co (W). By making the change of variable t = ix", dt = i. dx it follows that ux(x) = i4(x)'vx(x),
Io(Y )f2 dY .
IIu'AI+.
J R-+
f"'
Iw(t)I2 dt
as A
(45)
oo
and Ilu II,+t-, = 0(;-'). As before A(x, D)u;, = efi A(x, D)vA + A(x, D)vA,
where A(x, D) is of lower order, and by Corollary 7.13 applied to the tangential coefficients of B we have similarly B(x', D) u., = 0-B(r, D)vA + B'(x', D) vA
where B is of lower order in each entry (ord 8k; -< m11 + ti-I). Substitution of both expressions in (39) gives 11uA 111 +, < c'[11OAvz 111-, + II(vAvA ll,-.
+ Ilq'BvAII,-m-u2 + IICB1AIl1-m-1/2 +
(46)
11uA11,+.-1]
since 0q' = ci and CO = ¢. In view of (44) and the homogeneity of its/ we have IIlAvAII,_, = O(;-') as ). - oo, and as before II(PAvAIU,-, = 0(;.-1) as oo since A is of lower order. Further, by applying (7.31) to the tangential coefficients of nil and using the homogeneity property (20) of n.4 we obtain i.
B(x', D)vA = ; t/2-, e,x(x'.4o) M(A)X( go)n
where t have
(x', Co,
dt)w(01'
i
Since the diagonal elements of 1.'12-'M(A) are J.-'+'"k+112 we is the II4' r (x, (:o, (1/i)(d/dt))w(t)I,=oil, where 11 11
L2 norm for C'-valued functions on R", and IIC'BIA11I-m- 1/2 -+ 0 as A -+ oo. Hence letting i. - oo in (19) and using (45) we get from (46) that u2 Iw(t)IZ dt) I
1< C'
I dt
)w(:)I,=o
Since 0 e C0 '(W") is arbitrary it follows that 1112
for I
C o
w(t)12 dt)
1< c' n. x', Co' 11
(
Id i dt
for all x' e W'. C,
399
Elliptic systems on bounded domains in R"
In particular it follows that n4(x'o, o, (1/i)(d/dt))w(t)I,=o is injective as a map SDi+ - C', so we have verified that condition I of Definition 9.28 holds.
It remains to prove the implication (a) -, (b), that is, if the boundary value
problem (36) is L-elliptic then the operator Q = (.4(x, D), R(y, D)) is smoothable. We will do so first under the assumption that all terms bk, in the representation (32) have non-negative order ord b,/y, D) > 0. (R) Later on we show how to get rid of this restriction.
Let yo e M. We choose admissible coordinates in U a yo, with U' _ U n a52 being a coordinate patch on 852. As usual we identify the points in U with their coordinates in R"; in these local coordinates we have
y e U' c
cp_4(y, D)t/i = (p [n-V(y, D) + i(y, D)]*, for all gyp, 0 a Co (U), where
n.(y, D) = [Bki(y, D)],.,,
Bk,4y, D) = Y' nbtt(y, D')D., "
rzbkj(y,
R"-
D')v = (2n)' -" je')
(47)
nbk,{y,
dc',
and 4 consists of lower-order terms.
We may, of course, suppose that icbk,{y, t') has compact support in y e U' c R", because the regularizers are eventually defined by a partition of unity. In view of the text after Theorem 7.28 concerning the symbol spaces
S'', we can therefore do without a cut-off function in ' in formula (48) for the purpose of establishing estimates in Sobolev norms. In order to make the long proof of the implication (a) -+ (b) transparent, we first prove two lemmas. The points in R+ are denoted by x = (x', t), where
and t = x e R. If u(x', t) is a function on R+, we make use of the partial Fourier transformation a' defined by
a(', t) ='N'u(', t) = j'e'
.0 u(x', t) dx',
For integers k = 0, 1, ... , the norm on WZ(R+) is equivalent to the Fourier norm IIuI1
v=o
f(I + I
'12)k-"
f
0
v
d
a( ', t)
(see Theorem 7A.4), which in turn is equivalent to
IIuIIk = E k f 112"
fO'O
ddt "
z
a(g', t)I dt dc',
(48)
and this is a more convenient form to work with in the following lemma.
400
Boundary value problems for elliptic systems
Lemma 9.35 Fix a point yo a do on the boundary, let A(D):= n.rd(yo, D),
B(D) _= n-4(Yo D),
(49)
and as above choose admissible coordinates in a neighbourhood of yo. If (A(D), B(D)) is L-elliptic in W. then we have the a priori estimate (I >, lo) lull r+, , 1. Note that the choice of a ensures that the matrix Ii j°(1 + I2i)-'A-'(t) is a bounded function near = 0 while for large ItI, it behaves like A-'(ia). Now, we have
AoRf =f +Tf,
RoAu=u+ Tu,
403
Elliptic systems on bounded domains in 98"
where Tf = -a-'(1 + Ic
Since a 3 1, it follows that T is smoothing,
i.e. an operator of order - 1. I+1-`(98")
T:7/r'-.(W') -
and whence R is both a left and a right regularizer.
Step 2 In the half-space R+ we consider the L-elliptic operator with constant coefficients, 2(D) = (ird(yo, D), t_4(yo, D)) = (A(D), B(D)), where yo E 8Q is fixed and A(D) is the same as in Step 1. Let t+: W2(98") -+ W2(R+), m >, 0, denote the restriction of a function on R" to 98+. Also let fi: W2(98+) - W2(98"), m >, 0, be a linear, continuous
extension operator from R+ to W'. For all m in a bounded interval, it is possible to define f independently of in, i.e. the definition of ff does not depend on the particular space W2 to which f belongs (Theorem 7A.7). As before, a denotes the Fourier transform in R" and a' the partial Fourier transform in IB+. We define the operators
Rof =
:+t`r1-/'IsI°(1
IS/I°)-1A-1(S) f = 2+Rpf,
+
x > 0,
R19 =
where R is from (56), w(S', x.) is the canonical matrix function satisfying (53) and a':= max(mk + t j) + 1. The continuity of R0:= 'Y" +'(R+) -+ YY'-`(98+) follows from the continuity of R and the continuity of t+ and ji because IIRof III +,R_. < IIRflf III+c.R^ 1< cI II Pf III-..R" -- C211f III-..R
To show the continuity of R,: Yf'''to obtain estimates (55) of
IIR19IlI+c , Y Y
J=1 k=I
f
µY_
(57)
'1'(98"-') - *''+'(R+), we use the
IS,I2"
f Id w}k(', t)I21`tYgk12 dt
r
1 (I +
C k=1
C'II91II-°-112'
(58)
(Once again, one must consider two cases, I,;'l < I and Now define
91(f,9)'=Rof + R1[g - B(D)Rof]. We will show that the operator 91: #,"-`(R+) x Yl'''-
1.)
(59) 112(98"-1) -
''+'(98+) satisfies all the requirements imposed on a regularizer for 2, namely, that it is continuous and satisfies the left and right regularizing properties. First of all, it follows by (58), (57) and the continuity of B(D) that IIR1[g - B(D)Rof]ll1+c , 1 (by virtue of the restriction (R)), which proves the continuity of yj'.1+1-s(R+) x W'' 1/2(R"-1) I+1- 1/2 (R-1), T,: x (61)
Finally, to show that 91 is a left regularizer, we define the operator T, by 9Z o 2(D)u = u + T,u,
(62)
and must show that T, is an operator of order - 1,
T,: 1
`(f8+) -+
,Wrl+1+t(R+). First of all, it follows from the continuity of 2(D) and 91 that T is continuous from 71''1+t(R') to *I + I(R" ). Further, in view of (60), (62), we have
Qo9Zo2u=2'(1+T,)u=(I+T,)2u, so 21 u = T P-u, i.e. (A(D) T, u, B(D) Tu) = T,(A(D) u, B(D)u)
(63)
By virtue of the a priori estimate (50), we obtain
IITuII,+I+t , -min ord bk,{y, D'), x.k.j
y e M.
(72)
Now, suppose that (.ci, -4) is L-elliptic, i.e. V is elliptic and (.sat, 9) satisfies
the L-condition in Definition 9.281. Since this L-condition is invariant under multiplication of the boundary operator n-4(y; (c', (1/i)(d/dt)))I,=, with 0, we see that (.d, A9 o -4) is L-elliptic too, where we denote by the same notation A9 the diagonal r x r matrix with A" in the diagonal. In view of (72), the boundary operator AQ . :4 satisfies the restriction
q+ordbkj>q+min ordbkj>_ 0,
(R)
K, k. j
and we obtain by our previous reasoning that the map
A9 o _4) is
smoothable: `A, o (.sat, A4 -,R) = 1 + T,,
T,, e OP- '
(70')
(.a,AQo S)o9,=1+T
T,e0P_'.
(71')
Rewriting (70') we obtain
(91,o(1,A"))I+T, whence 91, (1, A4) is a left smoothing regularizer of (d, .4) with smoothing
operator T, e OP'. Also, the operator A°: yF--
i=(an) -
Vi-m-4-siz(m)
in (71') is Fredholm since A9 is elliptic, so it has a smoothing regularizer A. If we multiply (71') on the right by (l, A4) and on the left by (I, A-Q) it follows that (a,-4)°(`31,°(1, A°)) = I + T.,
where T,' _(I.A-')oT,o(1,A9)+(0,(1-A-QAQ) f)oR,.Hence`)i, (I,AQ) is a right smoothing regularizer with smoothing operator T, e OP-'. Thus we have shown that (d, -4) has both a right and a left smoothing regularizer, i.e. it is smoothable.
Remark In the proof of the implication (a) -' (b), we see that the regularizers constructed there have definitions which are independent of 1, i.e. they have
the same values on common domains of definition. Indeed, the Sobolev spaces W;' are function spaces and all operators in question either act directly on the argument or on the function value, or, as in the case of the extension
operator /i, can be taken independent of m for m varying in a bounded interval.
Elliptic systems on bounded domains in 11"
409
In conclusion, we mention that the proof of Theorem 9.32 applies without change to pseudo-differential operators, d(x, D), of the form
srt(x, D) = .d'(x, D) + /b(x, D),
where .c/' is a p.d.o. with compact support inside fZ and sdb(x, D) is a polynomial in D with support in a small neighbourhood of the boundary 00. (For the definition of support see Definition 8.4.) In §§10.7 and 14.6 we permit slb to be a more general operator which is polynomial in the normal derivative D. but not necessarily in the tangential derivatives. Exercises 1. Let T e D(X, X) bea Fredholm operator with ind T = 0. Show that there is a representation T = S + K where S is an isomorphism and K is a compact
operator. 2. Let T E (D(X, X) be a Fredholm operator. Show that there exists some p > 0 such that the d-numbers x(T + il) and j3(T + )1) are independent of
i. for0 0. In §10.1 we use the spectral theory of matrix polynomials to reformulate the L-condition in various equivalent forms, e.g. the Lopatinskii condition, the complementing condition of Agmon, Douglis, Nirenberg, the A-condition and other conditions (see Theorem 10.3). The L-condition for the Dirichlet problem is also discussed in detail in §10.2. In §§10.3 and 10.4 we prove some technical results concerning families of matrix polynomials. These results will be used in several places throughout the book, for instance, in §10.5 in order to prove a result due to AgranoviL and Dynin on comparing the index of two elliptic boundary problems, (sad, -41) and (d, R2), having the same elliptic operator sad (see Theorem 10.19). For a fixed p (the size of the system), we let
Ell" denote the set of elliptic operators, sat, with DN numbers s1,.. . , sp, t1, ... , tp and the set of L-elliptic boundary problems, (sa1,5d ), with DN numbers s1,. .. , sp,
t1, ... , tp, ml, ... , m,. Then we have a natural map defined by
9 (d, 9) w sat e Ell". If we have a homotopy salt, 0 0 that contains all the eigenvalues of L(A) there. By Chapter 2 there exists a y+-spectral triple (X+, T+, Y+) for L(2), and the base space of the y+-spectral pair (X+, T+) has dimension r, where r is the number of roots of det L(1) = 0 inside y+. We let TI+ _ QuInei&) denote the subspace of C °°(R, C°) consisting of the solutions of
L(i dt)u = 0 such that u(t) - 0 as t -> +oo. Recall that dim W+ = r and by Corollary 2.9 every u e'Tlt+ admits a representation of the form u(t) = X+er,r+c
(2)
for a unique c e C'.
Remark By Theorem 2.21, the matrix polynomial L(i-'2) has the spectral pair (X+, iT+) with respect to the eigenvalues in the half-plane Re A < 0. The (right) Calderon projector is
I P+ = Pr.
L-'(2)[L,(2) ... Li(A)7 dl
2xi
where LA(X) = Aj + A;+ A + - + A,d'-1, j = 0, ... ,1, and we know that the image of Py+ is equal to the set of initial conditions
col((!i
dt
J-0
ueAt+.
418
Boundary value problems for elliptic systems
Theorem 10.1 Let B(A) = D=o BjAj be an r x p matrix polynomial of degree p. The following statements are equivalent:
(a) For any y e C', there is a unique u e 97l+ such that Y.
B(i dt)ulr-a = (b) If (X+, T+) is a y-spectral pair for L(A), where X+ is a p x r matrix and T+ is an r x r matrix, then the r x r matrix AB BMX+T+ is invertible, i.e. M
det AB = det Y BMX+T+
0.
(3)
i=o
Proof As we remarked above, any u E 972+ can be represented in the
(
form u(t) = X+e"T+c for a unique c E C'. The equivalence of (i) and (ii) is clear since dt)u!
B(
o
=
BJX+T+
).c
Corollary Further, if p < 1 - I and we define the r x pl matrix
B = [B0...BB
0...0]
then (a) and (b) are also equivalent to: (b') The rank of B- P,,. is equal to r.
Proof Let (X+, T, Y+) be a matrix y-spectral triple for L(2). Then X+ is a p x r matrix, T+ is r x r and Y+ is r x p. Consider the r x pl matrix b P,. as an operator Cp' -+ C' and the r x r matrix F;=o B;X+T+ as an operator X+
[Y+...T+'Y+]. ff
P,+=
(4)
X+T+' where T is defined by §2.2(5), it is clear that the image of BP,. is contained
in that of Fi;o BMX+T+. Conversely, if y ==o BBX+T+c, c e C', then y=B col(X+T+)j=oc. Since the image of P,+ is equal to that of c = P,+c' for some c' E Cr', whence y = BP,+c'. This means that BP,+ and Ei=0 B;X+T+ have the same image, and, therefore, the same rank. This proves the equivalence of (b) and (b'). In the following theorem, L(A) denotes the cofactor matrix of L(A), i.e. the matrix polynomial such that L(A) L(A) = L(A)L(A) = det Also we let P',+ be the left Calderon projector (Theorem 2.15) L1(2)
P+ = P,+
21
ni r+
L-'(A)[I...A'-'I] d). L,(2)
Understanding the L-condition
419
Theorem 10.2 The conditions (a) and (b) of Theorem 10.1 are also equivalent to:
(c) The r x pl matrix (called the Lopatinskii matrix) G=
1f
tai ,.
B(.)L`(2)[1...A'-11] d)
has rank r.
(d) There exists a pl x r matrix S such that GS = I, and SG = P,,., where P',. is the left Calderon projector. (e) The rows of the r x p matrix polynomial B(A) - L(i.) are linearly independent
modulo p+().), where we have factored the scalar polynomial det L(i) as p(1.) p+(i.), p+ containing all the roots of det L(i.) = 0 inside;' and pall the roots outside I+.
Proof As a preliminary remark, note that by linear algebra the r x pl matrix G has rank r if and only if the corresponding linear map G: C°' C' is surjective.
(b) - (c) Let (X+, T+, Y+) be a matrix y+-spectral triple for LO). By the formula (ii') in Proposition 2.2, the Lopatinskii matrix is equal to v
G = Y B;X+T;
(5)
i=o
and we know that [Y+ - - - T'-'Y+] has rank r (see Definition 2.1(iv)). Hence
G has rank r if and only if the r x r matrix A = j,,0 B;X+T+ is invertible. (b) a (d) The left Calderon projector is equal to
P'r. _ Z -col(X+T+=o-[Y+... y+'Y+]
(6)
If (b) holds we let
S = _T-col(X+T+);=O'-(
B;X+T+J
(7)
0
then SG = P'.,., and by Corollary 2.11 we have GS = 1,. Conversely, if (d) holds then the equation GS = I implies that G has rank r. In view of (5) this implies that the matrix Ae = Lj=o B;X+T+ has rank r, hence it is invertible since it is a square r x r matrix. (e) Note that LO.) = det L(A)-L-'(i,); as remarked above, L(A) is the cofactor matrix for L(k) so it is a polynomial. If x e C' is such that (c)
xB(i.)L(i.) = p+().)' MO
(8)
for some I x p matrix polynomial M(i.), then dividing (8) by det L(i.) gives xB(i.)L-'(i.) = p-(i.)'' M(A), and the right-hand side is holomorphic inside
420
Boundary value problems for elliptic systems
y whence
x f B(A)L-'(A)[I ... A'- 'I] dA = 0.
(9)
r
Thus (c) implies that x = 0.
(c) Conversely, let x e Cr such that (9) holds. In view of (5) we see that x-D.=a B,X+T+ = 0. Multiplying both sides of this equation by T'k+Y+ for k = 0, 1, ... , and making use of the formula (ii') in Proposition 2.2 we obtain (e)
x
I
r
B(A)L-'(A)A'dA = 0
for all k = 0, 1, .... Hence xB(A)L '(A) has an analytic continuation inside y as a matrix function of A. Since L(A) = det L(A) L-'(A), this means that xB(A) L(A) vanishes at the roots of det L(A) = 0 inside y+, i.e. vanishes at the roots of p+(A) = 0. Hence (8) holds for some 1 x p matrix polynomial M(A), and then (e) implies that x = 0. Thus G is surjective, i.e. has rank r. Remark The matrix S is uniquely determined by the conditions in (d). The first equation implies that G is surjective, and if S and S, satisfy SG = Pr.
and SIG = P;. then (S - SI)G = 0 whence S - SI = 0. Remark From the proof of (c) a (e) it is clear that condition (e) can be formulated in another way: The only x e C' such that xB(A) L-'(A) has an analytic continuation inside y+ is x = 0. There is a natural correspondence between the conditions (a) and (c), i.e. depending only on L(A), not on the choice of spectral triple. First of all, note
that (5), (6) and Corollary 2.11 imply that G - Pr. = G. Also, for the map 0: CP' - 992+ defined by
1 fV. 2rct
we have
1-1
e'raL-'(A)' Y cj+iA'dA
j-o
B(i dt)1-o o
G,
and, in view of Proposition 2.5, 0 is surjective. Further, since ¢c = it follows that ker4)= ker P'+ by Corollary 2.11. Thus we have the commutative diagram
C,' -per' +
C°' -- wZ+
and that fact that PY+ and 4' have the same kernel means that we can identify im P'+ and 9R+ along the top row of the diagram. With this identification,
Understanding the L-condition
421
the map GI;.,,,: im P;. -+ C' is identified with the map B( Ii
)le=o:
+
d
-. C'.
Thus, condition (a) holds if and only if
Gl;.p : imP'. -+ C' is bijective. Now we apply the matrix theory to the L-condition for elliptic systems (d, 9). For the sake of emphasis we gather all the equivalent conditions together in one long theorem. From now on we usually omit mention of the contour i.e. we write instead of J),. because the value of the contour integral depends only on the residues in the upper half-plane, Im A > 0, not on the particular contour. y+,
In the following theorem, no assumptions are made concerning the transversal order, p, of the boundary operator (p >, 1 is permitted). Theorem 10.3 We suppose that the operator .d(x, D) is properly elliptic, let R(y, D) be a boundary operator as in (1), and fix y e i C1 , 0 # ' e Tq (d1Z). The following statements are equivalent:
(i) The initial value problem 1
".W y'
ddt))u(t)
c
= 0,
t>0
(10)
n+ (c', d )) u(t)le_, = g, has for every choice of g E C' a unique solution u e MLCl/td/d(). As usual, 9X+ u(t) = 0 (or, s i( dIdf) is the space of solutions of (10) such that lime in other words, corresponding to the eigenvalues of L(A) with a positive imaginary part).
.
(ii) The r x pl matrix G = G,,,, defined by
G=J
(11)
has rank r, where j + denotes the integral along a simple, closed contour y+ in the upper half-plane containing all roots of det L,4, (A) = 0 with a positive imaginary part. (iii) If (X+(Y, '), T+(Y, 4'), Y+(Y, c')) is a y+-spectral triplefor L, .(2), where
X+(y, l;') is a p x rmatrix, T+(y,l;')ar x r matrix and Y+(y,4')ar x p matrix (existence by Theorem 2.6), then
det Aa*{y,') = detB;(Y,')X+(Y,')T+(Y,')) # 0.
(12)
(iv) There is a unique pl x r matrix S = Sy,4, such that GS = 1, and SG = P+, where P+ = Py+ is the left Calderdn projector for Ly,t.(d).
422
Boundary value problems for elliptic systems
(v) Let us factor the scalar polynomial, det L,,,.(A), as p-(.l)-p+(A), where P+ contains all the roots above the real axis, and p- contains all roots below the real axis. If L'(1) denotes the matrix polynomial L(.1) = det LY,C-(A) - L- .(A),
then the rows of B,,,4.(A)-L(.1) are linearly independent modulo p+(A).
The first condition is of course the L-condition stated in §9.3. Condition (ii) is known as the Lopatinskii condition; the matrix (11) is the Lopatinskii matrix, and was introduced by Lopatinskii in his paper [Lo]. Fedosov used condition (iv) in a series of papers where he developed an index formula for elliptic boundary value problems [Fe]. The last condition (v) is called the "covering condition" or "complementing condition", and was introduced
and used by Agmon, Douglis and Nirenberg in their fundamental paper [ADN]. We call (iii) the A-condition.
Example The covering condition (v) is quite difficult to apply in general. However, for the case of a scalar operator d(x, D) (i.e. p = 1) it is perhaps the simplest of the five conditions. Supposing that the operator d is properly elliptic, let 1= 2r and let A,,.. ., A., be the roots of L(2) above the real axis, and factor the polynomial L(1.) as
L(2) = P M'P+M, where p+(2) = fi=1 (.1- i.;). Since p = 1 we have to consider only the r x 1 matrix y e 8f , 0:0 ' E Ty (oM) for linear independence modulo p+(A). By the Euclidean algorithm, we write the rows of B(A) as B(A) = n-4(y, (ce', A)),
.-1
bj(II) = Q!MP+M + Y bjk(y, ')A',
j = 1, ... , r
k=0
(b j are just principal parts of the individual boundary conditions) and have: The rows bj(A) are linearly independent modulo p+(A) if and only if the r remainder polynomials
.-1 j = 1, ... , r,
(13)
k=0
are linearly independent. This is equivalent to det[bjk(y, ')]j-1..... .
k=0,.....-1
# 0,
(14)
which in turn is equivalent to the Lopatinskii conditions by Theorem 10.3. When all boundary conditions are of order < r - 1 in the normal derivatives, the polynomials (13) constitute the principal parts of the boundary conditions (in this case we can set Qj = 0) and (14) become the most simple test for
checking the L-condition. For instance, the boundary operator for the
Understanding the L-condition
423
Dirichlet problem is
where a is the operator a/an, and (14) is then the determinant of the diagonal matrix Hence, the Dirichlet problem for a properly elliptic scalar operator d(x, D), p = 1, always satisfies the L-condition.
Remark When verifying the L-condition for particular examples of (sit,-4) it is not essential to include the factor 1/i. The transformation t --* it carries over the equations
L( d)u(it) = 0 (15)
into
B(i
)u(it)i..o = g
and it gives obviously an isomorphism of the solution spaces mud/dt) ^-
L(1/i d/dt)-
where `Mi(d,di) denotes the solutions of L(d/dt)u = 0 corresponding to the eigenvalues in the half-plane Re i. < 0. It is evident that if the first initial value problem in (15) has a unique solution u(t) for every g e C', then the second is also uniquely solvable and vice versa. 10.2 The Dirichlet problem
If d is a p x p elliptic operator such that r = ps for some s, we can pose the Dirichlet problem (.d, 9) where I
9=1
I a,,
I
I is p x p identity,
\Ian-1/ but in general it will not satisfy the L-condition. In fact Theorem 3.14 implies the following result.
Proposition 10.4 Let .d(x, D) be a properly elliptic operator and suppose that the number r is divisible by p, i.e., r = ps for some s. The Dirichlet problem (d, 2) satisfies the L-condition if and only if for ally e aQ and 0 # ' a T; (df2), L,X(2) has a y+-spectral monic right divisor of degree s.
424
Boundary value problems for elliptic systems
Proof The Lopatinskii matrix of the Dirichlet problem is
I Lre'(2)[I...Al-'I] d2 J
which is also the deciding matrix in Theorem 3.14.
Recall that L, .(A) = rtd(y, (', 2)). Suppose now that the principal part of d is homogeneous of degree 1, that is, with DN numbers s, = 1, t1= 0. In this case the leading coefficient Lr, s.(2) is A, = nsW(y, n(y)), which is invertible. Then det has degree p1. In order to pose the Dirichlet problem we need r = ps for some s, hence I = 2s. In this case we can strengthen the result of Proposition 10.4. Suppose there is a y+-spectral right factorization Lr.442) = Lr.4.(2)L, s'(2),
monic of degree s. Then Ly, must also have degree s with invertible leading coefficient, since A, is invertible and I 2s. Due to the homogeneity of trd we have xsal(y, (c', A)) (-1)'i sl(y, -A)), hence with
(-1)'Lq This means that whenever Lr, f.(2) has a monic y+-spectral right divisor of degree s for all y e 852, 0 0 l;' a Tr*(852), it also has a monic y+-spectral left divisor of degree s. In view of Theorem 3.16 we can now improve on Proposition 10.4. Theorem 10.5 Let sal(x, D) be a properly elliptic operator and suppose that the principal part of sal is homogeneous of degree 1= 2s. The Dirichlet problem
(all 2) satisfies the L-condition if and only if
I nsa(y, (c', 2))-'[I...2a-1I] d2 # 0
det I
(16)
2s-'1 for all y e 852, 0 96 t' a T, *(M).
One of the hypotheses of Theorems 3.14 and 3.16 is that there should be exactly ps zeros of det Lr,g,(A) = 0 inside y+. This is where it is necessary to
assume that the operator sal is properly elliptic. However, it is not hard to show that the invertibility of the matrix in (16) for all y e 852, 0 0 t' e Tq (852)
in fact implies the properly ellipticity of d. In checking the Lopatinskii condition we are confronted by the difficulty to decide which r of the lp (1= 2s) columns of (11) are independent. The theorem above shows that for the Dirichlet problem it is only necessary to check for linear independence in the first r columns. The same is true for
425
Understanding the L-condition
any boundary operator once the Dirichlet problem is known to satisfy the Lopatinskii condition. Theorem 10.6 Let the principal part of the operator sad be homogeneous of degree 1 = 2s and suppose that the Dirichlet problem (st1, 91) satisfies the
L-condition, i.e. the ps x ps matrix (16) is nonsingular. Then the ps x p boundary operator R(y, D) satisfies the L-condition relative to d and only if det J
it (y, (c',.1))nsal(y, (c',
for all y e i(, 0 # t' e
2))-1[J...)S_1f] dA # 0
(17)
T, (8SZ).
Proof Let (X+, T+, Y+) be a y+-spectral triple for (16) is equal to
The matrix in
X+
[Y+...T+ 1Y+]
(18)
X+T+i and the invertibility of (16) implies that both factors in (18) are invertible. Since the matrix in (17) is equal to P
Y BjX+T..[y+...7'3+ lY+] 3-0
the invertibility of [Y+
(19)
-71;-1 Y+] implies that (19) has nonzero determinant.
By (12) this is equivalent to the L-condition for (.0', ). As an example of an elliptic system for which the Dirichlet problems does satisfy the L-condition, we consider strongly elliptic systems. In the following definition, (c, d) denotes the usual inner product of column vectors in CP, i.e. if c = [c]j'..1, d = [dj] f=1 then (c, d) = dhc = P.1 CA-
Definition Let d(x, D), x e i2, be a p x p differential operator with DN numbers tt + tj where tj are nonnegative integers (i.e. si = t; > 0). Then sat' is said to be strongly elliptic at x e fZ if Re(ns1(x, ) c, c) 0 for all e R"\O and c e CP\0.
Clearly, if the operator sat is strongly elliptic at x then it is elliptic. We assume that Q is connected and n > 2 which means that f2 x R"\O x CP\0 is connected. It follows that if saf is strongly elliptic at all x e t2 then Re(nd(x, )c, c) has constant sign. Thus, either Re(nsad(x, )c, c) > 0
(20)
or Re(n.ct(x, )c, c) < 0 for all x e a e I8"\0 and c e CP\0. Without loss of
generality we assume that (20) holds. In view of the homogeneity of n-4(x, ) = [a ft(x, )] and the compactness of 11 and the unit spheres in 08"
426
Boundary value problems for elliptic systems
and C P, we see that Re Y ajk(x,1;)cji:k > const - Y
j
jk
for allxefl,l;ER°\O,CEC \0. For strongly elliptic systems we may consider Dirichlet boundary conditions of the form yE00, aAuj=g,{y), (21) where v = 0, ... , t j - 1, j = 1, ... , p (when t j = 0 there are no boundary conditions). Theorem 10.7 Let d(x, D) be strongly elliptic for all x e !D. Then the Dirichlet problem (21) satisfies the L-condition relative to sl.
Proof First we make a preliminary remark. Let Q(T) = f e-" `u(t) dt denote the Fourier transform of a function on the real line. Let 5°(R.) denote the set of functions u e C (IR+) on the half-line t >, 0 such that u and all its derivatives satisfy Idju/dtjI 0,
there is an open set U a xo such that y+ contains the zeros of det L,,(A) in
Imi.>Oforall xaU.
(*)
Boundary value problems for elliptic systems
428
Indeed, if we let fx()) = det Lr(i.), then by the argument principle from complex analysis we have
r=27ri y f,(;.) 1
.f=o(g)
di..
Since fro().) # 0 for all d on the contour y+, there is a neighbourhood U of x0 such that fx().) # 0 for all ). on the contour y+; by shrinking U, if necessary, it follows that
r
f.(;) di,
27ri
(25)
,I,. fx('
because the right-hand side is a continuous function of x, and integervalued, hence locally constant. But r is the number of zeros of f.(;.) in Im ). > 0 (by assumption), so the contour y+ must contain all of them, i.e. (*) holds. Conversely, if (*) holds, we conclude from the argument principle
(25), and the continuity of the right-hand side, that the number of zeros of fr(i.) in Im i. > 0 is locally constant. We do not assume that the degree of det L,r(i.) is constant, although for matrix polynomials that arise from an elliptic operator this will always be true, of course.
Example LE(A) = (e) + i)(). - i), 0 < s 5 1, is permitted; it has one zero with positive imaginary part, for all e. On the other hand, L,().) = (e). - i)(1- i), 0 < r < 1, is not permitted because it has two zeros with positive imaginary part when e # 0 but only one such zero when s = 0.
As above let M be a C ' manifold. For each x e M, there is the vector space 9nx of solutions of Lx(i-' d/dt)u = 0 such that u(t) - 0 as t -+ +00. Let yx be a simple, closed contour containing the eigenvalues of L.(;) in the upper half-plane Im i. > 0. As in the proof of Theorem 2.6, L.,(;.) has the yx-spectral triple (X+Ix, T+Ix Y+Ix) where
X+ Ix: 9 -+ CP
maps u I -e u(0)
T+I,,: 0 --e 9)I,`
maps u H
Y+Ix C" -+
maps
i dt u
c -, (27ri)-' J e"'Ls
(26) c di.
Lemma 10.8 Let Lx(i.) have the properties stated above. The family of vector spaces 91l+ = {Ulx } has a unique vector bundle structure over M such that 9R+ and Y+: C° -+ 9R+ defined on each the maps X+: W14 --+ CP1 T+: 10+ fibre by (26) are vector bundle homomorphisms.
Proof The Calderon projector for Lx(A) with respect to the eigenvalues in the upper half-plane is given by
429
Understanding the L-condition
I Px+
X
2ni
,.
L'(;:)[I...d1.
(27)
1'"'I
where yx contains the eigenvalues of LX(i.) in the upper half-plane Im i. > 0. Let xo E M. By assumption, see (*), there exists a neighbourhood U of xo such that the contour ys in (27) can be replaced by y o for all x c- U. Since the coefficients of Lx(i.) depend smoothly on x it is clear that the matrix P. depends smoothly on x. Hence P,' is a projection operator (Definition 5.53) for the trivial bundle M x CP'. By virtue of Proposition 5.52, the family of vector spaces im P+ = {im P,+) is a vector bundle over M. We have a map q : 91i+ - im P+ defined by thelCauchy data on each fibre u E']N+ i-+ 4'l = Col X
1
d J u(0) 1-1 a CP' i dt/ ) j=a
which we know is a linear isomorphism on each fibre (Theorem 2.8). We give 911+ the vector bundle structure that makes p into a vector bundle isomorphism. To prove that X+, T+ and Y+ are smooth we just have to show that X+ o (p-', rp o T+ = tp-' and (p - Y. are smooth. Recall that for any matrix polynomial, LO.), every u E 912+ has a representation of the form u(t) =
I
2ni
e'"AL-'(i)[I...i.'-'I]3'
d1.
+
where ?' is the Cauchy data of u (see §2.2 and recall that 1, is the short notation for f.,. ). It follows that X_ c, tp"' _ XIimP. , q' ° T+ ° (p -' _ and cp Y+ _ 1 where
_
21
L"'(i.)[I .
ni
+
1
Z_ L
col(ijl )j'=oi.L"'(i.)[I
2ni
3=
2ni
. .di.
IimP,
p x pt matrix
'I ] ' di,
pl x pl matrix
('+
J
di.
pl x p matrix
+
Since Y, F and # are smooth matrix functions on M, and im P. is a sub-bundle of M x C", it follows that X+ o 9-' E ..P(im P+, CP), rp T, ..P(im P..) and cp c Y. e 2(CP, im P+) are smooth.
cp
'E
The definitions we had previously in Chapter I for admissible triples of operators carry over to families of such admissible triples. A triple of vector bundle homomorphisms (X, T, Y) is called an admissible triple over M if X e 2(E, CP), T E 2(E) and Y e 2(CP, E), where E is a vector bundle over M. As before, E is called the base space of (X, T, Y). Two admissible triples over M, (X, T. Y) and (X', T'. Y'), with base spaces E and E'. respectively, are called similar if there exists a vector bundle isomorphism tp E 2(E', E)
such that X' = Xtp, T' = cp-'Tq and Y' = (p-'Y.
430
Boundary value problems for elliptic systems
Theorem 10.9 Let Lx(i.) = Y'j-0 A,{x)2' be a family of matrix polynomials with the properties stated above. For each x E M, let ys be a simple, closed contour containing the eigenvalues of L.,(;.) in the upper half-plane Im 2 > 0. Then there exists an admissible triple (X+, T, Y+) over M such that (X+(x), T, (x), Y, (x)) is a ;x -spectral triple of Q).) for all x E M. Any two such admissible triples are similar. Note: we call (X+, T+, Y+) a spectral triple with respect to the upper half-plane
for the family of matrix polynomials L(i.) = L,(i.), x e M.
Proof The existence of a spectral triple (X+, T, Y+) has been proved in Lemma 10.8 (the base space is E = 9R+). Now let (X+, T+, Y+) be another spectral triple for the family L(i.), i.e. an admissible triple over M, with base space E', such that (X+(x), T+(x), Y+(x)) is a i+-spectral triple of for all x E M. By Proposition 2.12, it follows that X +(x) = X_' (x) J. /IM, x
T'.
and Y+(x) _ -/tx ' c Y+(x) for all x c- M, where
l1x = row(T+(x)1'+(x));=o 3`s col(X +(x)7+(x));=o.
Obviously.Al e £(E', E) and is a (smooth) isomorphism. Hence (X+, T+, Y+) and (X+, T+, Y+) are similar.
Remark If (X+. T, Y+) is any admissible triple over M satisfying the conditions of Theorem 10.9 then its base space E is necessarily isomorphic to 912+ due to the isomorphism E -+ 912+ given by Ex a v X + (Y) ei1T
.Izl v E 1Rx .
Corollary 10.10 There exist vector bundle isomorphisms 91t+
im P+
and
912+
im P'+
where P+ and P+ are the right and left Calderon projectors with respect to the eigenvalues in the upper half-plane.
Proof The isomorphism 9114 -+ im P+ is given by the Cauchy data u '- col(uth(0))i=o on each fibre. As for the second isomorphism, consider the map ¢: C°' -+ 912+ defined by 1-1
C " 4c(t) =
ei« L-1(2)' Y c;+I;.' di..
(2ni)-1
+
J=a
Let (X+, T+, Y+) be a spectral triple with respect to the upper half-plane for
the family L(7.). Since ¢c = X+ e"'* [Y, T'' Y+] c it follows due to T+' Y+] = ker P+ (for injectivity of col(X+T,))=o that ker 0 = ker[Y+ the second equality, see Theorem 2.15). Since P. is a smooth projector, then
ker P+ is a vector bundle over M. By surjectivity of ¢, we have an isomorphism
C"'/ker 0 = 9R+
Understanding the L-condition
But also C°'/ker 4, = C°'/ker P+
431
im P;, so there is a natural isomorphism
912' a- im P. In §§10.4 to 10.6 and §15.1 it will be clear that the triple (X+, T, Y,) is
a key step in the construction of boundary operators satisfying the Lcondition for a given elliptic operator, this in turn is crucial for the proof of the index formula for elliptic systems in the plane in Chapter 16. In fact we would like to have spectral triples (X., T, Y+) which are C' matrix functions on M. (Precisely, X+(x) is a p x r matrix, TT(x) is an r x r matrix and Y ,(x) is an r x p matrix, with entries depending smoothly on x c- M.) One necessary condition for the existence of such a triple (X+, T+, Y+)
is obvious: 911' must be trivial, since there is the isomorphism from M x C'
to 0' defined by (x, r) E {x} x C' " X_ (x) er,r. :) VE
s
The triviality of 911' is also sufficient as we show in the next proposition.
Note that the condition that the number of eigenvalues of L-.,(;.) in Im J. > 0 be locally constant in x c- M just means that the number of them is constant on each component of M. Theorem 10.11 As usual, suppose that det L,(i.) 96 O for all real ; and let the number, r, of roots of det L,(i.) = 0 in the upper half-plane be independent of
x e M. Then there exists a spectral triple (X+, T+, Y+) with respect to the upper ha f plane for the famil y L(i.) = L.(i.) consisting of C' matrix functions on M if and only if the vector bundle M' is trivial.
Proof First, we claim that the base space E of any spectral triple (X.. T+, Y+)
is isomorphic to W. Indeed, there is a map 4): E -, 911' defined on each fibre by
v e Ex H X+(x)
eirT.lx)
v E Tl
which is a linear isomorphism on each fibre (Corollary 2.9). Since 4) is smooth it follows by Lemma 5.46 that N is a vector bundle isomorphism. Suppose now that 9R' is trivial. Then E is also trivial, i.e. there exist C' sections v ... , v, of E such that, for all x e M, the vectors vl(x), . . . , v,(x) form a basis of E. Relative to this basis, the vector bundle homomorphisms X+, T+ and Y+ are C' matrix functions.
Remark Exercise t at the end of this chapter gives an example of a family of matrix polynomials L,,(i.) for which the vector bundle !172+ is not trivial. 10,4 Homogeneity properties of spectral triples
Let d(x, D) be an elliptic differential operator in 0 with DN numbers s,, ... , so, t1..... t,,. The coefficients of the matrix polynomial L(2) = a.d(y,
2)) = i A,(y, 02 i=0
432
Boundary value problems for elliptic systems
depend smoothly on the parameters (y, S') e T*(cfl)\0 and have certain homogeneity properties in s' which follow from the homogeneity properties of rt.d.
We consider first the case where d is homogeneously elliptic, that is, = sp = 1 and t, = = tp = 0 (see Definition 9.25), because the notation is simpler then and some of the results take on a more complete
s, _
form. L(i.) has degree I with invertible leading coefficient, A, = ttd(y, n(y)),
and det L().) has degree x = pl. A finite spectral pair (X, T) for L(;.) is therefore a standard pair (Definition 3.1), that is, col(XT');=o is invertible. The homogeneity of ita/ implies Ly,C4.(i.) = c'L.(c 'i.), whence
j=0,...,1,YeEC
A,(y,ce')=c'-;.A.(y,
(28)
Let ST*(M) denote the unit cotangent bundle to 2f2 and r: T*(cf2)\0 ST*(as1) the map defined by (y, ')'--. (y, ' Lemma 10.12 There exists a spectral triple (X+, T. Y+) with respect to the upper half-plane for the family L(i.) = rt,d(y, (s', ).)). (y, S') e T*(aQ)\0, such that for any c > 0 X+(y, ce') = X+(y, ') T. (Y. ce') = cT+(y, S')
(29)
Y. (Y' cs') = c'Y+(y, S')
where X+ c -.V(?-'E, C'), T. E 2(r-'E), Y+ E -P(Cp. ?-'E), and t-E is the pullback to T*(af2)\0 of some vector bundle E over ST*(cf2). (E is necessarily isomorphic to RR.)
Proof By virtue of Theorem 10.9 there exists a y+-spectral triple (X+, T, Y,)
for LO.) with X+ e 2(E, C'), T+ E 2(E), Y+ e .Y(CP, E) for some vector bundle E over ST'(cM). Then we define X+(y, c') = X+(y, s'/IS'I),
T+(y ') = Is'IT+(). The triple (X+, T+, Y+) satisfies the homogeneity properties listed above and it remains to show that its satisfies properties (i), (ii'). (iii) and (iv) of
Definition 2.1 for all (y, S') E T*(af2)\0 given that it satisfies them on ST*(cQ). Condition (i) obviously holds. As for condition (ii') we have X+(y,
cs') Y+(y, cs') =
(2iti)-'
f
A
(i.) di.,
IS'i = 1, c > 0
for j = 0, 1, ... , for it holds when c = I and due to (28), (29) it then holds for any c > 0 (by making the substitution i. -+ c-'i.). Finally, conditions (iii),
Understanding the L-condition
433
(iv) hold since col(X+(Y, c>0
(30)
and row(T+(Y, ce')Y+(Y, cS'));o = row(T+(Y, ')Y+(Y, '));=o E(c),
c > 0 (31)
where F(c) = diag(c1I); = o and E(c) = c diag(c'I )j= o are p1 x p1 diagonal matrices which are invertible. For the matrix -° defined in §2.1(5), the homogeneity properties (28) imply (32) f(Y, cs') = E-'(c)Y(y, ')F-`(c)
where E(c) and F(c) are as defined above. Hence for the right and left Calderon projectors P. and P+, (4), (6) and (29) give P+(Y,
')F-'(c),
F(c)P+(Y,
P+(Y, cc') = E(c)P+(Y, ')E-'(c)
(33) (34)
Remark If we write P+ = [Pki], where Pki are p x p blocks (i, k = 0, ... , 1- 1), then (33) implies Pki(Y, c5') = ck-'Pki(Y, S'), whence Pki is the principal
symbol of a pseudo-differential operator on dig of order k - i. Similarly
for P.
From now on it is always assumed that the y+-spectral triple of L(A) is chosen with the properties (29). By repeating the proof of Lemma 10.12 we see that the triple (X_(y, c'), 7-(Y '), Y- (Y'
V. (Y' -c'),
-1)i-iY+(Y, -c')) (35) -T+(Y, is a y--spectral triple of L7 .(2), and has the (positive) homogeneity
and P_(y, S') are similar
properties (29). The Calderon projectors P+(y,
(with similarity matrix F(- 1)), and the same is true for the left Calderon projectors.
Let M be a boundary operator and consider the associated r x p matrix polynomial B(A) = n-4(y, (c', i.)) _
BB(y, J=O
The kth row of n59 is homogeneous of degree Mk in the variables _ 1.), whence the kth row of the coefficient B; is homogeneous of degree Mk - j. In other words, f')c-t,
c>0 (36) for j = 0, ... , µ where M(c) is the r x r diagonal matrix [c'"' Ski]. Now let B,(y, cS') = M(c)B;(y,
B;(Y, ')X+T+ 1=0
(37)
434
Boundary value problems for elliptic systems
(we omit the argument (y, S') in X. and T. to simplify the notation). By Theorem 10.3(iii), 9 satisfies the L-condition relative to sad if and only if is invertible for all (y, S') E T*(Q)\0. By virtue of (29) and (36) c > 0,
A +(y, ce') = M(c) A+(y,
(38)
so it suffices to verify invertibility of A+(y, S') when (y, ') E ST*(af2).
Remark If the rows of n-4(y, g) are not only positive homogeneous but also negative homogeneous in = (c',).) then the B;s satisfy the homogeneity property (36) for all c # 0. In such a case we also define N
Y Bj(y, ')X_T!
(37')
and it follows from (36) with c = -I and (35) that
A v(y, -c') = M(- 1) A-(y, ')
(38')
The next proposition shows that if u < I - I and n.4 is both positive and negative homogeneous then Ate, = A;, if and only if n-4l = n g, In other words, given the pair (X+, T+), the principal part ng of -4 is uniquely
determined by A. Recall that u is the transversal order; there is no restriction on the total order of 2 because the mk s are permitted to be any real numbers. Remark The condition that nr be both positive and negative homogeneous holds if -4 is a differential operator.
Proposition 10.13 Let -4 be a boundary operator with transversal order u < I - I such that ng( , ) is positive and negative homogeneous in S. Then we have ng(y; ' BB(y, S')A" where Bj = Air'
k=0
T'`+1'+Aj+k+i +A.0
k-0
Tk Y-Aj+k+i
(39)
j=0,...,1- 1 (and Bj=0ifj>u) Proof Since D: 10 B,(y, i;')Xt7 = A±, this follows immediately from the fourth corollary of Theorem 2.17, because col(XTJ) j=o is invertible. As we know from §10.3 the f a m i l y of vector spaces T l+ = { U ; } is an r-dimensional vector bundle over ST*(8i2). A boundary operator -4 that satisfies the L-condition relative to .W defines a trivialization
9n+ - ST'(af2) x C' ue
(1
1
b.: '-' BY, 4' i dtll u1r=o
This means that there is a "topological obstruction" to the existence of boundary operators satisfying the L-condition. Conversely, if it is assumed
Understanding the L-condition
435
that M + is a trivial bundle then there exists a boundary operator 9 satisfying the L-condition. In fact we have the following theorem. Theorem 10.14 Suppose that W+ is a trivial bundle. Let A: ST*(a1) -' GL,(C)
be any C' matrix function (for example, A = 1). Then for any mk c 18, k = 1, . . . , r, there exsts a boundary operator 9 with tranversal order p < I - 1 such that the kth row of the principal part 7r-4 is both positive and negative homogeneous of degree mk and
A' = A on ST*(an). Moreover, the principal part of .4 is uniquely determined by these conditions. Since A is invertible, .2 satisfies the L-condition.
Proof In Theorem 10.11 we showed that if At+ is trivial then there exists a
j+-spectral triple of L(1.) that consists of C ' matrix functions on ST*(a12),
i.e. X, T+ and Y+ are C' matrix functions on ST*(an) of dimensions p x r, r x r and r x p, respectively. This triple of matrices can be extended by homogeneity so that it satisfies the properties (29). Now let (X_, T_, Y-) be the , --spectral triple of L(1.) defined as in (35). In view of the first corollary to Theorem 2.17, the triple
X =[X+ X.], T=\T+ T) and YYY+
(40)
is a finite spectral triple of L(i.). Extend A to T*(afl)\0 by the formula A(y, l;') = then define matrix functions At: T*(an)\0 -+ GL,(C) by
and
A,. = A and A_(y, S') = M(-1) A(y, -1;') Now define r x p matrix functions BJ, j = 0, 1,
.
.
.,I
- 1, by the formulas
(39) where A± is replaced by At A. Let B j+ (y, l') and B; (y, l;') denote the +
and - terms on the right-hand side of (39), then due to (29) and (28) it follows that for c > 0 1-J-1
1
1.
k=0 '-J-1
=M(c) A+(Y, c'). Y c"T+(Y, )cY+(Y, ')A;+k+1(Y,
')c-I
k
k-0
= M(c)BB (y,
')c-J
and, similarly, Bj (y, cy') = M(c)B! (y, 5')c-J. Hence (36) holds for c > 0.
In view of A-(y,1') = M(-1) A+(y, ') and the equations (35), a similar calculation yields B, (Y,
and
=M(-1)B,
Bj(y, -c') = M(-1)B; (Y,
')(-1)-J,
436
Boundary value problems for elliptic systems
so that (36) holds for c = -1, and thus for all c A 0. Finally, let -4; be a r x p matrix of (- 1) classical p.d.o.'s on ail with principal symbol equal to B, (existence by Theorem 8.28), and then let 1-1
J=o
By construction the kth rows of 2 has order Mk (k = 1, ... , r) and A; = A3. This completes the proof of the theorem, for the uniqueness of n. follows from Proposition 10.13.
Remark 10.15 The kth row of BXy, ') has order m,t - j, where j = 0, ... . I - 1. If mt I - I for all k, then all these entries have nonnegative order. For elliptic operators in the plane (n = 2) on a simply connected domain it the "topological obstruction" mentioned above is proper ellipticity; that is to say, sit has a boundary operator satisfying the L-condition if and only if it is properly elliptic. Necessity has been proved in Remark 9.31, and sufficiency is proved in Theorem 15.9. It should be noted that triviality of 9R+ is the basic restriction for existence of boundary operators only because we have assumed that the image of the
boundary operator lies in the sections of a trivial bundle. If the boundary operators had been permitted in the form J
A C' (aC°)- $ j=1
where G; are vector bundles over aQ (and the j th component of -4 is of order mj, j = 1, . . . , J) then a weaker restriction on 9)V would have been obtained which is fundamental: The necessary and sufficient condition for existence of a boundary operator
9 is that 9R+ nt p* 1V for some vector bundle V over ail.
y, and aQ is the cotangent bundle projection (y, the condition on 9)?' means that the fibres 9R.v ;. = Vr are independent of One could, of course, allow the elliptic operator d to act in vector bundles too, i.e. a: C"(i2, E) C"(i2, F) where E, F are vector bundles over Cl with the same fibre dimensions. We leave the details to the reader. See also Here p*:
Exercise 2.
Remark The results of Chapters 9 to 14 can be generalized quite easily to elliptic boundary problems (4, -4) where the operators srt and . f act on sections of vector bundles. We have chosen to fix attention on systems so that the connection with the matrix theory of Part I is more readily apparent. The following is another version of Theorem 10.14. Note that the condition G P'- = G can always be achieved since we can replace G by G P". (where
P', = P. is the left Calderon projector).
437
Understanding the L-condition
Theorem 10.16 Let G be a smooth r x pl matrix function on ST*(00) such that at each point (y, ') e ST*(Df) we have G P+ = G and G: im P.. - C' is invertible. Then for any mk a R, k = 1, . . . , r, there exists a boundary operator 2 with transversal order u < I - 1 such that the kth row of it is positive and negative homogeneous of degree mk and
G=
1 ` B(2)L-'())U 2niJ+
(41)
-1J] d A
The principal part of -4 is uniquely determined by these conditions. Since the rank of G is r, then . satisfies the L-condition.
+ is a trivial bundle, due to the isomorphism G: im P+ -+ C' and the natural isomorphism UV ^-. im P+ (see Corollary 10.10). Therefore we can apply Theorem 10.14. Let (X+, T+, Y+) be a
Proof First note that
y+-spectral triple for L(J.) consisting of smooth matrix functions on T*(00)\0, with the usual homogeneity properties. Now, let A = an r x r matrix function on ST*(oM). In view of Corollary 2.11, we see that (42)
Since G is surjective, it follows that A is surjective at each point of ST*(i1fZ). However, A is a square matrix so det A 96 0, and A defines a smooth matrix function ST*(3Q) GL,(C). By Theorem 10.14 there exists a boundary operator 9 with A+ = A. The equation (41) follows from (42). The principal part of the boundary operator 9 is uniquely determined in view of Theorem 10.14.
We now return to discuss operators of general Douglis-Nirenberg
type. We have ,ts+d(x, [o,kC'k], So T(C) =
S(c)ttsi(x, l;)T(c), where
S(c) _ [bjkc'k],
Lv.,g,(A) =
which implies that the coefficients, A;, of L(2) are homogeneous of degree Si + tk - j in the (i, k) entry (k and i = 1, ... , p), that is, A;(y, c%') = S(c) Aj(y, t;')T(c)c-',
j = 0, ... , 1
(43)
(for any c e C). Once again we let t be the map from T*(8f)\0 to ST*(oM)
defined by t: (y, ') -' (y, Lemma 10.17 There exists a y+-spectral triple (X+, T+, Y+) for L(2) with the following properties for any c > 0
X+(Y, ce') = T-'(c) X+(Y, ') T+(Y, CO') = cT+(Y, ')
(44)
Y+(Y, CO') = cY+(Y, OS-1(c)
where X+ e 2'(2-1E, C"), T+ e 2(2-1E), Y+ a Y(CP, t-1E) and 2- 1E is the
Boundary value problems for elliptic systems
438
pull-back of E to T*(8il)\0 of some vector bundle E over ST*(i)il). (E is necessarily isomorphic to 9 l+.)
Proof The proof is by the same method as in Lemma 10.12, where
X+(Y, ') = T+(y, ') = I 'IT+(Y, '/I'I), Y+(Y,
') = I
'I Y+(Y,
Remark The equations (30) to (34) still hold provided we define
F(c) = diag(c'T-`(c))'-' J-0 and E(c) =
O-
From now on it is always assumed that the y+-spectral triple of L(A) is chosen with the properties (44). By the same method as in the proof of Lemma 10.12 it is clear that the triple
(X-(Y, '), T(Y, '), Y_(Y, '))
(T(-1)'X+(Y, -c'),
-T+(Y, -'), - Y+(Y, -')'S(-1))
(45)
is a y--spectral triple of Ly (A); it also has the homogeneity properties (44).
Let 9 be a general boundary operator. The (k, i) entry of rz0 is
homogeneous of degree mk + ti, so that the (k, i) entry of the coefficient B; of the corresponding matrix polynomial B(1) is homogeneous of degree mk + t; - j. In other words
Bj(y, ce') = M(c) Bj(y, ')T(c)c-j,
c>0
(46)
for j = 0,. .. , .u. We define A' just as in (37) and then (38) holds without change. If ng is positive and negative homogeneous we define AR as in (37') and then (38') holds. In general for DN operators we have 2r = a < pl, in which case the leading coefficient of L(2) is not invertible. We have a finite spectral pair (X, T)
X = [X+
X-j, T= (T+
]
and
Y=
/Y Y+)
but col(XTj)1=o is not invertible. A pair of equations of the form E: 10 B;X±Tf = A# is still solvable for B1 since -6r -row(TTY);=o is a left
inverse of col(XTj)jso but the solution is not unique (see §4.2). Hence Theorems 10.14 and 10.16 still hold, except for the uniqueness (and of course the condition of the homogeneity of n. is modified: n-4 has DN numbers
mk+tj). We end this section with the proof of a result which was needed in the
proof of Lemma 9.36. Fix y e Al In §9.4 we said that an r x p matrix w(t) = w(', t) satisfying the system
Understanding the L-condition
l dl 01
r 71M Y,
1d
0,
)== lr,
439
t > 0, t = 0,
is called a canonical matrix. In terms of the notation introduced in the present chapter, we have
W(5', t) = X+(Y, ')
S'))- I
The family of matrix polynomials L, .(2). (y. S') E ST*(M), satisfies the condition (*) at the beginning of §10.3, hence by compactness of ST*(as2) there is a 6 > 0 such that the roots of det L,.,;.(2) = 0 satisfy Im J. 3 b > 0 for all (Y, ') e ST*(al). Since the roots of det L>.,442) = 0 are exactly the eigenvalues of T+(y, ') (see Proposition 2.7), it follows that II T' (Y S) e
+r,1Y.1')Il
< C e-et
for all IS'I = I and v = 0, 1, ... . where C. depends on v but not on 5'. It follows that we have an estimate of the form °
dr atyt)-dt 0 we can find A,, e C'°(I x S2) such that sup p IIA.(r, x) - A,(T, x)II < e. (,.X)Et"t
Then .ol` Y .4,(T, x)D', 0 < r < 1, is a C'° homotopy of elliptic operators (ellipticity is preserved if c is chosen sufficiently small), and, further, we can
join sit' and d', i = 0, 1, by a linear homotopy K.d' + (1 - K).Q(', 0 < K < 1. Finally, we can join the three homotopies to get
K.d°+(I -'K).d°,
00
for all j = 0, ... ,1- 1. Now let R(l) e 0S0`-J(Bf, r x p) be a (-1) classical pseudo-differential operator on Oil with principal symbol BJ(T, ). Then the
boundary operator 9' = 3J 9 J)DJ, has the same DN numbers as
further, the pair (.W1, _V`) satisfies the L-condition for all r E I due to (51) This completes the and Theorem 10.3(iii), so it is a homotopy in
446
Boundary value problems for elliptic systems
proof of the theorem because in view of (51) with r = 0 we have A' = 0
,
Remark Just as in Theorem 10.20 if n,4° is positive and negative homogeneous
then the boundary operators 9T can also be constructed with this property. We let Di+ = L'(y, S'), M_ = Aj(y, ') in the formula §2.2 (14) to obtain a solution of the equations f-1
j-o
Bj(r, -)X±(r, .)Tj(r,
A1
(51')
of the form
Bj(r, ) = Os'
1-j-'
I
k=°
T+(r, )Y+(r, -)Aj+k+1(T, ) 1-j-I
+ A,-
Y Tk (r Y (r. )A,.k+1(r,
k=o
and then continue as before. Theorem 10.24 Let (.Cl(, a.), r E I, be a homotopy of boundary value problems in BE'-'-'. Then the index of (.d,, -4,) is independent of T.
Proof To simplify the notation let us consider the homogeneous case, i.e. S, _ =sp=1,t1 We have .r1, = Y A,(r, x)D'. and when s >, 1, s > max(mk + 1),
(.W, A): W2(i2, C°) - W2 1(Q, C°) x X
W2-mk-
k=l
is a bounded operator for each T. If we can show that its norm depends continuously on r then Theorem 9.12 will complete the proof. The topology on C'(i2) is defined by the semi-norms sup
II/pllc, =
ID7ro(x)I,
d = 0, 1, 2, .. .
17I Ed
Now, multiplication by a function rp e C'(f1) defines a continuous map W2(f2) -+ W2(0) with norm bounded by a semi-norm of cp, i.e. IIp'ull, CII/pllcdllull, for some d; we can take d = s if s is an integer and d = [s] + I
if s e R+ (see [WI], p. 64). Since D': W'(0) -, W2 1' O) is bounded, it follows that for any r° a I
d>, s-I+1, T
where 11
Il denotes the operator norm for maps lVs2(Q, C°)
C°),
Understanding the L-condition
447
s > 1. By assumption, the matrices A. are continuous from I to C'(a p x p), so the continuity of r r+ dL as a map I - 2'(W z, W2-) follows.
For the boundary operator, the entries in the kth row of the coefficientoperator A(`' belong to OCISmk-'(ai2), k = 1, ... , r, and then the continuity of r --' 337 from
I . '(W Z 1-112(ail,Cr), x
W2mk-112(an))
k-1
follows by Theorem 8.81. Since Dp is bounded from WZ(il)
W2 j-112(aQ),
this establishes the continuity of r F4 R. as a map
I -+ YWS X 2>k_1
WS-mk-1f2 2
In virtue of Theorem 9.12, there exists 6 > 0 such that ind(&4, -4,) = ind(.c/,, Mtc) when IT - r01 < b. By connectedness of I = [0, 1], it follows that ind(d1, c,) = ind(.do, Mo) Remark A somewhat different way to see the continuity of the maps r r- 9j "
is as follows. Let 1 = Y cp; be a partition of unity subordinate to an atlas (U;, c;) on OR and as usual let O; E Co (U;) be functions which are equal to In the argument we may replace .'i by cp;_4,`O; since the 1 on supp second operator differs from the first by an element of OP- °°(aQ) which can
be deformed to 0 by means of a linear homotopy. In that case it follows from the estimate (86) in Chapter 8 that
constllB,(r, ) - B,(ro,
for some d, where the norm 11 II on the left denotes the norm of operators WZ '-112 _. is some semi-norm in C'(ST*(ail)) (cf. Remark Iy2mk-'12 and I1 [( (.I) - 9;`0'11
8.88).
10.7 The classes V and To apply Theorem 10.24 we must be able to construct the homotopies sale. In general, this is problematic if the operators d are restricted to differential operators, i.e. symbols which are polynomials in ip. Consequently, we consider a larger class which keeps the polynomial nature in the normal direction, and in which homotopies are easily constructed. For simplicity, we consider only the case of homogeneous elliptic operators, i.e. with DN numbers s; = 1, t, = 0. Let a tubular neighbourhood of ail in R" be identitied with asl x (-1, 1), with ail x [0, I) c :.Q. Coordinates in ail x (-1, 1) are denoted by (x', x") and those in aQ by y = (x', 0). We also write ila = ail x [0, b). Let I be a positive integer; we consider operators of order I of the form sad = qt b + .d 1,
448
Boundary value problems for elliptic systems
where d' e OCIS'(R", p x p) has compact support in i2\S2i,2, and
.d b = j dj
(52)
j=o
where the coefficients, dj, are functions of x" a (- 1, 1) with values in
OClS'-j(ci2, p x p) that vanish for Ix,j > 2/3, and satisfy the following condition. For every coordinate patch U' on Of) and all cp, ' E Co (U'): pp .djli = Aj(x, D),
where Aj E S'-j(I8" x18"-', p x p) is (-1) classic.
Further, the leading coefficient salt is assumed to be a C' matrix function on Oil, rather than a pseudo-differential operator.
Remark S'-j(R" x R"-') is the set of symbols Aj E C z(P. x I8"-') satisfying the estimates
ID',Dl; j(x, s')I _< C,,(l +
Is'I)'-j-Isl
Vx E R", s' a
Rn- 1
and Aj is (-1) classic if it has a principal part nAj E C' (R" x18"-') satisfying Definition 7.21.
It follows by Chapter 8, Exercise 18, that .dj:= dj (& I is an operator of order I - j in the tubular neighbourhood M x (-1, 1). Since sfj = 0 when x" > 2/3 it follows that SSlb: W2 (D, C") -' W2 (fZ, C"),
S >' 1,
is continuous, i.e. db is an operator of order 1. The same is true of course for the interior part, .d', since it is an ordinary symbol. The principal symbol of of is defined to be the sum of those of de and
.d'. It is important to note that nsat can be made continuous for all (x, ) E f) x R "\O. We have n.d = n.d b + n.d'. By definition n.d' is already smooth for all # 0, but for the boundary term '-1 n -Vb(x,
j=o
ndj(x', S'; .))j + .ol(x, x");',
there is a possible difficulty at ' = 0. But the assumption on c.,tj means that x (-1, 1) (independent of we can regard it as a smooth function on s'), while for j = 0, ... , l - I the principal symbol of slj is homogeneous of x (-1, 1), so we degree l -j > 0 and it is smooth for (x', S'; x") a may define n.91j = 0 when S' = 0. In this way, n.de is continuously defined on
x (- 1, 1), including ' = 0. As usual, the operator d is said to be elliptic if det n.d(x, ) # 0 for all x e t2, 0 ; E R". Note that the assumption that s.-It be independent of implies
det3td(y,y',).)=am(x)i"'+...,
m=pl,
with am(x) independent of i;', and am(x) # 0 due to ellipticity with (0, 1). Hence the degree of det n.d(y, S', i.) in the variable i. is m = pl, and
Understanding the L-condition
449
sil is an invertible matrix. The continuity (and homogeneity) of xd implies
Idet nsl(x, )J 3 c > 0. The set of all p x p elliptic operators of order 1 is denoted L` (or sometimes t,,,,,).
The boundary operators that we consider are of the same type as before, i.e. N
J=O
where Mj is an r x p matrix with kth row having entries in OC1SO'-i(Ofl), k = 1, . . . , r. For convenience we assume k ( l - 1. The pair (sad, -4) is said to be L-elliptic if sl is elliptic and -4 satisfies the L-condition, i.e., we require only conditions (i) and (iii) of Definition 9.30. We denote the set of all such L-elliptic boundary value problem operators by OW-1. In the case that d is a differential operator, we showed in Remark 9.31 that proper ellipticity is a consequence of L-ellipticity (by virtue of the
negative homogeneity of icd) but in the present context all that we can conclude is that the number of roots of det xd(y, ', A) = 0 in the upper half-plane Im A > 0 is independent of 1;', i.e. equal to the number of boundary conditions. (Note: r = 0 is possible; for instance, the operator X/.' in Theorem 11.1 is Fredholm without boundary conditions.)
Now consider Theorem 9.32 for the operator (d, 9) e'8(l' "'. We claim that the theorem remains true as stated. Here we are dealing not with general
DN operators, but only with operators sad whose principal part is homogeneous of degree 1, i.e. si = 0, tj =1. Note: In the statement of Theorem 9.32 the letter 1 was used in a different sense, as an index for the Sobolev spaces. Lemma 10.25 Let d a W. Then for any cp a C°°(S2) we have [.d, cp] = stdcp - cps E OPa-'((i),
i.e. an operator of order 1- 1.
Proof First extend W to R" to obtain W E Co (R"). By definition, d = szdb + sad' where .4'e OS'(R") has support in f2\(2112 and
sib = ± x j=o
where .Wj, j < 1, has the properties mentioned above and .c1 is a C°° matrix function. Then st?bcpu = tpS.4,bu + dbu, where szlb = [-,j (c) and (c) - (d) are functional analytic and hold once again. It is also clear that the proof of (d) (a) is still valid due to the lemma above. For future reference we summarize some of the properties of the boundary value problem operator 2 = (sal, -4) in the following theorem. However, we
postpone until §14.4 the proof that the image is the orthogonal space of a subspace c C °°.
Theorem 10.26 If (d, 9) e
2 _ (d, . ): KIM, C")
wag '(s2, C') X x wZ mk -112(aQ),
s > 1, is Fredholm. The kernel is a finite dimensional subspace of C`'(S2, Cp) and
the image is the orthogonal space of a finite dimensional subspace of r
C (Q, C") X kx1 C -(On).
Thus, the index is independent of s. It is also independent of the lower-order terms in sal and in A If (sal, M)
the principal symbol of 9 has the form B,,4.(2) =
arB(y, (g', ),)) = Z;=O Bj(y, l;')V, a matrix polynomial in A. The results of
§§10.1 to 10.3 carry over to Sll;'' since they are essentially just results concerning the associated matrix polynomials depending on parameters. In particular, by Theorem 10.3, the L-condition for (d, 2) holds if only and only if
#0
detA,=det i=o
where (X+, T+) is a spectral pair of LyX(2) = nsal(y, (g',d)) with respect to the eigenvalues in the upper half-plane Im A > 0. The same is true of §10.4, except that there is no connection between spectral triples for the upper and lower half-plane as in (35) because nd is now only positive homogeneous in . In the following definitions, it is to be understood that we work with a fixed tubular neighbourhood of an. A homotopy of elliptic operators is a
Understanding the L-condition
451
family of elliptic operators in W,
srtt=db+0 sa1jD;,. Choose b < i so that
the support of sat' does not meet aft x [0, S], choose (p a Co ((- S, S)), 0 < p < 1, with qp(x") = I for 0 < x" < 6/2, and put
.d = .db + d',
where d° =
.ss1,(x', x"[1 - Tcp(x")])D,1,.
452
Boundary value problems for elliptic systems
The operators .WW have principal symbol arbitrarily close to that of d if the support of (p is chosen sufficiently small; since d is elliptic, it follows that .d, is elliptic for all 0 5 r WZ(S") and the restriction oper-
ator z+: WZ(S") - WWI) is also continuous. Let u e ker 7, then Ti+u = 0 and u = 0 in S"\ V. Conversely, let u e ker t; then u = 0 in S2\ P so it can be extended by 0 on all S"\ P giving us a function
u e W2(S") with Tu = 0 and :+u = u. Hence the restriction operator 4+ yields an isomorphism from ker T to ker t. Now suppose that N is a finite dimensional complement of im T, WZ(S") = im T ® N.
(56)
It is clear that t+(im T) = im T, and we claim that WZ(12) = z+(im T) (D a+N.
(56')
Indeed, by applying the restriction operator z+ we certainly obtain WZ(i2)=x+(im7)+4+N.Now, if tiet+(im7)nz+N then u`=a+Tw=t+u where w e W2(S"), u e N. Hence
Tw-u=0 in
S2;
by the assumption on T this means Tw - u e im T, whence u e im T n N = 0, so u = 0. Further, s+ is injective on N, for if u e N and a+u = 0 then u = 0 in
i2, so Tu = u e im T n N = 0. We conclude that im T and im T have the same (finite) codimension. Conversely, if we assume that in it has finite codimension then it is easily verified in the same way that im T must have the same finite codimension. The proof is complete.
Suppose that I= 2s and the Dirichlet problem (d, 2) satisfies the Lcondition. According to Theorems 3.14 and 10.9, the matrix polynomial L,,4.(1) = rrd(y, ' + An(y)) has a y+-spectral monic right divisor L; 4.(.%) of degree s, with coefficients depending smoothly on (y, r;') e T*(8Q)\0:
Ly.4,(.4 = Ly.t'(A)Lr t'(.)
Of course, it then follows that
is a y--spectral left divisor of
degree s with invertible leading coefl dent (= itd(y, n(y))). Since Ly.,C, ,) _ c t Ly, {.(A/c), it follows that
44'(A)=c'44. /c),
c>0,
456
Boundary value problems for elliptic systems
that is, the ;.j-coefficient is positive homogeneous of degree s - j (j =
0, ... , s - 1) and the ;.'-coefficient does not depend on
Note that
'.
L-o(%) = n,d(y, n(y))i.' and L,',().) = Consider now the following homotopy in x)1`)3':
LY.t,-t,,
+ irIs'l)'L> a
iric'1),
0
r < 1; (57)
thus L° = L and Lr.0
+il 'I)
L
where a(v) = n,d(y, n(y)). By Theorem 10.30, this homotopy may be lifted to L', i.e. there exists a homotopy from s.Vt in L' such that 4° = d and
n.d1 = a(y)'A' on c?Q,
(58)
for some invertible C' matrix function. By (57) the Dirichlet problem (s, 2) satisfies the L-condition for all 0 0, thus r = 0 and no boundary conditions are necessary. This means that .4+ is a Fredholm operator by Theorem 10.26. To determine the kernel of .4+, which we know is in C=(fl) by Weyl's Lemma, we choose a positive density (measure) in fl which in the
collar is the product of one in ail and the Lebesgue measure, dx". If u e C'(S2) then since sd + = (p (D" + iA') we have
2 Im(sd+u, u)n = i-1((d+u, u) - (u, sd+u)) = 2 Re((pA'u, u) + i-'((cpD"u, u) - (u, (pD"u)).
Applications to the index
467
For the first term, if we use Girding's inequality on the operator A' (see Chapter 8, Exercise 10) we obtain
xp))m dx
Re((pA'u, u)n) = f 1 (p(xp) 0
f -c"
f
dx
(p(x,,)Ilu(-,
0
-cllulln,
(1)
For the second term, an integration by parts gives us u) - (u,
_ - 1' q
au
ax
dx -
u
q (xp) u,
au
-\11
Ox. an
an
dx
8
_ - f 1 Ox (q (x )u, u)an dx + o
Ilullan
u)an dx, fo
- cllulln
-cllulln
(2)
Adding the inequalities (1), (2), and noting that
Im .4+ = -Re(1 - (p) A(1 - (p) is bounded from below, it follows that 2Im(,V+u, u)
-cllulln
(3)
Now we consider d... As before the ellipticity of .sd is clear. Also L_(,t) = x.sa?"_(y, (c', A)) =
-A +
and we have exactly one characteristic root i I
y e 8d2, ' I,
g' # 0, in the upper
half-plane, so r = 1. The solutions of the equation
L (i dt)u = 0 are u(t) = const e-'14'1, thus 9Q = span{e-'I I } and (,sat'_, s_) obviously satisfies the L-condition. By Theorem 10.26, it follows that (.sat' , ._) is Fredholm and ker(d , R_) c C°°(f). Analogous to the above calculations, we obtain for u e C°°(0) that 2 2 -Ilullan -CIIuII0, (note the term - Ilullaa appearing with a negative sign). If R_u = ulan = 0
2Im(sz?°_u,u)
468
Boundary value problems for elliptic systems
then
2 Im(d_u, u) >, - cllulln
(4)
In view of (3) and (4), the kernels of sal+ + itI and (sat + itl, M_) are equal to {0} if 2t > e. Suppose that (sl+ + itl)u = 0 and u 0. Then we have
0 = 2 Im(d+u + itu, u) = 2 Im(d+u, u) + 2t(u, u)
> 2 Im(d u, u) + c(u, u) in contradiction to (3). The same proof holds for (sal_, M_). Replacing alf by salt + ill, we can assume in what follows that the kernels are equal to {0}. The densities on t l and on ail introduced above allow us to construct the adjoints of d+ and (s+0_, B_) with respect to the scalar products on Q and on if . By virtue of Theorem 10.26 we know that the images of r1 and (sal_, -'_) are the orthogonal space of a subspace c C'. Assume now that v c- C m (S2) is orthogonal to the image of &/+, that is
(v, d+u)n = 0
for u c- C'(01).
Integrating by parts (or using a Green's formula) we get (sa* v, u)n + (v,
0,
Vu E C'°(Q).
For u e Ca (i2) we obtain (.ul* v, u)n = 0, and since Co (i2) is dense in L2(11) this means that sat *v = 0. Hence (v, u),-.Q = 0 for all u e C'°(Q), so v = 0 on
dig. Since -sal+ is of the same form as a_ it follows if the number t is chosen large enough that v = 0. Thus sst+ is an isomorphism if t is sufficiently
large, and in any case ind sl+ = 0. Suppose now that v e C"'(i2), h c- C°°(oi2) is orthogonal to the image of (jV_, M_), that is, (v, A(_u)n + (h, u),,n = 0
for u E C"(().
Taking u from C0 '(Q) and integrating by parts we obtain that 0 = (v,sat-u)n = (sd*_v, u)
Vu e Co (i2).
Since C0 "(fl) is dense in LZ(Q), it follows that sl* v = 0 and because -.a*-
is of the same form as d+ we get v = 0 and then h = 0. Thus (d...,._) is also an isomorphism if t is sufficiently large and in any case ind(d_, _) = 0.
Let p and q be any nonnegative integers. In the sequel, we refer to any B (d_, M-), that is,
direct sum at+ e .. e W+ a (at_, R_) e at-1,
(0
10 eze".0
as a trivial elliptic b.v.p. operator. Here p is the number of copies of d+ and q is the number of copies of (.o(-, M_). On the boundary ail, the principal symbols ofd are acl+(y, (c', i)) = i. + ilg'I and ns.)1 (y, (c', A)) = -i. + so we make the following definition.
Applications to the index
469
Definition 11.2 An elliptic operator st c- (9(p+q)x(p+q) is said to be trivial on
the boundary if
td(y,
ill;'(lp 2)) =
),
(-)
YE 09
,
where a: OU -, GLp+q(C) is an invertible C°° matrix function. Such an operator always has a boundary operator satisfying the L-condition, namely, .4 = [0 Iq].
We also say that the boundary value problem operator (d, .) E M(p°q) X (p+9) is trivial on the boundary.
Our aim is to show that the direct sum of an arbitrary b.v.p. operator (sal, -4) eVtip ° with a suitable trivial r x r elliptic b.v.p. operator (with q = 0) is homotopic in !5(g(P0+r)x(p+r) to a b.v.p. operator (sa/i,R1) that is trivial on the boundary (Theorem 11.4). The index problem for (.rli, 611) can then be reduced to the index of an elliptic operator on a compact manifold without boundary (Theorem 11.5), to which the Atiyah-Singer theory can be applied. The basic idea for the construction of this homotopy is to begin on the level of matrix polynomials, Wl$', and then apply the homotopy extension theorem 10.30 for elliptic operators.
Lemma 113 Let E be a finite dimensional complex vector space, and let A E £P(E) with no real eigenvalues. Let P = (27ri)-' J+ (12 - A)-' dl be the Riesz projector corresponding to the eigenvalues of A in the upper half-plane.
Then forallc>0 A,:= (1 - T)A + t(icP - ic(1- P)) has no real eigenvalues for 0 < t < 1 and the Riesz projector for AT coincides with that of A,
dt.
P=(2R1)-i f (IA-AT)-'dl,
Proof Since P is the Riesz projector corresponding to the eigenvalues of A in the upper half-plane then, with respect to the direct sum E = im P ® ker P, we have
A=
0 A_) A+ 0
where A+ e 2(im P), A_ e 2(ker P) with spectrum contained in the upper and lower half-planes, respectively. Correspondingly, we have
At-\((1 - t)A+ + ict1+ 0
0
(1-r)A--icrl_
where I +,1 _ denote the identity operators in im P and ker P. Let A+ = (I - r)A+ + ictl+ and AT = (1 - r)A_ - icrl_. Clearly, the spectra of A+, At. lie in the upper and lower half-plane, respectively, so that A, has
Boundary value problems for elliptic systems
470
no real eigenvalues for 0 < r < 1. Moreover, (2ni)-'
f
(I).
- A,)-' d1 =
r ((1.1- A+)-
(2xi)-1
J+
+
0
1
0
1 d2
(Id - A`)'/
In the proof of the next theorem, one should keep in mind that if L(2) =12 - A is of degree I with no real eigenvalues, then the Riesz projector with respect to the eigenvalues in the upper half-plane Im A. > 0 coincides with the Calderon projector. See, for instance, the third corollary to Theorem 2.17. Also, the right and left Calderon projectors coincide; see Chapter 10 (4), (6) with l = 1 and A, = 1. It is important to note in the proof of Theorem 11.4 that it is the existence of the boundary oprator 9 satisfying the L-condition that makes it possible to deform the elliptic operator sat to one which is trivial on the boundary. (See the remark after Theorem 10.27.) Once again it follows that there are "topological obstructions" to the existence of boundary operators satisfying the L-condition.
Theorem 11.4 Let (d, 9) E'B@EP v (i.e.. is of order 0 in all variables, including the tangential variables). Let 2 = sal+ I, E Lr x r, where sal+ _ cp (D + iA') + i(1 - (p)A(I - (p) is the elliptic operator defined in Theorem 11.1 (i.e. 2 is a trivial elliptic operator as defined just before Definition 11.2). Then the operator
`[d is in
J'
[-4
(5)
0])
ti P °,) x (P+,) and when the support of pp is sufficiently small it is homotopic
in that space to a boundary value problem operator (sale, 91) that is trivial on the boundary; in fact, .1211 = s 1b + sal i where
a'- Bali = ip
r(D + iA')Ip
1
(6) I
and -41 = [0 1,], where a(y) is some invertible C1 matrix function on Q. Proof By Lemma 10.28, we may assume that the coefficients of V = sib + d'
do not depend on x near the boundary. Thus, for some 8 > 0, we have where a(y) = nd(y, n(y)) is an invertible p x p C' matrix function on 852, and A E OCIS'(Of2, p x p). Further, we may assume that the support of safi does not meet dig x [0, 6]. We divide the proof into two steps. In the first step, where we make an initial deformation of the operator d, the boundary operator is not required for the construction.
Applications to the index
471
Step 1. Let A be the principal symbol of A and let
P(y, ') =
1 f 2ni
(IA - A(y,
d1.
+
be the Riesz projector for L(4) = a ' nsat(y, ( ',1.)) = IA - A(y, S') corresponding to the eigenvalues in the upper half-plane. In view of the remark following Lemma 10.12, the matrix P is homogeneous of degree 0 in la'. Let A, = (I - T) A + r(i ll;' I P - i ll:' I - (I - P)). Since det L(A) # 0 for all a e 08, Lemma 11.3 implies that L,(i.) := Ii. -A, has no real eigenvalues when 0 < r < 1. Then by Theorem 10.30 applied to the homotopy a L,(1.) in a'$, there exists a homotopy sat, in (91 with solo = sat and n.sat,(y, (c', 2)) = a-L,(i),
0 mpJ,
0 < X.
J=O
and s,tj = d/y, D') a OCIS'-j(fl, p x p). Also, .d,, = A, = I is just a matrix function, not a pseudo-differential operator. Set ,
satj:= i CjkdkR' k, k=0
where R is a_parametrix of A'. Note that sat j e OClS°(852, p x p) and by (17) we see that sltj has principal symbol A j(y, c'). Corresponding to (15), we have ,4b
= Y s?j-(D - iA')j(D + iA')'-j,
0 < x < 5,
(27)
j=o
apart from terms of order - oo in ail and order 0. When f = 0 we have u = X+ e'^T' c for some c = in the
base space of the spectral pair. To satisfy the boundary conditions we must have c = (A;)-'g. Thus the Poisson operator for (1) is given by Kg = X. el`^T'(Ay)-'g'. It is also possible to express the Green operator G in terms of the spectral pair Tff)) as we show in §12.2. 487
488
Boundary value problems for elliptic systems 12.1 Extension of C°° functions defined on a half-line
If f is a function defined on R, we let 2 "f denote its restriction to the half-line I8+ = {t; t > 0}. The restriction map t+: C'°(R) -' C'°(18+) is continuous, and we define
Cr(I +) = 2+C" (R),
9'(18+) = 2+So(68),
where Y = .9'(R) is the Schwartz space on R. These spaces are easy to characterize using the following lemma. Lemma 12.1 Let m j e C, j = 0, 1, 2, ... , be a given sequence of constants. Then
for any r > 0 there exists f e Co ((-e, e)) such that
f '(0) = m j
for all j.
Proof Choose g e Co ((-E, e)) such that g(0) = I and dk
g(0) = 0
for k > 0
(for instance, choose g such that g(t) = 1 when -E/2 < t < E/2). Now, our aim is to define the modified Taylor series f(t) _ Y_ g(t/Ej)tjmjlj!
(2)
J=o
with suitably chosen Ej. By taking t/Ej as the new variable we find that ldk dtk
g(t/EJ)t'mili,
Ck
k,
and we choose 0 < ej < 1 such that Ck Jef-k 2 for k S j - 1. Then the kth derivative of the jth term in the series (2) has magnitude k, so the series (2) and all series obtained by termwise differentiation are uniformly convergent. Hence f e C'°(R), and its support lies in (-e, e). Corollary 12.2 C °`(18+) is the set of all functions u e C'°(R+) such that the limit lim
r-o.
dk dtk
u(t) exists
for all k = 0, 1, 2, ..... 9'(68+) is the set of functions u e C'(A+) such that u and all its derivatives are rapidly decreasing as t - + oo, i.e. k
U(t)
const (I + t) -',
t > 0,
11 kk
for all j, k = 0, 1,2,.... Proof The necessity is clear. To prove sufficiency let u e C °°(R+). By Lemma
12.1 there exists f e Co '((- 11)) such that f U)(0) = u(i)(0+). Now let
B VP's for differential operators and connection with spectral triples
U(t) =
u(t),
t>- 0
If(t),
t 0 and U(t) = 0 when t < -1. The dependence of U on u is neither linear nor continuous because ej depends on the constants mj. For the sake of completeness, we also show that there exist linear, continuous extension operators /3: C'°(R+) - C "(R) and /3:.5"(R+) - .5"(R). Note that in the construction of /3, the constants cp are fixed, i.e. they do not depend on the given function u. Theorem 12.3 There is a continuous, linear exension operator (1: C"(R+) C'(R) such that :'fl = identity, i.e.
flu(t) = u(t),
t > 0,
where the topology on C'(R) (resp. C "(R+)) is that of uniform convergence of each derivative on compact subsets of R (resp. R+). If f has compact support then so does fif. Moreover, /3 restricts to a continuous, linear extension operator 3:.5"(Q8+) --+ .5"(R).
Proof We choose a rapidly decreasing sequence cp, p = 0, 1,-, such that m
Y pkcp = (-1)k, D=1
k = 0, 1, ...
(3)
(see Corollary 12.5 below). Also let tp be a CI function on R with qO(t) = 1 for 0 < t < I and tp(t) = 0 for t > 2. Define the extension of u by fu(t) = u(t),
t>0,
fiu(t) = Y- cp.(P(-Pt)u(-Pt), P.1
Then because p
t < 0.
or) the sum is finite for each t < 0; because Y-n 1 pkl cpl < oo,
all derivatives of flu converge as t - 0-; moreover, these limits agree with those for t - 0+ in view of (3). Thus, if we define flu(0) = lim,.p. u(t),
then the derivatives of f3f exist and are continuous everywhere on R, whence fife C'(R). The continuity of /3: C'(R+) . CZ(R) also follows from Ya 1 Pkjcpj < oo. This proves the first statement in the Theorem. The second statement follows immediately since flu(t) = u(t) when t > 0 and
flu(t) = u(-t) when t < -1.
Now we turn to the construction of the rapidly decreasing sequence cp satisfying (3). The following lemma is due to [Se 1].
490
Boundary value problems for elliptic systems
Lemma 12.4 There are sequences {ap}, {bp} such that by is negative and decreases to - oo as p -+ oo and k = 0, 1, 2, ... ,
Y (bp)kap = 1 , p=0 and
P-0
k = 0, 1, 2, ...
I bpIkI apI < co,
(4)
Proof Set by = - 2p. Then the solutions a() of N
E (bp)k'xp = 1, p-0
k = 0,..., N,
are by Cramer's rule and the Vandermonde determinant equal to
-IT>t(bj-b;)
xp - f1i
>r (bj - bt)
where by = 1 and bj = bj if j 96 p. Numerous factors cancel so we obtain xv
where
P
P-1 1+2j
Apjo2j-2p =F1
N
=jap+t n
B( °
1 +2j
2j-2p.
p;o 2J+2-p = 2-tp2-3P 12 and, using the fact that log x 0 and zero for t < 0. Then
L(D)u° = (L(D)u)° + L`yu,
(6)
where L`:= i-'[b.. D'-'S].` and 6 is the Dirac distribution at t = 0. Proof We have u°(t) = u(t)'©(t), where 0 is the Heaviside function: 0(t) = 1,
t > 0 and 0(t) = 0, t < 0. Note that (Du)° + u(0)' i -'b,
because D = id/dt, and by induction on j one shows that J
DJ+1u° = (Dju)° + i-' Y Yj-ku-Dkb k-0
forj=0,1,....Then t
L(D)u° _ Y- AJDJu° J=O
J-1
j=0
Aj((D'u)° + F'
t
0
AJDju
k-0 1
+ i-'
YJ- 1 -kuDka)
j- 1
Y- AJYj-1-ku'Dkd
j1 k=0
_ (L(D)u)° + L`yu where I-1
j
L`411 = i-' L. F, AJ+1q1j-kDkb = r1[6 ...D'-'b]'.T-V, j=0 k=0
for 41 = [Vj]0' e C'. There is a close connection between Lemma 12.6 and many of the formulas
of Chapter 2. Assume that L(i.) has no spectrum on the real axis, and let u e SAti , that is, u e Y(f8+) and L(D)u = 0, t > 0. Then
L(D)u° = 0 + i-'[6 ...D'-'6]'.°yu
B VP's for differential operators and connection with spectral triples
493
and taking the Fourier transform in t (-+ T) we get L(T)fi°(T) =
Hence by solving for a(T) we find that 0°(t) = (27r)-'
eilt&(T) dT (7)
=(21zi)-1 J
(the integral is oscillatory). For t > 0 we may deform the contour of integration to the upper half-plane (by Cauchy's theorem) to obtain
u(t) = (2rri)-' I
di.,
t > 0,
(8)
where we have set 41 = 711. The Calderon projectors for L(i.) corresponding
to the eigenvalues of L(i.) in the upper and lower half-planes are the projectors in CP' defined by
=(2ni)-1
Pt
I
f
-1I
)L1[J....i_1I].d.
see §2.2(7).
Choose corresponding spectral triples (X+, T. Yt) for L(i,), then we have
Pt see 2.2 (8),
Tt'Y+]f°,
[YY... T3'Yt]
(9)
1,
(10)
0,
(11)
see Corollary 2.11
[Y{...T$'Y+] see 2.2 (12)
and hence (P')2 =
P+,
(P-)2 = P- and P-P+ = P+P- = 0. For any
W e CP', if we define tt a 9121 as in (8) then
;u = lim [u(t) . D'-'u(t)] = PAI. t-O
In particular, given u e TIL' then (8) implies that !u = P+711. Thus, we see once again that the image of P+ consists of the Cauchy data of solutions UCW W.
From now on we assume that L(i.) has invertible leading coefficient; therefore L(A) has no spectrum at infinity (see §4.2). Because we are also assuming that L(A) has no spectrum on the real axis, then P+ and P- are complementary projectors, P+ + P- = I (see the remark at the end of §3.2).
494
Boundary value problems for elliptic systems
For future reference, note that L(A) has an invertible leading coefficient with no spectrum on the real axis if it is the matrix polynomial associated to an elliptic operator .sad a Ell'.
Proposition 12.7 For '1l a C'', let OD
errrL-'(T)[I...T'-'I]--W&dt
v(t) = (2ni)-1 J
(the integral is oscillatory). Then v coincides for t > 0 with an element v+ a WZi and when t < 0 with an element v_ a ¶lla . For t = 0 we have the jump relation
j = 0, 1, ..., I - I.
DJv+I,=o, - DJv-I,_o- ='?I,
Proof For t > 0 the path of integration can be deformed to a contour in the upper half-plane, while for t < 0 it can be deformed to the lower half-plane, so we obtain v(t) =
e'`'' L-'(A)[] ...A'-'1]
1
2ni
_
°I& dA. 0(t)
+ 1
2ni
f-
where 0 is the Heaviside function. This gives v(t) = v+(t) + v_(t), where Div+Ir=o - DIv_I,=o- = P+qj + P-ill = V. Corollary If V e CO then 9l = yu for some u e 9JtL' if and only if v_ = 0, i.e. if and only if W e im P+. The Cauchy data of solutions in ¶O and tU are complementary subspaces of C°'.
Next we consider the fundamental solution of L(D). Lemma 12.8 We have L(D)E = 8, where
E(t) = I f 2n
.
I e"x L- '(A)
e1iz
(12)
27r
Also, yEL` = P+ and y-EL` = - P-, the Calderdn projectors. (Here ydenotes the Cauchy data operator taken from t < 0.)
Proof By Fourier transformation t - r, we have L(T)E(T) = 1,
so that E(r) = L- 1(-r). Thus
E(t) = (2n) -'
errs L- 1(T) d r
J and the formula (12) follows from Cauchy's theorem as in the previous
BVP's for differential operators and connection with spectral triples
lemma. Now, since (EL')
EL` =
495
EL`, then
eut L-'(T)[I...T'-'I]-,Ydt
1
2ni
and by Cauchy's theorem
EL =
J
I
e''' L-'(;)[I.. A'-'I].T
2ni {
-
'-.f e'"L-'(A)[1
@(-t)
-
2ni
(13)
By taking the Cauchy data from t > 0 and from t < 0, we obtain the second statement in the lemma (see (9)). It follows from (12) that
yEf °eimP-,
for all f e °.9'(R,),
where f ° is defined as in Lemma 12.6. Indeed, since
Ef°(t) =
f
T
E(t - s) f(s) ds,
0
then fort > 0 we have
i-'Ef°(t) = 1 f f e'u-,r' L-'(;.)f (s) ds d2 tai
_ I
2ni
X,
-
eW-Su L-'U)f (s) ds di.
eW-OT. Y, f(s) ds
J0
-f
X_ eru-srr- Y_ f (s) ds
so applying the Cauchy data operator y we obtain
i-';Ef° _ -col(X_T! );=o.
f
e-sT- Y- f(s)ds.
(14)
0
Thus 7Ef°eimP-. Remark In deriving (14) we used the fact that
X,T+Y,+X-T!Y_=XT'Y=
1
f
2ni r
).1L-'(i.)d).=O
for j = 0, ... ,1 - 2, since L-'(i.) is of order -1 as 121 -+ ac. (Recall that L(2) has invertible leading coefficient.) Here r is a contour enclosing all the eigenvalues of L().).
496
Boundary value problems for elliptic systems
Theorem 12.9 Let f e °.9'(R+) and W e Cr'. Then the system
yu=V, t=0,
L(D)u=f, t>0,
(15)
has a solution u e °,9'(18+) if and only if PA/ = 7Ef °. Moreover, the solution is unique and is given by
u=-a+Ef°+r+EL`gl.
(16)
Proof Suppose first that the system (15) has a solution. By Lemma 12.6, we have L(D)u° = f ° + L`;u.
If we apply the fundamental solution operator E, then
u°=Ef°+EL`yu
(17)
and, by taking the Cauchy data from t > 0,
yu=7Ef°+;EL`yu. Since ;7EL`=P+ by Lemma 12.8, then,/=yEf°+P'4'/, whence P-1/=7Ef°. Conversely suppose that P-W! = yEf °, and define w = Ef ° + EL`4/.
(18)
Then L(D)w = f ° + L`'W and if we let u = t+w then L(D)u = f + 0. (Note: t+L` = 0 since t+D'tb = 0 for all k.) Hence
yu = Ef° + 7EL`?I = P-W + P+?/ = ?/. Finally, the uniqueness of the solution follows from (17). The proof is complete.
Remark If w is defined by (18) then w(t) = 0 for t < 0 since
t < 0,
L(D)w = 0,
and y-w = y'Ef ° - P-yY = 0. This shows the correspondence between equations (17) and (18), that is, w = u° where u = t' w. In view of Theorem 12.9, P-714 is fixed once L(D)u is known. Once a basis
is chosen for im P' = C', the operator P7 can be regarded as a boundary operator.
Corollary If f e
and -1 a im P' then the system
L(D)u = f, t > 0, has a unique solution u e
P+;u = *, t=0
given by
u=z'Ef°+t'EL`)l/.
(19)
Proof By applying the operator t+ to (17) we obtain u = t+Ef ° + t+EL`;u. In view of (13) and (11) it follows that t' EL`P - = 0 and hence (19) holds.
BVP's for differential operators and connection with spectral triples
497
Conversely, if u is defined by (19) then L(D)u = f + 0 and
yu= yEf°+yEL`qi =yEf°+P+9l=yEf°+9l. Since yEf ° e im P-, then P+yu = 0 + V. Proposition 12.10 Let y-u = Iim,-.°_ [u(t). D'- i u(t)]T be the Cauchy data taken from t < 0. If f e °.9'(08+) then y-Ef ° = yEf °, and Ef ° e °C'-'(68).
Proof Since L(D)Ef ° = f ° = 0 when t < 0, then -Ef° e im P-. Lemma 12.8 gives
i-'Ef°(t)= -21r:
fo'O
f-
eu'-s)AL
when t < 0, and by virtue of (14) it follows that y-Ef ° = yEf °. The fact that Ef ° e °C'-'(08) is now clear since Ef ° E C' for t 9& 0 and we have just shown that the Cauchy data taken from t < 0 and t > 0 coincide.
We now consider the boundary operator, B(A) = Yj=0 8;A', an r x p matrix polynomial with P < 1. Let
At =At =
B;Xt T+
(20)
J=O
and assume that the A-condition holds, i.e. det A. # 0. If g E C', the system
L(D)u = 0, t > 0,
B(D)u = g, t = 0,
has the unique solution u = Kg in °.9'(08+) given by Kg(t) = X+ e"1r. &+-I 99
t > 0.
(21)
The linear map K: C' - °.9'(08+) is called the Poisson operator for the system. Note that u = Kg is the unique solution of L(D)u = 0 in .9'(18+) such that yu = Sg, where
S=
(22)
is a pl x r matrix. Also, we write B(D)u = B`yu, where B` = [B°. . BB 0. . . 0]
is an r x pl matrix, and we define the pl x pl matrix S' by the equation S' + SB` = 1P I.
(23)
We claim that S'P+ = 0. Indeed, S' = I - SB' and it follows that col(X+T+);=o - col(X+T+);=o A+' A+ = 0,
hence S'P + = 0 because P+ and col(X+ T+ );=o have the same image. For future reference we summarize the properties of S and S' in the following proposition.
498
Boundary value problems for elliptic systems
Proposition 12.11 Let P+ denote the Calderon projector for L(1) with respect
to the eigenvalues in the upper half plane Im A > 0. There exist unique matrices S, S' of dimensions pl x r and pl x pl, respectively, such that
(i) B`S=1 P+S=S;
(ii) S' + SB` =1,1, S'P+ = 0
Proof The existence of the matrices S, S' has been shown above. Now, suppose that S is any pt x r matrix satisfying the two equations in (i). Since P+ and col(X+T+ );=o have the same image, the second equation implies that S = col(X+T+)j-=
for some operator M e .(C', fit+) where Dt+ is the base space of (X+, T+). Now, if we multiply this equation on the left by the matrix B` we have
B`S = A+M. Since BS =1, it follows that M = (A+)-'. This proves the uniqueness of S, and uniqueness of S' then follows from the first equation in (ii).
In view of (16), we can write u in terms of L(D)u and the Cauchy data yu. By means of the equation (23) we will then write u in terms of L(D)u and B(D)u.
Theorem 12.12 Let f E Pb°(18+) and g e C'. The unique solution ofL(D)u = f, t > 0, B(D)u = g, t = 0, is given by
u = z+(I + EL`S'y)Ef ° + z+EL`Sg.
(24)
Proof We already know that there exists a unique solution for given f and g, so it remains to verify the formula (24). Let V = yu. Since P-'P1 = yEf °, see (14), we find that S"fl = S'P-1V = S'yEf°. (Recall that S'P' = 0.) Also, B(D)u = g so we have BV = g. Now, in view of (23), we see that ql = SV + Sg, and (24) follows from (16).
Note that K = EL`S, where K is the Poisson operator (21). Indeed, (13) with t > 0 gives
EL'Sg=
eraL-1(,1)1...,1+-11].°1°da Sg,
1
2iri
+
t>0.
In view of (10), (22) and Proposition 2.3, it follows that
EL`Sg = X+ eitT. A+'g = Kg. The formula in Theorem 12.12 can therefore be written in another form:
u = Ef ° + K(g - B(D)Ef °),
t > 0,
(24')
To see this, we use (23) and obtain EL`S'yEf 0 = EL`yEf 0 - EL`SB`yEf °.
B VP's for differential operators and connection with spectral triples
499
If we recall that yEf ° e im P- then (11) implies 0,
_ -KB(D)Ef°, since B(D) = B`y, so (24') holds. Definition 12.13 The operator °So(R+) -i'So(68+) defined by
Gf = +(I + EL`S'y)Ef °, is called the Green operator.
Note that u = Gf is the unique solution of
L(D)u = f, t > 0, B(D)u = 0, t = 0 (25) Theorem 12.12 says that every solution can be written in the form u = G f + Kg.
Proposition 12.14 The "singular part" of the Green operator,
Gi f = :+EL`S'yEf o, is given by
G, f = iX+ eUT* A+1 A_ .
f `O
aY_f(s) ds,
t > 0.
0
Proof In view of the calculations above, we have G, f = -KB`yEf 0, t > 0, and then by (14) i''G1 f = KB` col(X_ Tj );=o J
e- IT- Y_ f (s) ds. 0
Recalling the definitions (20) and (21), we obtain the desired formula for G,. For the sake of completeness we write out a few more details about the
Green operator and then give an example. Recall that the equation L(D)u = f, t > 0, has the particular solution u = Elf ° =
f- Go(t, s)f (s) ds,
t > 0,
0
where G0 (t , s)
=
X_ e'(r-s)T- Y_
t < s
X + e ru->>T. Y+
t>s
We call Go a pre-Green operator, it has the following properties:
(a) L
dl (
CI
(b)
1d
d dt)Golt=s` -
(i dt)
j
0 ifj=0,...,1-2 ifj =I - 1
(26)
Boundary value problems for elliptic systems
500
for s > 0, t > 0. Note: (b) follows from
ifj=O...; 1-2
X+T+Y++X_V. Y_ =XTJY={0
J-
Remark 12.15 Since sp(T+) lies in the upper half-plane and sp(T_) in the lower half-plane. G°(t, s) and all its derivatives are bounded by a constant times a -61'
for some S > 0. Consequently, f - Ef ° is an operator
PY(R+) - °.q(R+) (This was implicitly assumed in the definition of the Green operator.) We would now like to determine a function G(t, s) such that
u=Gf= I
G(t,s)f(s)ds 0
is a solution of the system (25). In view of Proposition 12.14, we see that G(t, s) = G0(t, s) + X+ e"* N(s)
(27)
where N(s) = A+'0_ e-`3T- Y_ e °.'(68+). Note that G has the same properties (a), (b) as G° but in addition (c) B(
for all s > 0.
-)G(t, s)1,=o. = 0, .
Indeed, to verify (c), we obtain by differentiation of (27)
ld
Id
B(i dt)GIi°O. = B(i
k dt)G.,,=o +
B,X+T+
N(s)
= -( BjX_T )-a-uT-Y +(BJX+T+).N(s) = 0,
since At = j=o BjX±Tt. Example Suppose that the matrix polynomial
L(.?)=ird(l;',A)=1)2+AIA+AO arises from a second-order elliptic operator ,V(D), D = i-' d/dt, for which
the Dirichlet problem satisfies the L-condition. This means that Xt is invertible and L(.) has yt-spectral right divisors V. - St, where St = Xt TT X
(see §3.7). For the system
L(I
dt)u = f,
t > 0, u(0) = 0,
(28)
B VP's for differential operators and connection with spectral triples
501
we then have 0± = X* and the Green function is G(t, T) = Go(t. T) + X. eitT, X +'X _ = Go(t. T) + errs . e -
Since X_ Y_ = (tai)-'
"s_ X _
e'rtT-
Y_
Y_
L-'(i.) di. = (S_ - S+)-' we obtain that the solu-
tion of (28) is
" Go(t, r)f (t) dt - e"s.
u(t) = J
f " e -rts_(S+
0
- S_)_'-f(T) dr
0
where Go is given by (26) and hence (S+
G0(t, t) =
fe
S_)-'
t -1. Let x" = e"', where log x is defined to be real when x > 0. The function on R defined by
x".=.x" ifx>0,
x+=0 ifx 0.
This is obvious when x 0- 0, but near x = 0 there is something to show. Let cp a C0 '(R), then
/d
s
\\
`-x4,(P) = -<x.',(P'> dx
x°lp'(x)dx =
ax'-'cp(x) dx 0 f,
_ .
where to get the third equality we integrated by parts and used the fact that Re a > 0 so the boundary term vanishes. We wish to extend the definition of x' to a e C such that the property (1) is preserved as far as possible. For cp = C0 '(R) the function
aH<x+,cp>= f:", x"V(x)dx 0
is analytic when Re a > - 1. The definition of <x+, (p> is then extended by analytic continuation. In view of (1) we have
<x+, p,> = -a<x+ ', rp)
if Re a > 0,
(2)
thus for Re a > -1 and any integer k > 0 we have <x+, cp> = (-1)'<x'++*, q 502
>/((a + 1)...(a + k)).
(3)
Behaviour of a pseudo-differential operator near a boundary
503
This shows that we can extend <X+, cp> analytically to be a meromorphic function in C with simple poles at the integers - k - I. Thus, x+ is a distribution of order - I - Rea.
In addition to x+ we define the function
X_ =0 ifX>0,
X_ =1XI° ifX - 1. This is the reflection of X+ with respect to the origin, <X° , (P> _ <X+, O>
where O(x)= (p(- x) .
The definition of x°- may also be extended by analytic continuation to any a e C not a negative integer. Consider now the gamma function
x°-' e-x dx,
r(a) = J
Rea > 0.
0
The same calculation which gave (2) for q a Co (11) also holds for rp(x) = e-", so we obtain
Rea>0.
I-(a+1)=al-(a),
(4)
Thus, we can extend r(a) analytically to a meromorphic function in C with simple poles at the integers -1 (with respect to the weak topology on 9'). In view of (2) and (4) we see that <X+, tP'> _ < - X°+ 1, (D) when Rea > 0. Therefore, X+ can be extended analytically to all a e C such that d dx
In particular, since Xo+ is the Heaviside function (= I when x > 0 and =0 when x < 0), then X+k = 60-1),
k = 1,2,....
In the next theorem we show that the Fourier transform of Xt a .9'(R) is equal to a constant times the boundary values (i; + i0)-°-' of the analytic function
C-°-' on the real axis. The existence of these boundary
values in the sense of distribution theory is proved in Theorem 13.2 below. Theorem 13.1 The Fourier transform of the distribution X" is the distribution eTirzla+11/2(5
+
i0)-a-1
Boundary value problems for elliptic systems
504
Proof Due to the reflection x
-x (see above), it suffices to find the
Fourier transform of y+. Let Re a > - I. When 1. > 0 the Fourier transform of a y+ is
fo"
+ is)-°-'
Ya e-xu+''.' dx/r(a + 1)
z° e
dz/r(a + 1)
J0
+ iS)-a-'r(a + 1)/r(a + 1)
=(+
is)-a-'
where the second integral is taken along the ray with direction 2 + i and z° is defined on C\R_ so that 1° = 1, and after that we used Cauchy's theorem to show that the integral can be taken along R+. Now, if we observe that il;)-a- i = e-"xia+ 1 /2( - ii.)-°- ', (). +
then let i. -. 0+, and note that a-z"X0+ - y+ in ,°', we find that the Fourier transform of C+ is given by e-""°+",-( - i0)-°-', as required. This proves
the theorem when Re a > -1. The proof is then finished by analytic continuation, since we have shown that
<X+, ) =
e-ix(a +1W2,
rp E Co (l),
when Re a > -1 and both sides are entire analytic functions of a E C.
In Theorem 13.1 we implicitly used a simple result on existence of boundary values on the real axis of analytic functions in the sense of distribution theory. Let I be an open interval in R and let
Q={zEC;z=x+iy.xe1,0 0 and we define F(z) as in (7) and if the path from zo to z is taken first horizontal, then vertical, we obtain
Cl(Im z)' -" - C log(Im z) + C,
I F(z)I I F(z)I
if N > 1, if N = 1.
Since log t is an integrable function of t near 0, we obtain after N + 1 integrations that f (z) = G("+')(z) where G is continuous in C1 and analytic in Cl This proves that
lim f( + iy) = lim d"+'G(- + iy)/dx"+' = d"+'G(-)/dxx+' in 1'1x+1 (I ).
Corollary 133 If f (x + i0) = 0 then f (z) = 0. Proof Take y > 0 and q e Ca (I) and define + ivy) dx = Jc(x - Re w-y) f (x + i Im w-y) dx.
K(w) =
Let d be the distance from supp (p to eel. Then K is an analytic function of iv when 0 < 1m w < b/y and IRe wl < d/y because in that case x + wy a Q for all x e supp (p. When Im w - 0+ we know that K and all its derivatives tend to 0 since f (x + i0) = 0. Hence if we extend K by 0 when IRe wI < d/y
and Im w < 0, then K remains analytic because it satisfies the CauchyRiemann equations. By the uniqueness of analytic continuation it follows that K = 0 identically. This implies f = 0 in 0. There is another proof of Theorem 13.2 which gives a formula for fix + i0). We use the fact that if f2 has C' boundary then 2i f c¢/0- dx dy = J n
4,(z) dz,
, e CI(G), f2 cc G,
(8)
en
which follows easily from the Gauss-Green formula (see [Ho 1], p. 60) in the plane, applied to the vector function (cli, icli). Set for (p a Co+'(I ) I(x, Y) = Y_ cpul(x)(iy)Jlj ! hex
Then D(x, 0) = cp(x) and 204)/Of = (a/ax + i 3/ay)t = (p(N+ u(x)(iy)x/N!
506
Boundary value problems for elliptic systems
Fix Y with O< Y 0. We let Cz(R+) denote the set of functions f e Cm(R+) such that f and all its derivatives can be extended continuously to O. Note: In §5.9 we had another definition of smoothness of a function defined
on a closed set. Theorem 13.4 implies that the two definitions coincide for R+. Also see Exercise 3 at the end of the chapter. Theorem 13.4 There is a continuous, linear extension operator fl: C'(R+) C'(R") such that +# = identity, i.e.
/3f(x, x") = f(x, x"),
X. > 0.
where the topology on C'(R") (resp. CX(R+)) is that of uniform convergence of each derivative on compact subsets of R" (resp. R,).
Proof We just have to modify the proof of Theorem 12.3 in the obvious way to take into account the parameters x' e R"- I. Choose a rapidly decreasing
Behaviour of a pseudo-differential operator near a boundary
507
sequence cp, p = 0, 1, ... , such that V
Y p'kcp=(-1)",
P.1
k=0,1,...
(10)
(see Lemma 12.4). Also let qp be a C °° function on R with rp(t) = I for 0 0, 00
Qf(x',x")= Y_ cp'(p(-Px")f(x', -px"),
X. m+k+1.
In the next theorem we characterize when v has a C" extension to the half-space x" >, 0. This result is important for §13.2 where we study the transmission property for pseudo-differential operators. Note that if b has an asymptotic expansion b - F bt, then, by definition, b = Y X(4n)-bj + rN, j 0. has a limit of 0 as R - cc. Hence it follows from (11) that all derivatives of v are bounded when x. > 0. Hence vlx",o and its derivatives have continuous extensions to x" >, 0. (b) Necessity of (13). Suppose that vlx,, o has a C < extension to x" > 0. By Theorem 13.4 (or just Exercise 3 below) we can find w e C0 '(R") equal
to v when x" > 0. Then v - w is a distribution with compact support contained in the lower half-space x" - t -m(2n) - 1
is a difference of two terms involving v and w. For the first term, we make a change of variables and find that t-m(27[)-1 1, Im C > 0.
(16)
It follows that the functions t -mBm(t)), t > 1, are uniformly bounded on compact sets in the upper half-plane Im C > 0. It can then be shown by Cauchy's formula that this family of functions is equicontinuous, hence there is a sequence t,k -+ oo such that the limit F(S) = lim tk mB,(tks) k - ae
converges uniformly on compact sets in the upper half-plane, and, therefore,
F(5) is analytic there. We claim that the boundary values of F on the real axis are F(S" + i0) = b°ol(i;").
Let I e l be an open interval not containing the origin. In view of (9) with
510
Boundary value problems for elliptic systems
N = 0, we have for rpeCo(I) J(P(x)F(x + i0) dx =
J(x. Y)F(x + iY) dx + 2i JI
F(x + it Y)rp'(x)iYdx dt. J0
For each k the boundary values of tk-"'B (tks) also satisfy a similar equation. By virtue of the estimates (16) we can then let tk -+ oo and then use (15) to obtain that f 0 and then conclude that b1 satisfies (13) and so on. Thus, (13) holds for all j = 0, 1, 2, ... . 13.2 The transmission property Before considering the transmission property (Definition 13.7), it is illuminating to consider first a related situation involving the kernel of a pseudo-
differential operator. The Schwartz kernel of an operator b(x, D) a OS' is the oscillatory integral K(x, y) = (2n)
f eu=-r.c, b(x, )
(17)
where b e S"(R" x R"), and it is singular only on the diagonal x = y. If we introduce new variables x' = x, x" = x - y, = l:, we obtain
(2n)-"
f
e++=°.t-' b(x',
c") dc"
(18)
and now the singularity occurs at x" = 0. We now allow the possibility that the x' and x" variables have differ-
ent dimensions, and change notation so that RZ" becomes R' with the variables x = (x1. ... . x") split into two groups x' = (x 1. ... . X.-k) and x" _ (x"_k+ 1, ... , x"). Note that in Definition 7.1 and for many properties of symbols it is irrelevant that there be as many x variables as S variables. Lemma 13.6 If b e Sm(R" x Rk) and v is defined by the oscillatory integral v(x) = then we also have
f
b(x, S")
x = (x', x"),
(19)
J v(x) =
Jeurl(x1. ") ds"
(19')
Behaviour of a pseudo-differential operator near a boundary where 6 e
S'(R"-k
511
x R') is defined by
b(x', i") = (27r)-' JJeie'
b(x, 0") dB" dx"
(20)
and has the asymptotic expansion
b(x',
S") ^
Y(-i'
1=1
D'-D' b x
Proof This is essentially Propositions 7.5 and 7.8 in another form, in the same way as (17) is related to (18) (for the case n - k = k). Therefore we retrace the steps in the proof only briefly using the current notation. Assume first that b e .5o in the variables x, S". Then v e .9' and (19') means exactly that (2n)kb is the Fourier transform of v with respect to x", b(x',
(27r)-k
Je_0'.e'I v(x', x") dx"
(27t)-k Je10-e'' b(x, 0") d9" dx",
_
i.e. (20) holds when b e Y. Now, for any b e S', the formula (20) makes sense
as an oscillatory integral and defines b e x Rk) such that each semi-norm of b is bounded by a semi-norm of b in S'. It follows as in the St(R"-k
proof of Proposition 7.5 that the relation between (19) and (19') holds for all
beSt.
To obtain the asymptotic expansion we follow the second proof of Proposition 7.8. Substituting q" = 0" - ", the equation (20) takes the form &(x', c") = (2n) -k f j"et" .o-I b(x, " + tl") dq" dx".
(21)
Expanding b(x, S" + q") by Taylor's formula in powers of q" we obtain b(x
") =
Ialsx- I
f)=
a,-b(x,
(tl
+ rN(x, S", q")
(22)
where (q"):
N
rN(x, IQI=N
aI
I
o
(I
-
t)N-'Ds-a24-b(x, " +
try") dt.
By the Fourier inversion formula (2rz)-k JJe"x.4" ay-b(x,
d j7" dx" = (- 1)10I a{-b(x', x", S"')Ix=o,
and then by substituting (22) in (21) we get the asymptotic expansion for b, once the appropriate estimates for the remainder terms are verified as in the second proof of Proposition 7.8.
Boundary value problems for elliptic systems
512
Remark When n - k = k, Lemma 13.6 can be obtained from Propositions 7.5 and 7.8 by the same change of variables that transformed (17) into (18). Letting x" = x' -Y', the formula (20) becomes b(x',
(2s)-k JJei _YE)b(x, x' - y', 0") d0" dy',
which is the ordinary symbol b corresponding to the amplitude function a(x', y', 0") = b(x', x' - y'. 0").
Remark Later on, we apply Lemma 13.6 with k = I and x = (x', x") = (x', x"). Note that if a function v has the representation (19') with k = 1, then Theorem 13.5 enables us to decide if v e C' (R+). We turn now to discuss the transmission property. Let A e Sm(R") and let
i2 c R" be a bounded open set with C °° boundary. For any u e C '( define u° = L2(R") by u = u in il, u° = 0 in R"\Q. Note that sing supp A(x, D)u° c sing supp u° c Of). Definition 13.7 Let + denote the restriction map 1'(R") 1'(il). An operator A e OSm(R") is said to satisfy the transmission property with respect to ail if for every u e C z(fl), we have
r+A(x, D) u' a C that is, A(x, D)u° and all its derivatives have continuous extensionsfrom it to C1
In general Anu = t*Au° defines a map Cm(i2) the transmission property then we get a map
9'(Q), but if A satisfies
An: C "(0) - C'°(fl). It is clear that if Q is any tangential differential operator and P is any differential operator then PAQ also satisfies the transmission property. We could also formulate an apparently weaker transmission property:
r+Au° a C"(fl) if u vanishes to some fixed order on ail. But it will be clear from the proof of Theorem 13.8 that this weaker version actually implies Definition 13.7. Thus, if A satisfies the transmission property then PAQ satisfies a weakened transmission property, and hence the full transmission property, for any differential operators P and Q.
The question of whether or not A has the transmission property is obviously local (by virtue of a partition of unity and quasi-locality of A) so we need only consider the case when it is defined by x" >, 0 and IM is defined by x" = 0. Thus, the problem of reformulated as follows: If A(x, D) E OSm(R)
has compact support (Definition 8.6) and if u e C'°(A+), under what conditions is it true that A(x, D)u°Ix">0 e
(23)
Behaviour of a pseudo-differential operator near a boundary
513
Theorem 13.8 Let A e S' be a (- oo) classical symbol (see the remark after Definition 7.21) with asymptotic expansion A -
A;, where A; is homogeneous
of degree m - j, and let the operator A(x, D) have compact support. Then A(x, D) satisfies the transmission property (23) with respect to 8F8+ = U8"-' if and only if AZ)(x', 0; 0, -1) = eix(R,-J-IaI)AX))(x', 0; 0, 1) (24)
for all j, a and P. Here we have used the notation Ap))
OP.Aj.
Proof First we prove sufficiency of the transmission conditions (24). Let u e C°°(l
); since A has compact support, we may assume that u has
compact support. Then we can write u° as the inverse Fourier transform in the x" variable of the "symbol" p(x', l;R), u°(x) =
21
f e`"""p(x', c,,)
where
0 P(x', R) = fo e
is
x = (x', xx),
-;x" ' u(x) dxR
(25)
(26)
the Fourier transform of u° in the x" variable. For l;R # 0 we may integrate
by parts to obtain N-I
' -kDpu(x', 0) + rN(x', R),
E
P(x',
k-0
where SS
KK
rN(x', bn) = bn
N
u(x) dxR. 0
x R'), that is,
It follows that p e S-'(R"
(27)
RI)-'-a"
R)I < Cx.p(l + I ID4. DI.p(x', X
and there is the asymptotic expansion P(x',
R)
-i J=o
be
(28)
1 -kDRu(x', 0)
(where it is tacitly assumed that the terms on the right-hand side are multiplied by a cut-off function Now, when we calculate A(x, D)u° we must take the Fourier transform of (25). By the Fourier inversion formula in the xR variable we obtain
d°() = Je_i d , dy,
Boundary value problems for elliptic systems
514
It follows that A(.)., D)uo =
f
2n
where
q(x, a) = (2n)' -"
q(x, S") dg
(29)
JJei'.S') A(x, -,)p(y', ba) dS' dy'.
Substituting z' _ (y' - x') R
(35)
is independent of the choice of H(z) (or the number R).
Proof If H,(z) and H2(z) are two such functions then H = H, - H, satisfies the conditions of the lemma with h = 0, i.e. oo in rIR and H(t) = O(t-2)
H(z) is analytic in IIR, H(z) = O(z") when z
whent -ooinR.
Now consider the function V(z) = z2H(z)(1 -
iez)-"-3
on fIR.
Since cp(z) -' 0 uniformly on {z; Im z > 0, IzI > p} as p oo it follows from the maximum principle that cp attains its maximum on arlR. Since 11 - iezl >, I + e Im z ,> I on aFIR, we obtain iez)-"-3I
sup Iz2H(z)(t -
M,
nit
where M = supen* Iz2H(z)I < oo. Letting a -+ 0 we conclude that Iz2H(z)I < M
on fR, i.e. H(z) = O(z-2) when z -+ oo in FIR. By Cauchy's theorem, the integral of H(z) along the boundary of the region Im z >, 0, R < Iz1 < p, vanishes. Hence by letting p -,. oo we obtain H(z) dz = 0.
LHR
This proves that (35) gives a unique definition of J, h(t) dt. Remarks
(i) If h(t), t e R, is continuous and h(t) = O(t-2) when t - oo, then we can choose H = 0 in Lemma 13.10, thus
J h(t) dt =
h(t) dt. J
Behaviour of a pseudo-differential operator near a boundary
519
(ii) If h is a rational function with no real pole then J+ h(t) dt is equal to 21ri times the sum of the residues of h in the upper half-plane.
The definition (35) of the operator J+ is a good definition in the sense that it depends continuously on parameters: Corollary 13.11 Same hypotheses as for Lemma 13.10. If F(z, s) is an analytic
function of z when Im z > 0, for 0 < s < 1, and F is a bounded, continuous of (z, s) then J+ h(t)F(t, s) dt is a continuous function of s. Proof J+ h(t)F(t, s) dt is defined by (35) with h replaced by s) and H by Hence the corollary follows by the dominated convergence theorem. Before stating Theorem 13.12 we must determine what conditions on the symbol A(x, g) a S' ensure that A(x; satisfies the conditions of Lemma
13.10 with respect to . Suppose first for simplicity that A is homogeneous of degree in, and does not depend on x:
A(ct) =
c ? 1.
To apply Lemma 13.10 we need an asymptotic expansion for A as Since ±1), then by Taylor's formula about we obtain
A(. ', ) = I
. A(0, ± 1)'+ RN+1(
I"'-lal
(36) oo.
(37)
IaISN
The remainder in Lagrange's form is 'a
1)
RN+1(S',
1
Y IaI=N+1 a!
(1 - t)N
1) dt
o
and hence (38)
IRN+i(S',Sa)I
for
1''J.
The expansion (37) is not adequate for the purpose of applying Lemma 13.10; the terms are not analytic functions. However, the function -lad has an analytic continuation to the upper half-plane if we define it to be I at 4 = I and e'K4'-1a1) at S. = -1. Thus if it happens that
A(0, -1) = e'"-IaI' a,A(0,1)
when a = 0,
(39)
then we obtain an asymptotic expansion for A:
b") _ 7 Ia1 S N
n -1
RN+i( '> ")
(40)
where RN+,( ', is of order m - N - 1 as S -+ oo (when i;' is fixed, or at least bounded). If we choose N = m + 1, so that RN+1 = O(s 2), then in
Boundary value problems for elliptic systems
520
view of Lemma 13.10 we can now define r
and R = I1'I
with H(z) = Y-1QI4N z'-"'I .8,A(0,
Note: To be precise in (40) we should write (c,, + i0)' instead of A -lal to indicate that we are dealing with the boundary values on the real line of a function analytic in the upper half-plane. Remark Differentiating (36) with respect to ', we find that 0 4 ' , A is homogeneous of degree m - Ia' I. But then (39) implies that
s;
-'a'
&4.A(0, 1).
Now, if we differentiate this equation with respect to c,,, it follows that (39) holds for all a, not just when a" = 0. Recall that a (- co) classical symbol A(x, t) e S' with asymptotic expansion A - F A j, where A; is homogeneous of degree m - j, is said to satisfy the transmission conditions if all derivatives AM(p) satisfy the conditions (39) at x" = 0, that is
r, 0; 0, -1) = Ae(x )
e"'(m-j-IkUAjap(x'0; 01) ) , ,
(24)
for all j, a and P. As usual, we have used the notation A(') _ M ?xA j. Note that there is some redundancy here: (24) holds for all a, $ if it holds when
a"= 0, $'=0.
Theorem 13.12 Suppose that A e Sm(R") is (- oo) classic and has an asymptotic expansion A - At, where A, is homogeneous of degree m - j, such that the transmission conditions (24) are satisfied. If v e Co (Rn-1) and u = v(x') 0 S(x,,)
then A(x, D)ulx,,,o has a C°° extension to the closed half-space x" > 0, and lim A(x, D)u = Q(x', D')v, X.-O+ where Q(x', l;') =
A(x', 0, -f+ 2x
(41)
do", and Q e Sm+'(18"-') has the asymp-
totic expansion Q - Y Q j, with Qj(x', ')
A j(x', 0,
being
2R J
homogeneous of degree m + 1 - j. Proof As mentioned earlier, the fact that A(x, D)ulx,,,o has a CaO extension to the closed half-space follows from §13.2, so it remains to verify (41). Since we already know that A(x, D)u and all its derivatives converge uniformly on compact sets x' c- K c R` as x -* 0+, it suffices to verify that (41) holds pointwise.
The conditions (24) ensure that we can define !+ A(x', 0,
do".
Indeed, since we may apply (40) to each term Aj in the asymptotic expansion we
Behaviour of a pseudo-differential operator near a boundary
521
obtain A(x', 0, '> bn) -
j
I=I . Al=)(x'> 0; 0, 1)2/2!
2
j+Ial m+ 1 a"=0
(42)
1s'I Hence Lemma 13.10 is applicable with
when
Y
H(z) =
_m - j - I=I . A(ar(x% 0; 0,
j+Ia1-< m+I
a.=0
and R = I. To compute A(x, D)u we take cp a Co ((- 1, 1)) with ! (p(t) dt = I and let uc(x) =
Since u, - u in .9', then A(x, D)u = lim, - 0 A(x, D) u, where
A(x, D)u, = (2n)-"
Je() A(x,
dS
We have )I
f iw(t)I dt.
Im
0,
provided that e in + I and set A
j, 0, and vanishes when x" < 0. (c) When m is an integer >0, then v is the sum of a function U which is in C when x" >, 0 and vanishes when x" < 0, and a multiple-layer, i.e.
v = U + Y_ vj(x') ®81(x"), ,,m where v; a C°°(R"-). Hint: Since v = 0 for x" < 0, then it certainly has a C °° extension to x" < 0. If we change the signs of x" and of c", it follows from Theorem 13.5 that b;(x', c") can be extended to a homogeneous analytic functions of l;,, in the lower half-plane Im l;" < 0. Now apply Theorem 13.1. 8.
Let u e C`°(& ), and let A e OSm(R11), m < 0. Show that A(x, D)u°a C"(Ir)
ifm+k 2). Hence it is identically 0 which
528
Boundary value problems for elliptic systems
implies g, = eu/an. To solve the Cauchy problem (3) is, therefore, equivalent to solving the equation (6). In particular, it follows that S, is injective, for if
S,g, = 0 for some g, then 0 = S°0 + S,g, implies that the Cauchy problem (3) with g0 = 0 has a solution u. But uniqueness for the Dirichlet problem implies that a-0 and hence g, = 0. Note that since S, is injective and has index 0 it must in fact be invertible (which amounts to surjectivity of the Dirichlet problem). Now consider a general boundary operator of transversal order 51 for the Laplace equation. Au = 0
in a
b°u + b,au/an = h on ail,
(7)
where b° and b, are differential (or pseudo-differential) operators in ail. Let S ' denote the inverse of S,, which is a pseudo-differential operator. Due to the equivalence of (3) and (6), the Dirichlet problem Au = 0 in S2,
u = g on an, is solved by taking g, = Si'(1 - S°)g. (Thus the operator in (5) is S = S, '(1 - S°).) To solve the boundary problem (7), therefore, is equivalent to solving the system of pseudo-differential equations (b° + b,S1'(1 - S°))g = h.
It is not hard to see that this is an elliptic p.d.o. on an exactly when the boundary operator in (7) satisfies the L-condition. This example is the prototype for the procedure in Chapter 14: knowing that a boundary problem (d, 9,) (in this case the Dirichlet problem) satisfies the L-condition then the Fredholm property for (d, .42) (i.e. the problem (7)) can be reduced to the Fredholm property for a p.d.o. on ail. This was also the idea behind the formula of Agranovic-Dynin in §10.5. It is important to be aware, however, that - in contrast to the case of a single equation - the Dirichlet problem for p x p elliptic systems (p > 1) does not necessarily satisfy the L-condition (see §10.2). Nevertheless, there still exists a sort of "reduction to the boundary" using the Calderon operator which we now wish to define.
Let us look at the problem locally and suppose that d has constant coefficients in the half-space it = R+. If u e C '(68+, C°) we write the Cauchy
data
iu =
where !;u = Jim D;u(x', x")
(and recall that D. = i-' a/ax"). The elliptic differential operator d can be written in the form
d= j d1Dj. i=o
where d; is a differential operator of order l - j on 00 = li"-' and .d, is an invertible matrix. If u e Cx(R+, C°), we let u° = you, that is, u° denotes the locally integrable function on R" which is equal to is when x" > 0 and equal to 0 in the lower half-space x" < 0. Then we obtain in I8"
du° _ (du)° + d`u,
(8)
529
The main theorem revisited where 1-1
j
j[=O
k=0
ql=[*j]j=pEC°o(R'-1,CD"),
-Qfj+1qlj-k®Dk.(,
where S is the Dirac distribution in x,,. Since .sad has constant coefficients, it
has a fundamental solution E(x - z); applying it to both sides of (8) we obtain the following: if du = 0 then u = Esl`yu when x,, > 0. This is the desired generalization of the Green's formula (2). Now, by applying the Cauchy data operator y, we obtain
* = P'l,
where P = yEjad`,
(9)
and 'W = yu denotes the Cauchy data. The operator P is a pseudo-differential
operator on P°-1, called the Calderon operator. (Since d has constant coefficients it is not hard to show that P is a projector, with image being the space of Cauchy data of the solutions of the homogeneous equation safu = 0. The principal symbol of P is equal to the Calderon projector P+ which was
introduced in §2.2 for the matrix polynomial L(A) = n.d(y, ', .1)) with respect to the eigenvalues in the upper half-plane Im A > 0). Now let 9 be a boundary operator, 1-1
J=O
The equation Ru = g can be written in the form -4`7u = g where I
=
[_V°'''-41-1 J. By applying the operator yE once again to equation (8), it is clear that the boundary problem
du= f when
Vu=g when
is equivalent to a system of p.d.o.'s on I8"-1 of the form
*=v+P'l,
V`'1 =g,
(10)
where v = yE f ° and Ell = yu. This might seem like too many equations, for instance, the Laplace equation with Dirichiet boundary conditions leads to three equations for two unknowns. But, as it turns out, v is not arbitrary since Pv = 0 (see Chapter 12 (13)). In this chapter we treat in detail the general case when sad and -4 have
variable coefficients and 0 is any bounded domain with C°° boundary. Roughly speaking, the same kind of reduction to the boundary indicated by (10) is still valid, modulo operators of order - co. Some of the specific results that will be proved in this chapter are as follows. Let (sW, -4) E BE'-". It was shown in Chapter 9 that the operator r
$: W324 C') - W'2 1((, C') X
kX =1
W'2 ,nk-1/2{
),
s i 1,
defined by Qu = (du, au), has a smoothing regularizer, i.e. an operator which is a right and left inverse of 2 modulo operators of order -1. In this
530
Boundary value problems for elliptic systems
chapter we prove a stronger property: there exists a parametrix for 2, i.e. an operator r 13: W'i i(fl, C,) x WZ ",k-1/2(852) - WW(Q, C°) k=1
which is continuous for every s > l and is a right and left inverse of 2, modulo
operators of order - co. The essential elements in the construction of this parametrix are the Green formula (35) which expresses the solutions of du = f as a sum of multiple-layer potentials and the Calderon operator (36). We will then use this parametrix to show that the image of 2 is defined by C W relations, i.e. it is the orthogonal space of a (finite dimensional) r
subspace of C°°(f1, C") x X1 C'(00). In the first two sections, §§14.1 and k=
14.2, we study some spaces of distributions in R+ which are needed later on in the chapter. The Calderon operator is defined in §14.3 and the construction of the parametrix 3 is given in §14.4 when s91 is a differential operator, and then extended later on in §14.6 to the wider class of elliptic boundary value problems that was introduced in §10.7. In §14.5 we apply the formula for the parametrix $ to obtain an interesting result on the index. 14.1 Some spaces of distributions on R+ The following notation is used throughout this section. Let R+ denote the
open half-space, x" > 0, and R+ the closure, x" > 0. If F is a space of distributions in R", we let
F(R+)
(11)
denote the space of restrictions to R+ of elements in F, and F(R+) the set of distributions in F supported by 18+ (similarly for F(08+) of course). We use this notation for the spaces F = .P, .9', WZ, and so on. An exception
to the notation is made for the cases F = 2' and F = CO'. .9'(R+) already has a meaning as the space of distributions in R+, so instead we let .9(R+) denote the space (11) of extendible distributions. Similarly, Ca (R+) already
denotes the set of C' functions with compact support in R+, so we let Co (R"+) or C(o)(R") denote the space (11) of restrictions to R+ of functions in Co (R").
Remark F(18+) is a subspace of 2'(R"), while F(R+) is a subspace of 2'(R+) which may be identified with the quotient space F(R"+) = F/F(R"_).
Recall that W32 = W2(R") is the set of distributions u e "(R") such that the Fourier transform G is locally integrable and
IluIt = JIa()l2(1 +
oo.
(12)
The main theorem revisited
531
As we know, C0 '(R") is a dense subspace in the Hilbert space W2(R") and the spaces 22R") and WZ'(R") are dual with respect to an extension of the sesquilinear form (u, v)0=Juu dx,
u,veCo(R")
For all s e R, we have .9 c W2 e .9', the inclusions being continuous. Let W2(R+) denote the space of restrictions to R". of elements of W2(R"), with norm
Ilull,=infllUII
(13)
the infimum being taken over all U e W2 which are equal to u on R. When s > 0, this definition of W2(18+) agrees with the definition in the appendix
to Chapter 7, and (13) is an equivalent norm, due to the existence of a continuous extension operator, Theorem 7A.7.
Also, let 2(R"+) denote the closed subspace of W2(R") consisting of elements supported by R. Our first aim is to show that Co (R+) is dense in W2(R+). (Thus the definition of W2 is equivalent to the one given in the appendix to Chapter 7.) The following lemma and corollary are needed for the proof.
Lemma 14.1 For q,, u e 9
IIWllllull.,
IIc' - ull: where 11911 ,.i,i
(14)
di;
=f
Proof Since ((pu) * = rD * ti, we have 2 dSj1/2 Il(Pull, =
But, as easily seen, (1 +
LJ
0 + Isl2Y 1f 4(S - n)u(n) dnl
Isl2),1z \
(1 + I - nI)ISI(1 +
InI2)2/2,
s E. R,
and, therefore, LJ
IJ (1 + I -
gl)111I0(s
dn12 dyliiz, - n)IId(n)I(l + 1,112)'12
or, in other words, Ilgnull, is bounded by the L2 norm (in i;) of the function J K(S,
dn, where v(,) = I4(q)I(1 + 1,112)'12 and
K(i , n) _ (I + I - gl)I3I.Iw(S - n)I n) ds and j K(S, n) do are bounded by IIc1I we see that (14) Since f follows by Schur's Lemma 7.26 applied to j K(S, q)v(,) dn. Corollary 14.2 Choose cp e C0 '(R") such that cp(0) = 1, and set cpt(x) = cp(ex). Then for all u e W2 we have
IIc' -u-uq,-+0
ase-+0.
(15)
532
Boundary value problems for elliptic systems
Proof The norm 4 0 "1 11 . isi =
=
f
di;
e-"14( /e)I(1 + I
JIG)I(l +
di;
is bounded by a constant independent of e for 0 < e 0. Then u * cp, a Ca has support in R+ and its Fourier transform is For the norm (12) we have flu * (P. - ulls =
f
14(e) -1121z2()I2(1 +
which approaches 0 as a - 0 by dominated convergence since O(sie) 0(0) = I and 11 < 2. It follows from Theorem 14.3 that a continuous linear form on WZ'(F8+) is uniquely determined by its restriction to Co (R+), so we may identify it with an element of 9'(R+). The following theorem characterizes the dual space of Wi'(R+).
533
The main theorem revisited
Theorem 14.4 For any s e 08, the space C(o,(B+) is dense. in Wz(R, ), and C'(R" ) is dense in WZ-'(R+). The spaces WZ(68+) and : s(QS+) are dual with respect to an extension of the sesquilinear form
(U. v)0 = Juidx.
u e C(ol(U8+), v e Co (P+).
Proof Since Co (Q8") is dense in Wz(R ), it is clear that C(10)(18+) is dense in Wz(Rn,). The fact that Co (l8 ) is dense in W2 s(l8+) was proved in Theorem 14.3.
Now, if u is the restriction to R", of a function U E Co (l8"), and we regard as a C' function on 13" with v = 0 for x, 0, and we obtain ( n+
iA)-k = i-k f-OD e-ix"a"
ez""(xn)k-' dxnl(k
- 1)!
k = 1, 2, ... .
In fact, it is clear from the proof of Theorem 13.1 that the following lemma holds.
Lemma 14.6 Let A zz> 0. For any s e C, (bn ± iA)s = et' 12 Un(et lx" X-s-'(xn)),
(16)
where an denotes the Fourier transform in the x,, variable.
Now set A t ='n ± id, where A _ _ (1 + differential operator A f(D)'u
Ib'I2)1'2. Let se R. Then At(D)':.9' - So is the pseudo-
i)'. Here 5 is the Fourier transform on R" and is an isomorphism a:.9' -' .9' and a:.9" -> .So'.
Note that At(D)':.9' -' .9' is a continuous map and it has the inverse A±(D)-', so it is an isomorphism on Y. The equation (17) also defines an operator At(D)':.9' -+ Y', where (g" ± is the distribution in .9' defined by (18).
Remark Recall that if f e C' is slowly increasing in the sense that an estimate of the form IDf(x)I < CQ(1 + IxI)"° holds for all a then the product
The main theorem revisited
535
f - T is defined for all T e Y' by 0 by virtue of (19) since x_ has support in x" < 0.
Theorem 14.8 The isomorphism A_(D)' _ (D" - i)s: Wi(R') - L2(R ). Thus, for all u e Wz(R+) we have Ilulls = IIA+(D)sul1L=aai), and
W$(l+) = {z+U I U E.9' such that t+A+(D)'U E L2(08+)}. Here t+ denotes the restriction of distrubitions to f8+.
Proof The first statement follows from (21) with 1= 0. To compute the adjoint of A_(D)': L2(R+) - WZ'(R+), we use Theorem 14.4. Let v c- Co (IF+) and let u be the restriction to l+ of a function U E Co (f8") Then as in the proof of Lemma 14.7 (U, A-(D)sv)o = (A+(D)'U, v)o
and since the supports of v and A _(D)'v lie in R+ we have (t+U, A-(D)'v)o = (t+A+(D)sU, v)0.
This implies that t+A+(D)'U depends only on u = r+U. Further, by Theorem 14.4 the equation holds for all U e WZ(R") and v e
where
(, )p now denotes the duality bracket between WZ(R") and Wis(l
).
Thus,
(A-(D)')*t+U = t+A+(D)'U,
U E W2(R").
For any u e Wz(P' ), we define A+(D)'u:= t+A+(D)'U, where U E W32(f8") is any extension of u, and it follows that (A_(D)')*u = A+(D)'u, i.e.
(A-(D)s)* = A+(D)'.
537
The main theorem revisited
Since A _(D)' is an isometric isomorphism, the same is true of the adjoint. The proof is complete. For any function v E L2(R"+), we define v° e L2(R") by
v° = 0
v° = v in 68+,
in R'\18+.
This leads to the following corollary.
Corollary 14.9 For every u E W' (R+) there exists U E W'2(R") such that
U=uinR+and l1Ull,=llull, Proof With v=A+(Dyu, we set U=A+(D)-'v° a V2 (R"). Then A+(D)'U=v° is an extension of A+(D)'u, and by definition we have
t+A+(D)'U = A+(D)u. Since A+(D)': W52(!",) - L2(R",) is an isomorphism by Theorem 14.8, it follows that :+U = u. i.e. U is an extension of u. Moreover, I!U!I, = IIA+(D)'U!10 = 11v°Ilo = IIA+(D)'ullo = Ilull,.
where the last equality again follows from Theorem 14.8. Remark Recall that Ilull, in Corollary 14.7 refers to the norm (3), rather than the (equivalent) norm given in the appendix to Chapter 7.
Let us consider the case when s is an integer. Proposition 14.10 When s is a nonnegative integers = k, k = 0, 1, 2, ... , then
A+(D)k = ik
(1 + lD'12)(k-1)12(- 1)'
bJ OX.
!=o
When s is a negative integer, s = - k, k = 1, 2, ... , then
fe-t(d)tk-'u(x', x" + t) dt/(k - 1)!,
A+(D)-'u = i-k
u e 9.
(22)
o
Proof The first statement is obvious. Now, when s = -k is a negative integer, we have
A+(D)-u(x) = (2n) = (21r)1
where K
Jei' "
(+ j'ei(s'.K(6',
d xn) * u(S', x") dS',
+ i (x"- )k-'/(k - 1)! then
A+(D)-ku(x)=i-k(2tt)'-"
f J
0
e'
'. t')
tk-'u(S', x"-t) dt dl;'/(k-1)!
-.O
and (22) follows if we make the change of variables t - - t.
We end this section by determining the dual space of Clo,(R+) and of Co (R+). The topology on the space C4o1(R+) is defined by the semi-norms which are restrictions to R+ of the semi-norms in C0 '(R"). while Co (R+)
inherits a topology as a subspace of Co (R"). Recall that Co (R+) denotes the set of functions f e Co (R") with supp f c R+; it can be identified with the set of functions in C'R"+) with compact support which vanish of infinite order when x" = 0, because any such function can be extended by 0 when x" < 0, thus giving a function in Co (R").
If Te 1'(R") then by means of the definition of the support of a distribution and a partition of unity one can show that = 0
if supp T n supp tp = 0,
i.e. = 0 for every tp a C0 '(R") which is equal to 0 in a neighbourhood
of supp T. In fact, a stronger result is true (see [Ho 1], p. 46): If T e 9' and is of order < k, then = 0 for every cp a C' such that D°rp(x) = 0 when x e supp T and
IxI)'':
we have IIuII(,,,) =
a L2(08"+)).
I
Here z+ denotes the restriction of distributions to 08+. Writing can be written in the form
we see that every u e u = uo + D"u".
(25)
where uo a FI1 "' -`-1(R+), is,, e H' -' -`(R+} and the norms of uo and u,, are
bounded by the norm of u. In fact, with uo = - i m1 + 1. (Note: When m2 < m1 the proof is trivial since Hm", c H"2. `2 by virtue of (23).)
First case: m2 < m1 + 1. We will prove by induction that
j = 0, 1, ... , 1.
D' u E Hmx -;. sx,
(28)
Let us first prove (28) for j =1. Note that D"U E
Hm'-a,,.s' -l'l
since u e HM"'", which implies D°u E HM"+' -'--" if a" < l and Jal < 1. More-
over, H" +'-'-" c
Hm1-t''1 by virtue of (23) since m2 < m1 + I. It then
follows from du E Hm2-1.31 that DAu E H'"1-'''1 since we have just shown
this is true for all the other terms in du. Now we argue by backwards induction. As our inductive step, suppose that (28) has been proved for some
j + I < 1; then we must show that it is true for j. By Lemma 14.13 this is equivalent to showing that E HM'1-'-1,'1+1 and Dj,+ 1u e H"x-'-1.'1 But the first inclusion follows since u c- HM"-s' r_- H"'1-'''x+', while the second
inclusion is just the inductive hypothesis. Hence (28) holds for all j; in particular it holds for j = 0. Second case: m2 > m1 + 1 . We have sdu e H'2-',12 c H n' + 1-1'" -1, so the argument just given with m2 replaced by m1 + 1 shows that uEHm' +1,3 -1,
There is a positive integer k such that m1 + k < m2 < m1 + k + 1 and we can repeat the argument up to k times to get
u e H" +k.s' -*, so we are back in the first case, and we conclude that u ca
H"'2"2.
We can also put the hypotheses in a form which is suitable for localizing on manifolds. Corollary 14.16 Let the hypotheses be as in the Lemma. If m1 + s1 = m2 + s2 and for some qp e C(10)(R"+) we have
cp 'u E H'"''"(R+)
and
then it follows that Qpu E H"'x''x(R+).
cp
E H"'x-'''x(IR+),
The main theorem revisited
543
Proof The commutator qid - dcp is of order I - 1 so (psalu - dtpu e
Hm,-1+1.s, C Hm,-lrl.s,-1 If m2 t In1 +I then c3f(puEH'2-',s2, and the inductive step in the first case above still works with u replaced by v = If 1?12 > m, + I then drpu E Hm' -1+ 1.s1 -' and the second case proceeds as before.
We denote by r' the restriction map (''u)(x') = u(x', 0),
u E C1o,(R+)
Proposition 14.17 If m > 1/2 there exists a continuous extension of t' to a map i': Hm.s(R") -- Wmrs-1/2(R"-1).
Proof First let us verify this fact for I8" in place of R+. Let U E Ca (R"). As in the proof of the Lemma before Theorem 7A.9, one shows that if v = :'U then it(') = (2r)-' ! dy", whence li(S')12 < C2
Now if we multiply by (1 +
+ Is12)m(1 +
IS'I2)s+m- 1!2 and integrate, it follows that
11 ?'U 115+m-112 0, we have by (23) that III°I6-,-k.k)I, 0, and the same is true of the second term due to (47') with m = 0. Lemma 14.25 Let S e OS `(R"). For any multi-index a we can write
DOS = E SOD$
(48)
1014161
where Ss is a p.d.o. of order
-1, and P. = 0 if a" = 0. It follows that S is
The main theorem revisited
551
11"(W) for every integer k > 0, and all real numbers continuous H' t, 1. (Also, see Theorem 14.34 below.)
Proof By commuting S with the derivative Dj we have
DS = SD + S0 where So = [DjS] a OS-'-'. This shows that (48) holds when lal = 1. The general case follows by induction on lal. If k is a nonnegative integer then to estimate the norm IlSull,,k we just have to estimate iiD°Sull, for all Jul < k,
a = 0. In view of (48) we obtain IID'Sull, I - 1; see Theorem 7A.10 (where different notation was used there). The boundary operator has the form 1-t
-V(y,D)_
.
J-0
(y, D')D,j,
552
Boundary value problems for elliptic systems
where .4j = a and bi, E OC1S'"°and the boundary conditions R(y, D) u = g can be written as
a`;u = 9,
where £C = [V0..., ,-, ]
The matrix p.d.o. 2` is a block DN operator on O3Q,
-4` = p 'A. with the I x p matrix R' = [bk;];=......D in the kth row and jth block (k = 1 , ... , r, j = 0, ... ,1 - 1) having order < mk - j. Also, the Calderon operator
P = [Pi] is a block DN operator with the p x p matrix Pj in the ith block-row and jth block-column (i, j = 0, ... ,1- 1) having order 5 i -j. Proposition 14.26 Let .sf e Ell'. Then the boundary value problem (d, -4) satisfies the L-condition if and only if the principal part, nR`, restricted to the image of nP()-,1;') is bijective for all (y, ') E T*(di2) \ 0.
Proof This has already been proved in §10.1, see the corollary to Theorem 10.1. For convenience let us recall the proof of sufficiency here. Notice that 7r_0 _
[B0...B1-1]
where Bj = n-4j are the coefficients of the matrix polynomial B(i.) _ n.V(y, (c', i.)). Let L(.)=n.al(y, (1c', ).)) and choose a spectral triple (X+, T+, Y+)
for L(i.) with respect to the eigenvalues in the upper half-plane Im ). > 0. Since nP is the "Calderon projector" from §2.2 we have nP = P+ = Now it follows that
see §2.2 (8).
1Y+]Y1
o BjX_T+. Since A.+ is invertible the surjectivity where as usual A _ of no when restricted to the image of nP follows immediately from T+ ' Y+].T T. On the other hand, if n.Vv = 0 for some v in the image of nP then [Y+ . T,71 Y+]Yv = 0 and hence v = nPv = 0. This completes the proof of sufficiency. surjectivity of [Y+
In virtue of Theorem 14.20 we have PZ =_ P, and then by Corollary 14.23 (with p replaced by p1) there exist DN operators on M = Oi2,
S=[S,,],
S,kEOClS'-r°(l3Q,p x 1),
i=0,...,1- l,k= 1....,r,
S'_[S;;],
S;jEOCIS1j(& ,pxp),
t,j=0....,1-I
such that
OS =- I
S'+S.`-lo,,
PS =- S;
S'P - O.
(50)
(51)
The main theorem revisited
553
In the next theorem we use the operators S, S' to construct a parametrix for (d, -4). This construction is of course similar to the formula for the solution in Theorem 12.12. Note that the conclusion of the theorem is that
$2=I+K,
243=I+K'
where the operator K: WZ(Q, C°) -+ W2(S2, C°) is continuous for any t (s >, 1) and the operator
W` 2(Q, C°) X W2(aS2, Cr)
K': WZ(s1, C°) x WZ(ai2, Cr)
is continuous for any m and t.
Theorem 14.27 If the boundary value problem operator i' = (d, 9) is Lelliptic then the operator W 32-M CP) x kX w2 mk-1/2(m)
w2(n, C°)
defined by 3(f, g) = a+(I + E.rfS'y)Ef ° + z+E.gf'Sg is continuous for every s >, 1; 3 is a parametrix for Q, that is,
$Q=I+K,
243=I+K',
where K: W2, (f2, C°) -+ C°°(Q, C°) is continuous and
K' =
(K, K3
KZ): WZ(S2, C°) X
K,
Q'(aQ,
C') -
X
C°)
C(ai2, C')
is continuous.
Proof The continuity of 3 is an easy consequence of Theorem 14.24, taking
into account the orders of the blocks Si, and S;t in the matrix operators
S and S'. From (51) it follows that S'(1 - P) + SR' = I - R, where R e OS- '(OR pl x pl). Then, substituting this equation in the Green formula (35), we obtain
u° + T u° = Ef ° + Esd"(S'(I - P)yu + SB`yu + Ryu).
(52)
Further, by applying the trace operator y to (35), it also follows that yu + T u° = yEf ° + Pyu, that is, (I - P)yu = YEf ° - 7T u°, and substituting this result in (52) and then applying the restriction operator
:+: 2'(l8") -' 2'(fl), we obtain u + Ku = :+(I + Esr(`S'y)Ef ° + 3+Esi'Sg,
(53)
where Ku = a + Edc(S'yT u° - Ryu) + i + T u°. As usual, we may assume that T,, a OS-'° since T, can be replaced by T,4/ where 0 e C0 '(fl) and 0 = 1 on
A It follows that K is a continuous map W2(Q, C°) - C'(S2, C°) because OS- I and T, eOS-1.
554
Boundary value problems for elliptic systems
The equation (53) has the form u + Ku = 63(f, g), which shows that $3 is a left inverse of .P modulo the operator K, i.e. $Q = I + K. It is also an approximate right inverse. In fact, with the notation
u = 1( f, g) it follows by (30), and the fact that sd is a local operator, that
du = (I + T,)(f ° + -d S'yEf °) + (I + T,) at`Sg in 12, Since :+.sl' = 0 then .olu=f+K1f+KZg in 0,
(54)
where K, f = t + Tf ° + :+T,.dcS'yEf ° and K2 g = r+T,.slt`Sg. Since we can
replace T, by T4 a OS- ', then K, is a continuous map WZ(Q, C°) -+ C Y'(a C°) and K2 is a continuous map Q'(0f2, C') -. C `e, (SZ C°). With u still equal to "43(f, g), we have
yu = 7Ef° + PS'yEf ° + PSg = (I + PS')yEf ° + PSg, whence
Ru = M`yu = g + K3f + K4g,
(55)
where K3f = R`(l + PS')yEf ° and K4g = (RIPS - 1)g. In view of (50), it follows that RIPS - 1, so K4 is of order - oo and is a continuous map from C'). -9'(of, Cl) to It remains to prove the continuity of K3. To do so, first observe the following fact which follows from the equations (50), (51). If we multiply the
equation SM' + S' - I by M`P, and recall that RIPS - M IS = I, we obtain Re + RIPS' -= 3'P, that is,
M`(1 + PS') _ Rep. By adding and subtracting R`.yEf ° we can write K3f in the form
(56)
K3f = (RV + PS') - Rep) -yEf° + M`'PyEf°, and then Proposition 14.21 implies that K3 is a continuous map from W2(S2, C°) to C°°(aQ, C'). The equations (54), (55) show that Q'3 = I + K'.
Remark 14.28 We can write down a formula for the DN principal part of S in terms of a spectral pair, (X+, T+), with respect to the eigenvalues in the upper half-plane Im t > 0 for the matrix polynomial L(A) = njo'(y, (S', i.)). By taking the DN principal parts of the two equations in (50) it follows from the uniqueness in Proposition 12.11 that irS = col(X+ T;.) j=o
where A+ is defined as in Chapter 10 (36). In view of Theorem 14.27 we have another proof of the Fredholm property for elliptic boundary problems. This follows from Theorem 9.11, since the embeddings C m(S2) a W2"(S2) and C °°(af) c W'2(ai2) are continuous for all m, t, and WZ+'(S2) c WZ(S2) and W'2+'(M) c 912(0Q) are compact. Since u = 9352u - Ku, the continuity properties of ¶3 and K show that the following
555
The main theorem revisited
Regularity Theorem holds: If u e W2(0, CP) and r
sd u e W 2''(f?, C"),
(u e kX= W2
112 (M),
s
1,
1
then u e Wz(f2, C°). In particular, the kernel of 2 is contained in C°°(i2, C"). As mentioned earlier, the properties of the parametrix, 1, make it possible to show a further result, namely, that the image of i? is the orthogonal space
of a finite dimensional subspace c C. For the proof we need to use (33), (34) which when written out more fully becomes '-1
(u, sd*v)n = (adu,
J
v)n + i-' E
E ('VJ+1YJ-ku, Ykv)00 1=0 k-o
(57)
for all u, v e C °°(S2, C°). Here we have used the notation
(f1, f2)a'=
f fi '' f2 dx,
(91,92),90'=f 91 k92 da a0
where dx is Lebesgue measure in R" and dv is the measure on OM induced
by the standard Riemannian structure in R". As usual, the superscript k indicates the Hermitian adjoint, i.e. kf2 is the conjugate transpose of the vector function f2. Recall that sd, is a differential operator of order I - j on M. By introducing the p1 x p1 differential operator Aei
-42
.42
0 41,
we can write the equation (57) in the form (u, sd *v)n = (W u, On + I -'(97u, Man
(58)
and since st1 is an invertible matrix function, rather than a differential operator, the operator f has an inverse -' which is a differential operator. In the proof of the following theorem, one should recall that as usual we identify the dual space of the Hilbert space Li(ft) with itself by means of the inner product (, )0, and the dual space of W2(8f2) with Wj x(011), t e R, by means of an extension of the inner product ( , )an.
Theorem 14.29 If the boundary value problem (49) is L-elliptic then the boundary value problem operator 2 is a Fredholm operator for every s 1. The kernel of 2 is in C'(0, C") and the image is the orthogonal space of a finite dimensional subspace of C°°(i2, C°) x _X k=1 C°°(8f2), that is, there exist a finite number of junctions 11, ..., fB e C '(R C") and g1,. .. , gp eC °°(8f2, C')
such that if (f, g) e W'2 '(f2, CD) x X.=1 WZ "/2(m) then f = .4u, g = -'u
556
Boundary value problems for elliptic systems
for some u E W2(Q, C°) if and only if
Jf.hfdx+Jg.hg.d(Y=0 n
i=1,...,$.
an
Thus, the index is independent of s.
Proof As mentioned above we have already shown in §10.7 that 2 is a Fredholm operator. To complete the proof, it suffices to show that the image of 2 is defined by C°° relations when s = 1, because then it holds for any s > I by virtue of the Regularity Theorem mentioned above. Let
(v, w) be an element of the annihilator of im 2 in the dual space of L2(O, C°) X ) I then by Proposition 14.17 the Cauchy data
yv = lira [D{,v]j=o X._O*
are well-defined. We claim that these Cauchy data must be in C 1. Indeed, by (58) and (59)
(au, w)a = i-VYu,Yv)an,
du a Ca0(S2, C').
Since Eu = Q'yu, then
(acff-i1, w)an = i-1(9l, Yv)an for all qi e C 10(852, C°`). By transposing the pseudo-differential operator .` and the differential operator T-1, we would obtain an expression for yv in terms of various pseudo-differential operators acting on w. Since w is C°°, then so is yv. Now, because .sc is an invertible matrix, we can use the equation d*v = 0 to solve for the higher-order normal derivatives Div, j > 1, in terms of the
Cauchy data; hence all normal derivatives of v are C' on afl. If we apply Exercise 3 of Chapter 13 locally, then use a partition of unity on 852, it
The main theorem revisited
557
follows that there exists i e Co (i2\i2) such that on ail for all j. Then d *i vanishes to infinite order on 00 and we extend v to S2 by setting v = u on i2\S2. After this extension we have v E L2(R"), supp v is a compact subset of f2, and .d*v a C0 '(fl). Since d* is elliptic on i2, it follows that v E C0 '(fl). Indeed, the adjoint of the equation .dE = I + TT implies that
v=E*4*v-T*ve C since E*.d *v E C°° and T, *v = T, :liv E C x (where 0 E C0 '(fl) is equal to Ion
supp v). By restriction to i2, we obtain v e C'°(a C°). Thus, we have show that the annihilator, N = (im of the image of P is contained in C"i2, C°) x X k=, Cx(ai2). We already know from §10.7 that £ is a Fredholm operator so N has finite dimension. One could also argue as follows. An application of the closed graph theorem implies that the inclusion is a continuous embedding, i.e. the topology on N as a subspace of the dual of L2(Q, C°) x X kW,-1°-112(m) coincides with the topology on C'°(f2, C°) x X k=1 C"(a0). Since the latter space is a Montel space (i.e. closed, bounded sets are compact in the C°° topology) and the former space is a Banach space, it follows that N is a locally compact Banach space, hence it has finite dimension.
Since im i is closed it follows that im i is the annihilator of N, i.e. (f, g) a im i if and only if Sn
f
g1 hg'dr=0
('n
for all (f, y') e N. By taking a basis for N, the proof of the theorem is complete.
14.5 An application to the index Let M be a compact manifold without boundary and suppose M is a union
M=f2, vi2_, with S2+ = fl and common boundary oil = f2+ n i2_. For example, we can identify Ii" as a submanifold of M = S" by means of stereographic projection from the south pole, and then M =f2, u Q_ where Q.. = Q and 0_ = S" \i2 Another possibility is to let M be the boundaryless double of Q. Let n be the unit normal along ail that points into Q , then -it points into Q_. Choose a tubular neighbourhood of ail such that ail x [0, 1) c f2,
and ail x (-1, 0] c i2_, i.e. x" > 0 in 0, and x" < 0 in i2_
.
Theorem 14.30 Suppose that A: Cz(M, C°) C'(M, C°) is a p x p elliptic differential operator on M of order 1, and denote b .sit, = +A the induced
operator C'(f2t, Ca) - C"(ilt, C°). Let Rt = L.i=o 9j DJ,, be boundary operators such that (,dt,Rt) satisfies the L-condition. Then
ind(.d,,R+) + ind(,d_, ._) = ind A - ind R
(61)
Boundary value problems for elliptic systems
558
where R e OCIS°(8Q, pt x p1) is an elliptic operator of order 0 with principal symbol aR: ST*(di2) - GL,,,(C) given by (recall that pi = 2r) nRO',
[col(X+T+);=o'(A._ir.)-'
col(X_T
(62)
where A E;=o Bf X±Tt and B! = ate; . Here (Xt, TT) are spectral pairs for L(i.) = rrA(y, ' + An(y)) with respect to the eigenvalues in the upper and lower half-planes, respectively, depending smoothly on (y, 5') e T*(Ofl)\0. (As above, n is the unit normal along cQ that points into 52+.)
Proof Without loss of generality, the order of -4t in each row is equal to I - t so that the boundary operators are continuous maps -4t : W'2(11 , C")
L2(iiS., C'). Denote by M the disjoint union of i2+ and L. By taking the direct sum, 2 = L'+ ®1 ?_, of the operators P3 = (salt, Mt): W'2(Qt, C°) L2(Qt, CP) x L2(852, C') we obtain
2: Wi(M, CP) - L2(M, CP) x L,(tQ, C' ® C')
(u+, u-) H (d+u+, .01_u_; R+u+, a-u-), where W;(M, CP) = WZ(52+, C") x W2(S2_, C"). Denote by J: W' (M, CP) W'2(M, C") the natural embedding u f-+ (u1n. , uln_ ). Then obviously (63)
where j: L2(M, CP) x L2(00, C'(1) C') - L2(M, CP) is the canonical projection. Here we have identified L2(M, CP) is
L2(M, CP)1 i.e. (u+, u_) a L2(M, C")
identified with u e L2(M, CP) such that urn, = ur.. In that way, if
u e W'2(M, CP) then (dd+u+, dd_u_) is identified with Au and (63) follows.
Let E e OS-'(M, p x p) be a parametrix for A, and let P+ = P_ = -y_E.nf`_ be the Calderon operators for respectively. Here ytu denotes the Cauchy data operator taken from inside Sg, that is, ytu = limX.-ot [Du]. Then we may construct parametrices `:2t: L2(f1t, CP) x L2(aQ, C') W2(s1t, CP) of .?t) as in Theorem 14.27,
't(f,9),= rt(I + E.u(cS'±it)Ef° +:tEzf'Sig,
(64)
where S±, St are defined by (50), (51) for the boundary operator yit. It follows that the direct sum 3 _ + ®')3_ is a parametrix of 2. Now define an operator G: W' (M, C")
X W'2 i-'r2(c'S2, C°)
J=O
(u+, u-) H y+u+ - 7-11By virtue of Lemma 7A.1 I we have ker G = im J, i.e. ; +u+ only if there exists u e W'2(M, CP) such that uIn° =ut. Let
j: L2(.Q, C'@ C')
_u_ if and
L2(M, CP) x L2(852, C' E) C')
be the canonical embedding (g,, g2) r-' (0, 0; g,, g2). We claim that G o o/ is a DN elliptic p.d.o. on 00. Indeed, by (64)
=7+E.saI+S+g, - y_Ed`_S_92 = P+S+91 + P-S-92,
559
The main theorem revisited
that is, Go
of = [P+S+
(65)
P-s-1
a pl x pl matrix operator. By Remark 14.28 the DN principal part of P± St is
n(PPS±) = 7xPP col(X±T±)j-o(A't)-' = col(X±Tt)j=o(AR,)-', (Note that 7rPP is the Calderon projector corresponding to the eigenvalues in the upper and lower half-planes, respectively, so im nPP = im col(X±TJ)j' =10.) In view of (65) we see that G o 43 oj is a DN operator, with DN principal part ir(G o 13 -j) equal to the right-hand side of (62) on ST*(an). Since im 7rP+ 6) im irP- = CP',
it follows that Rr(G e 3 o ) has invertible values, that is, G o 3 o1 is DN elliptic. Now choose A' E OCISk(8fl) such that Ak: W2(8S2) _ WS (00) is an isomorphism for all s, let A = [bi;A-'+J+'/Z];; J=0 and
R==(AoG)o3oj.
(66)
Then R is a 2r x 2r classic p.d.o. of order 0 on OR with principal symbol (62). We now have the following commutative diagram: 0
-
1-1
(C")
WZ(M,
W2(M, C") --------
X L2(011, C") - 0 0
211)
TR
1A
0 - L2(M, C") -y L2(M, C") -+ L2(8f2, C' ED C') -- 0 L2(311, C' (D C')
where the rows are exact and the identities (63) and (66) hold. Thus, the formula (61) is a consequence of the following proposition. Proposition 14.31 All species E, F, M, N, U and V in the diagram below are Banach spaces and the maps are continuous, linear operators. Let 2 e 2'(V, U) be a Fredholm operator and 3 E 2(U, V) be a regularizer for 52, and define A = o o $ o J, R S o $ o j, so thit the diag m is commutative: 0 '
E E- V -+ F --' 0
Al
,
24P
I 0 t-- M -- U -- N .- 0 Suppose that the top row is exact, and the bottom row is split-exact (i.e. it is exact, and im 1 = ker o is complemented in U). Then A is a Fredholm operator if and only if R is a Fredholm operator. If A is Fredholm then
indA=ind2+indR.
(67)
Boundary value problems for elliptic systems
560
Proof Before proving (67) we show that it is possible to make some simplifying assumptions. First, we claim that without loss of generality ind S = 0. Indeed, suppose that V e 2(V', U') is another Fredholm operator, with regularizer $'e 2'(U', V'). By introducing the operators
$=
( Q
2)C-Y(V®V',U(
U'),
we obtain the following diagram: 0
E
V® V' S- F ®V' -- 0 ®U'
0
M
U
®UI
N
i0
where S' = S 81,,. and ,, = a p 1 are the extensions of S and 1 by direct sum with the identity on V and on U', respectively, and J is equal to J composed with the inclusion V - V ®V', and a" is the extension of a which
is equal to 0 on U'. Further, A and R are defined as before to make the diagram commutative, i.e. A = 3 o 2 o J and R = S o 43o, . Note that A = A
andR=Rc
3'. Now ind
ind2+ind52'andindR =indR+ind¶3'=
ind R - ind 2' so that
indD+indR=ind2+indR. Thus, (67) holds for the original diagram if and only if it holds for the new diagram. To make ind A = 0 it suffices then to choose a Fredholm operator
$' with ind 2' = - ind Q. For instance, if ind Q = k - I we just take any linear operator V e 2(C', Ck) because ind Q' = I - k (which follows from the isomorphism C'/ker 2' im 2'). The second point to notice is that the validity of (67) does not depend on the choice of ¶3 in the definition of R, for if $1 is another regularizer of 2 then
¶3 - I$, is a compact operator, whence R - R, = S o (3 - 431) -j is also compact, so ind R = ind R1. Furthermore, since addition to 2 of any compact operator does not affect
(67), the two preceding observations show that we can assume that 2 is invertible and $3 = 2-'. Then by using 2 we may identify U and V, so 2 and 13 become the identity maps, which we assume from now on. Further,
the bottom row of the diagram being split-exact, we may assume that U = M ® N, that a is the canonical projection (m, n) i- m and that d is the canonical inclusion n i- (0, n). Then we have A = a o J and R = S aj.
F- 0
0-+ E A
0+-Mi
R
With these assumptions, we now claim that
j(ker R) = J(ker A).
(68)
The main theorem revisited
561
If n e ker R then, by definition of R, we have ax a ker S = im J, so
in = Je,
for some e e E.
Then 0 = s j n = aJe = Ae, so e e ker A, This proves g (ker R) c J(ker A). Conversely, if e E ker A then 0 = Ae = jJe, whence Je = in for some n e N, which implies
Rn=Sin=SJe=0,
and this proves the other inclusion in (68). Now, since j, J are injective, (68) implies that dim ker R = dim ker A
(69)
in the sense that either both dimensions are infinite or both are finite and equal.
It remains to show the cokernels of A and R have the same dimension (either both infinite, or both finite and equal). Suppose im A has finite codimension, i.e. M = im A ® Mo with dim Mo < oa. Since S is surjective, then
F=S(MED N)=S(imAED 0)+S(M0®0)+S(0®N). By definition of R we have S(0 (D N) = im R, and it is easily verified that S(im A ® 0) c im R because im J c ker S. Therefore F = im R + S(M0 (D 0). In fact, F = im R ®S(M0 (D 0), (70) for if f e S(MO (D 0) n im R, then
f = S(mo, 0) = Rn = S(0, n),
for some mo e Mo, n e N,
whence (mo, - n) e ker S = im J, and (mo, - n) = Je for some e e E. Therefore, mo = oJe = Ae, and since Mo n im A = 0 it follows that mo = 0. Hence f = 0, so (70) holds and im R has finite codimension < dim Mo. In fact, it is clear that S is injective on Mo ® 0 (again because ker S c im J), so codim im R = dim S(M0 ®0) = dim Mo = codim im A.
(71)
Conversely, suppose im R has finite codimension, i.e. F = im R ® FO with dim FO < oo. Since S is surjective, there exists a finite dimensional subspace Vo c V such that S is an isomorphism from Vo to FO. We claim that
M = im A ®Mo,
where Al, = o(Vo).
(72)
First we show that the sum is direct. If mo e im A n Mo then mo = Ae = ox for some e e E, x e Va. Since A = of then we have
aJe = ax,
a(Je-x)=0, Je-xekero=im,j, so Je - x = in, for some n e N. and then, by applying the operator S and
562
Boundary value problems for elliptic systems
using the fact that SJ = 0 and R = Si, we obtain
-Sx = Rn. Thus, Sx e F° n im R = 0, so x = 0. Hence m° = 0. Finally, we show that
M c im A e M°. Let m e M. Since F = im R ®F° then in view of the definition of V° we have
S(m, 0) = Rn' + Sx,
for some W E N, x c- V°.
Since R = Sj and ker S = im J, then (m, -n') - x = Je, for some e e E. Applying the operator o gives
m=ax+iJe=ox+Ae so m e M° + im A, and the proof of (72) is complete, whence im A has finite codimension.
It follows that A is a Fredholm operator if and only if R is Fredholm, and (69), (71) show that ind A = ind R. 14.6 The main theorem for operators in MV' and the classes Sm.m x R") the set of all A E C'(R" x R") such that Denote by IDrDOA(x, )I < CC,p(l +
(73)
ISI)m-a^(1 + IS I)'" -la'I
S'-" is a Frechet space with semi-norms defined by the smallest constants which can be used in (73). As before we define 1 e'(',4i A(x, f)6(S) A(x, D)u(x) = (2n)"
(74)
which is a continuous operator A(x, D):Y -+ Y. The operator A(x, D) is sometimes denoted by Op(A). For m, 1< m2 we have and for m', 4 m2 we have Sm.mi e c= If A(x, ') E Sm(R" x l8"-1) is If A e then D'D'A E independent of " then A e 9°m((R" x G8"). The following relation is also important: Sm.m'
C Sm+v.m'-v
p i 0.
(75)
General symbols or amplitudes are defined as follows. We say that a(x, y, ) E
x 1l
x R") if the derivatives have the bound
ID{D'Dya(x, y, )I < Ca.p.7(1 +
I
'Um
-la I,
x, y,
e R",
for all x, fi and y. Then we define the operator (Au)(x) =
JJ'e_4) a(x, y, S) u(y) d y ds,
u e .9',
(76)
(the integral is oscillatory) and as in Proposition 7.4 we have a continuous map A: 9 -+ Y. The operator A is sometimes denoted Op(a). The following proposition is the analogue of Proposition 7.5.
563
The main theorem revisited
Proposition 14.32 Let a(x, y, ) E S'-" be a general symbol. If we define the ordinary symbol by A(x, S):=
I (2n)"
e'(x-,.e) e-ux-r,4) a(x, y, 0) dy d6,
(77)
with each semi-norm of A bounded by a semi-norm of the then A(x, ) E symbol a. Further, both definitions (74) and (76) of the operator Au agree, that is, Op(A) = Op(a).
Proof Substituting z = y - x, 7 = 0 - S and regularizing, we rewrite (77) in the form
A(x, )=
1
e-±' < s e R, so we obtain when Jai + 1#1 < k that ID,'DxA(x, )I COnst<S>m-lal.<S'>m
x
a
+ I >mI -°'-N dz dtl
JJ-M
Im-,-I-N` 2.
denote the symbol matrix of d in the local coordinates, A(x, 5) = q((k&(x)x, S).
(85)
Then A e S", A(x, ) =d(x, ) in a neighbourhood of supp u and IS1'A(x, g) -' is bounded for large I fl. Since D., A E S'- ', 0 for every j, then by Lemma 14.35 for every j and E(x. )A(x, we can choose E e S is the identity for large l;. By Theorem 14.33 it follows that E(x, D)A(x, D) _ hence R is continuous from Wz+'-' to 4V23+r I + R(x, D) where R e Thus
!lull,.,
IIEAuII,+1 + IiRull,+,
C'(IIAuij, + ilull,+,-, )
Choose ,p a Co equal to I on supp it. It follows from (84) that 11Aull, _
567
The main theorem revisited
IIA(PuII, < IIgDAuII5 + Cllull,+,-, and then, since rpAu = gpslu, we obtain the desired estimate (83).
We now wish to construct a regularizer $ in local coordinates near dQ The definition of for the boundary value problem operator (s9, ) '43 will be the same as in §14.4 but its properties will be weaker. (Compare
Theorem 14.40 with 14.27.) Before we can look at the properties of 3, however, we need to establish the transmission property in the present context.
Consider a local coordinate patch where .91 = s/°. As in (85) we extend the symbol of I to the whole space to get a symbol of the form
A(x, ) =
i
i=0
A,(x, ')s',
where the coefficients have the properties stated after (79) and ItI'A(x, s)-'
is bounded for large ICI. As in Lemma 14.35 we define E(x, 5)-' where X E Co is equal to 1 in such a large set that (I Ee
and D;,E a
for everyj. In view of Theorem 14.33 it follows
that
A(x, D)E(x, D) = I + T(x, D)
E(x, D)A(x, D) = I + T,(x, D),
(86)
where T T e
Thus the operator E(x, D) is continuous from H3'(98") to to and T,(x,D) and T(x, D) are continuous from H-"'-'(R") for all s, t. The following lemma shows that the symbol E(x, l;) has a nice asymptotic
expansion as " - x which makes it straightforward to establish the transmission property for E(x, D). Lemma 14.37 Let
,... = s" + i(IS'l2 + 1)12
(=A+ in the notation of §14.1).
Then E = and by Lemma 14.35 it follows that H-' a Let the hypotheses on the symbol A(x, i;) be as above. Then for any N there is an expansion of the form
N-1
E(x, S) = F Ej(x,
+ R,v(x,
(87)
J=0
x98").
where Et e S'(P" x 98"-') and RN e
Proof Without loss of generality A, A(x.
i=0
I. Then we expand A in powers of
Am,
where A, = I and A; e S'-1(98" x98`). We claim that it is possible to choose E ; such that for N = 1, 2, .. . N-1
A(x,
J-0
Et(x.
I+
-1 !a0
RN J(x.
C,'}!-1-N
(88)
568
Boundary value problems for elliptic systems
where RN,, a Sj+N(R" x R"). For N = I this means that E. = I and R,,j = A, _ I -j. If the identity has been established for one value of N, then for N + 1 we have
r-I ').-,-j = I + Y RN.j(x,
x
A(x, ) Y Ej(x, j=o
')=-j-N
J-0
+
j=o
Aj(X> S)EN(x+
and to put this in the desired form we choose EN ,= - RN, o so that the term j = 0 in the first sum cancels with the term j = I in the second sum, and then set
RN+I.j=RN.j+i + A, for j = 0, . . . , l - 1 (and RN,, = 0). Thus (88) holds for N replaced by N + 1. Now, multiplying the identity (88) by E(x, S) _ (1 - X(S))A(x, 1;)-% we obtain (87) where N-r r-I ')2-1-N RN(x, S-)= -X() Y- Ej(x, Y RN.j(x,
')E''-(1
J.0
j-0
The second term belongs to RN, j E S'-J" and n-N-j E S-'-N.N so it follows that RN a
S-'-'
since (1 - X(4))A(x, 4)-' E S-"0,
S-N -j. The first term lies in Co C S
As usual, if f e LZ(R") we denote by f ° the function which is equal to f in R+ and 0 elsewhere. Also :+ denotes the operator of restriction to x" > 0. Corollary 14.38 (Transmission Property) Let the hypotheses of Lemma 14.37 hold. If f e .9' then
II:+E(x,D)f°II+r.,1
C,.,il1+ flls > 0.
(89)
We also have (see (87))
Ili+T,(x, D)fOll+ II:+T,(x, D).f°flt,+r.n
C,,,II=+f
(90)
when s ? 0.
Proof With the expansion (87) the proof is quite simple. We have N - I
E(x, D) f ° _ Y- Ej(x, D')E-'-j(D) f ° + RN(x, D) f °. J=O
It is clear from (22) that the restriction of 3`(D)u, u e .9', to the half-space x" > 0 is determined by the restriction of u to that half-space. The same is, therefore, true of Ej(x, D')?-'-J(D)u. Since Ej(x, D')8-'-j(D) is continuous from H'+'"'(R") to H'-'(R"), it is clear from the definition of the norm (24) To estimate the that we also have continuity from H'+'-'(R;) to
569
The main theorem revisited
remainder term we choose N >, s, and then I[ RN(x, D)f 0II(,+t.,) ". C II f °111,-N.,+N) 1< C IIf °I11o.5+t)
eS-t-NN
and where the first inequality is due to Theorem 14.34 since R can be the second inequality is due to (23). Since the elements in considered as L2 functions of x" with values in WZ+'(68"-') it follows that Ilf°i11o.5+,) '< II 1+f II(o.,+8) and the latter norm is 0 is the density on ail in local coordinates. Now replace u by
(f, 9) in (103), where rp a Co has support in the same coordinate patch. Note that IF(u)I < CIIuN(j-1.-t) < CIIuPII,. -I) and then by the continuity of ' t (see u = (P '
Theorem 14.40) it follows that
IF(u)I < const(11111(o. -r-1)
+
II9k II -,+1-mk-3/2). k
By taking qp = 1 on supp 0 and using Lemma 14.42 instead of (99), (100), we see that the inductive step from t to t + 1 in Lemma 14.41 remains valid. Hence iv is C'; since 0 is arbitrary and a is C', it follows that w is C' on M. Then it follows as in the proof of Theorem 14.29 that v e C-(ffl +, C"), whence v e C °`(a C°).
Part V An Index Formula for Elliptic Boundary Problems in the Plane
In general, the matrix function A' ' does not provide any topological information about the boundary problem because its definition depends on the choice of y+ spectral pair (X+, T+); see the remarks after Theorem 10.19. For elliptic systems in the plane, however, there is a choice which is essentially canonical, and the winding number of det A along ST'(3Q) is then a topological invariant of (sl, -4). In Chapter 15 the topological index (or symbol index) of (sad, 9) is defined in terms of this winding number. In Chapter 16 we prove the index formula, i.e. that the analytical index coincides with the topological index. Chapter 17 gives some examples of the use of this index formula as well as a complete
homotopy classification of elliptic systems in the plane with 2 x 2 real coefficients.
For simplicity we treat only the case when the region i2 is simply connected.
577
15
Further results on the Lopatinskii condition
In §15.1 we examine the Lopatinskii condition and the results of Chapter 10 in further detail for differential operators in the plane. In §15.2 we review
the definition and properties of the winding number of a continuous nonvanishing function on the unit circle. The topological index of an elliptic boundary problem (sat, 9) is then defined in §15.3 under the assumption that the region f1 is simply connected (i.e. conformally equivalent to the unit disc by the Riemann mapping theorem), and we show that the topological index is additive under the composition of boundary value problems. In §15.4 we show that there is no loss of generality in assuming that sad and 9 have real matrix coefficients, which turns out to be an important simplification in the proof of the index formula. 15.1 Some preliminaries By a region we shall mean a nonempty connected open set in the complex plane. Let f) a o82 be a bounded region with C°° boundary. We denote by Ell' the set of properly elliptic differential operators
A,(x) a,
mt(x, a)=
x = (x1, x2) E
(la)
(21st
with principal part homogeneous of degree 1. As usual, the coefficients A, are smooth p x p matrix functions on (I Note that we have expressed .4 in terms of o2 rather than D; for the examples in this part of the. book it is convenient to omit the factor 1/i. In contrast with §10.5 we let BE'-" denote the set of all boundary problems (s4, 9) where sat e Ell' and -4 is a differential
boundary operator of the form
Ay, a) =
Y bx'(Y) &
, Irk=
L14-1,
y e all
(l b)
1
satisfying the L-condition relative to sat.
Remark More general (pseudo-differential) boundary operators had been permitted in the previous chapters for the purpose of being able to construct 579
Boundary value problems for elliptic systems
580
homotopies of elliptic boundary problems. For systems in the plane, however, this turns out to be unnecessary (see Proposition 15.2).
Let n(y) be the inward-point unit normal at y e 8Q, and r(y) the unit tangent vector to 8il such that r(y), n(y) is a positively oriented basis
of 182. We identify ST*(asl) with the unit tangent bundle ST(8Sl). If (y, ') e ST*(8il) then
so ' = tr(y),
I'I = 1,
and ST*(asl) is the disjoint union of two copies of ail. Let (sd, 9) E BE'-', i.e. sd is an elliptic operator of the form (la) and a is a boundary operator (1 b) having order m,F in the k th row, k = 1, ... , r. that is, In the notation of §10.1, we let L,(2) = Ly(A)
= nsd (Y, r(Y) + An(y)) = Y A j(y) 2.',
J-o
y e 8i1,
(2)
where Aj(y) = Aj(y, r(y)). If (9:t (y), Tt (y), Yt (y)) is a y t -spectral triple of L,,(A) then we may define a y+-spectral triple of L,,g.(A) as follows (see Theorem 2.21) (X+(Y, c'), T+(Y, '), Y+(Y, '))
(X+(Y), T+(y), Y+(Y))' = r(Y) - a- (Y), -T (Y), (-1)t -Y (Y))
-r(Y)
We also define (X_(y, c'), T_(y, c'), Y_(y, ')) by the equation (35) in §10.4. Associated with the boundary operator 9 there is the matrix polynomial it (y, r(Y) + An(y)) = Z B;(Y) AJ,
y e all
(3)
i=o
where B;(y) = BJ(y, r(y)) and u < max mk, and we let
A(Y) _
to
( Y)X±(YY)
(4)
It
A a(Y,
') =
B;(y, ')X+(Y, 4')T+(Y, ')
(5)
=o
Note the following relationship between the matrix functions (4) and (5): A (y, r(Y)) =
A,+#(Y, -r(Y)) = M(-1) Aj(Y),
(6)
the second equality being valid for a differential operator 9, because 7r1(y, cl;) = M(c) - n.(y,1;) when c = -1. We know from Theorem 10.3 and (38) of §10.4 that (sd, 9) satisfies the Lopatinskii condition if and only if ±r(y), which we now see is equivalent to the det A' (y, 4') 0 0 when condition for all y e ail. (7) det Am(y) 0 0
Further results on the Lopatinskii condition
581
Remark 15.1 If the coefficients of L,(i.) are real matrices then we may assume that
t _(y)) = (X+(y), TT+(y)). Further, if the coefficients of BM(A) are also real matrices then we have A(y) = (y); in that case (sat, 9) satisfies the L-condition if and only if
det A'(y) # 0
for ally s Oil.
We now specialize the results of §§10.5 and 10.6 to elliptic systems in the plane. Denote by BE'-"
the set of elliptic boundary problems (.d, -4) fulfilling the L-condition such that -4 has the same order in all rows, mk = p (k = 1, . . . , r). Proposition 15.2 Let nt t c- 1, he a homotopy of elliptic operators in Ell'. If -4
is a (differential) boundary operator satisfying the L-condition relative to s! = da such that the principal part of .4 is homogeneous of degree I - 1 (in all rows) then there exists a homotopy of boundary value problems
(d, A) E
1, t E I, such that (.at, Ro)
(.W. 4).
Proof This is the same proof as in Theorem 10.23, except that it is necessary
to verify that the boundary operators can be chosen to be differential operators. By the remark after Theorem 10.23, there exist (unique) r x p matrix functions Bj(t, ) on T*(1Q)\0 such that
I-I Y Bj(t.
j=o
X, (t,
Since theDNnumbers of1areml =
)Tt(t, ) = Al
=ttt,=l-
0, ... , I - 1, for all c (positive and
then Bj(t, y, ce') = c'- I -jBj(t, y, negative). We let
GI-1-j vj
1-I `4,(y, D) _
j=p
(8)
Bj(t, y, t(y))
aT
r- t -j i ' can
which is a differential operator. Since A;, = All then (,ol 4,) satisfies the L-condition and in this way we have constructed a homotopy
4) e
Also, by construction, the principal parts of :4o and -4 are the same (see Proposition 10.13), thus (sat, -4o) x (sa(, 9). (The notation is defined in the text preceding Lemma 10.18.)
Remark 15.3 If the operators d and 9 have real matrix coefficients then we can find -4, also with real matrix coefficients such that (.at,,. 9,) satisfies the L-condition for all t e I. Indeed,
(X-(t, .), T-(t, .), Y_(t, ')) = (X+(t. .), T+(t, .), Y..(t, )), is a ; --spectral triple of L,..(;.) and the solution of (8) (which does not depend
Boundary value problems for elliptic systems
582
on the choice of (Xt, Ti)) can be written in the form Bt(t, ) = 2 Re A4
I-I T+ (t, ')Y+(t, ') Al+k+1(t, 1-
k=0
)}
(9)
which is a real matrix function. The same remark applies to the following theorem.
Theorem 15.4 Let (d, £) e BE"' with u
1. Then
(d,
(d, R)
0/0t)u-t+i.4)
(10)
for some (differential) boundary operator P such that (d, l) e BE'-'-'. In fact, since .
is a differential operator there exists 3P with Aj = A-t.
Proof This is just a restatement of Theorem 10.20 for systems in the plane. There exist unique r x p matrix functions R1 on ail such that 1=0
R1(Y) Xt Tt = >J B;(y)X± Tt. 1=0
y e ail
(11)
that is,
+A, It follows that the differential operator 0'-i-1 ai
r-1
iv= Y R1(Y) Qri- i -1 ani i=o
satisfies (11), that is
A. =Ate. Since R is a differential operator and t 0 is real-valued we also have if I - 1, so f is null-homotopic. Now we define the degree of a continuous matrix function, M: S' -+ GL,(C).
Here GL,(C) denotes the group of invertible r x r matrices with complex entries. The degree of M is defined to be the degree of the complex-valued function, det M,
deg M = [arg det M(z)]s,. The following proposition shows that two continuous r x r matrix functions
on S' with invertible values are homotopic if and only if their degrees coincide.
Theorem 15.10 Two continuous matrix functions, M: S' - GL,(C) and N: S' GL,(C), are homotopic if and only if deg M = deg N.
Proof For scalar functions (r = 1), this follows immediately from Corollary
15.9 since f -g if and only if f -g-' - 1 and deg f =deggeo- deg(f - g - 1) = 0. Thus, we assume r > 1. The degree of a continuous nonvanishing function
is locally constant (Theorem 15.8), so it is clear that if M and N are homotopic to each other then deg M = deg N. To prove the converse, it
Further results on the Lopatinskii condition
585
suffices to show that M is homotopic to the matrix function det M(z) I
N(z) =
1
Without loss of generality by approximation), M is a rational matrix function of the form M(z) = ,;_ _ k A jzj, and then it may be assumed that M(z) is a polynomial in z (by consideration of zkM(z)). Now, M(z) has the Smith canonical form, E(z)D(z)F(z), where D(z) is a diagonal matrix polynomial and E(z) and F(z) are matrix polynomials with constant nonzero determinant. By means of the homotopy E(tz), 0 < t < 1, E(z) is homotopic to the constant matrix function E(O) e GL,(C), and, since GL,(C) is pathconnected, it follows that E(z) (and similarly F(z)) is homotopic to the constant matrix function I (the identity matrix). Hence M(z) is homotopic to the diagonal matrix D(z) = diag(di(z))i=,. Then by a succession of rotations we see that D(z) is homotopic to N(z). For example, if r = 2, the matrix polynomial
\
r
Cdl(z) d2(z) is homotopic to l dl(z)-d2(z)
11
by means of the homotopy Cdl(z)
Xcost in t
Vd7(z)
1)( csin t cost),
cos
for 0 0 and Im 2 < 0, and we let T? :1 denote the vector bundle over S2 is a direct sum 971L=
with fibres 971x.
586
Boundary value problems for elliptic systems
Proposition 15.11 Let 52 be a bounded and simply connected region with C°° boundary. Then the vector bundles V1 + and 9R are trivial. Hence there exists admissible pairs (X f, T±) over 11 such that (X±(x), TT(x)) is a yt-spectral pair of L.,(A) consisting of matrix functions with entries in C°°(Q).
Proof The coefficients of st can be extended smoothly to a neighbourhood 52' S2. In view of Exercise 1 at the end of the chapter, we may assume that Cl' is also a simply connected region. Also, by continuity, we may assume that det nd(x, ) # 0 for all x e Cl', # 0. In this way we obtain a vector bundle lfl t over 52' which is trivial since 52' is contractible. (In fact, by the Riemann mapping theorem 52' is conformally equivalent to the open unit disc, see [Ru 1], p. 283.) Hence its restriction to 52 is also trivial. The last statement in the proposition follows as in Theorem 10.11. Note that LX(A) is defined in terms of the basis (1, 0), (0, 1) for R2 whereas the matrix polynomial L,(A) of (2) is defined in terms of the basis r(y), n(y).
Writing r(Y) = (rl(y), r2(y)) and n(y) = (-r2(y), r,(y)) we have L,(A) = ird(Y, r(y) + An(y)) i = E A;(Y)(r1(Y) - Ar2(Y))`-'(r2(Y) + Ar1(Y))' J=0
= (ri(Y) - Ar2(Y))' L,(w -1(A))1 where rp-1(A) = (r2 + Ar,)/(t, - AT2). In other words. L,(A) is the transforma-
tion of L,(A) under 9(A) = (-r2 + Ar,)/(ti + Are) as defined in §2.4. Let (X t (y), t± (y)) be a y'-spectral pair of L,(A). Since (p maps the upper half-plane to itself, then by Theorem 2.21 X±(Y) = X±(Y),
(14)
T±(Y) = (ri(Y) TT(y) - r2(Y)I)(r2(Y) TT(Y) + r1(Y)I)-1
is a yt-spectral pair of 4(2).
In the next proposition we assume that (d, R) a BE'-", that is, the principal part of the boundary operator 9 is homogeneous of degree p in each row, it (y, ) = Et., _" Ba(y)s°. Analogous to the considerations for L.,(A), there is the following r x p matrix polynomial " B,(A) = n4(y, (1, A)) = E B,(y)2 ,
y e 8C2,
=o
where B;(y) = B"_j,j(y), and we see that B,(A) defined by (3) is the transformation of the rectangular r x p matrix polynomial B,(2) under ip(A) = (-T2 + AT,)/(T, + Ar2). We let "
Al (Y) = E B;(Y)X±(y)T±(Y) X60
(15)
where (Xt(y), TT(y)) is a yt-spectral pair of L,(A), y e M. Note that the
Further results on the Lopatinskii condition
587
matrix functions Am(y) should be distinguished from A'(y, ') which was defined in (5).
Proposition 15.12 Let (Xt(y), TT(y)) be yt-spectral pairs of L,(a), and (X±(y), TT(y)) the corresponding yt-spectral pairs of Z,(.1) as defined by the
equations (14). Let -4 be a boundary operator whose principal part
is
homogeneous of degree p. Then
A:'(y) = A:(y)-(tl(y)I + i2(Y)TT(Y))-". 91 Hence det A±(y) # 0 if and only if det A(y) 0 0.
y e aft
Proof This follows from Lemma 2.22.
Corollary 15.13 Let the hypotheses be as in Proposition 15.12. Then the following are equivalent for all y e aft:
(i) The map u r. 9,,
(ii) The map u r--. B,(-
)ul,.o from gut
to Cr is invertible;
ult=0 from 9RL, to C' is invertible.
Proof In view of Theorem 10.1, (i) and (ii) are equivalent to (i') det AR*(y)
0
(ii') det Amit(y)
0
and (i') and (ii') are equivalent by Proposition 15.12. Proposition 15.11 has the following important consequence. Theorem 15.14 Let f2 e 182 be a bounded and simply connected region with
C boundary. A properly elliptic operator sad boundary operator satisfying the L-condition.
in 0 always has a differential
Proof Let d e Ell', choose (Xt(x), TT(x)) as in Proposition 15.11, and then define the spectral pair (14). There is a natural differential boundary operator associated with the spectral pair (Xt(y), T. (.v)) since the equations Y- BB(Y)Xt Tt = I.
f=o
have the (unique) solution
[Ago ..=[R.....T;.'k.] '-1.[P-The boundary operator
I-1 f=o
a'-1 -j aJ
bo) ail - 1 -i ani
therefore satisfies A,± __ I, and we have (d, 3°) a BE"-'.
588
Boundary value problems for elliptic systems
Since 0 is a bounded, simply connected region in 182, the boundary 1Q is
diffeomorphic to S'; see [GP], p. 64. Let (p be an orientation-preserving diffeomorphism from S' to dig. Then the degree of a nonvanishing continuous
function f e C(3C) is defined to be deg(f 9)), where the degree on S' was defined in §15.2. In the sequel we also use the notation deg(f o (p) =- [arg f(y)]cn
If M is a continuous r x r matrix function on 00 with nonvanishing determinant then the degree of M is defined to be the degree of the complex-valued function det M, i.e. [arg det M(y)]... In order to write down the definition of the topological index, we choose a C =' matrix 'r `-spectral pair (X t (x), Tt (x)) of L.O), x e S2, as in Proposition
15.11. The definition, however, does not actually depend on the choice of such a pair (see Lemma 15.17). Definition 15.15 The topological index of a boundary value problem (d, 9) is defined as follows:
ind,(d, 54) = r(2 - 1) - Zn [arg det A ,(y)]en + n [arg det A,(y)](,n where (X±(x), T±(x)), x e f2, is chosen as in Proposition 15.11, and then (X± (y), T+(y)), y e OR is defined by the transformation (14) and A.±(y) is defined by (4).
At first sight it might appear that the definition of ind,(.d, R) depends only on the value of the coefficients of d on the boundary ?fL but in fact the choice of (X+, T+) and hence of (X+, T+) depends on the coefficients through the region f2.
If the operators d and
.
have real coefficients, we may assume
A_(y) = AA+(y) (see Remark 15.1) and then we have 1
ind,(,o(,. ) = r(2 - 1) - - [arg det A+(y)],,n
(16)
The next two lemmas show that ind,(d, -4) is well-defined, independent of the choice of -,,'-spectral pairs.
Lemma 15.16 If (sd,. ) e BE"', i.e. the principal part of the boundary operator -4 is homogeneous of degree p in all rows, then the quantity
[arg det A(y)]m is independent of the choice of a Cm matrix y -spectral pair (X,(x), TT(x)) of L,,(i.), x e f2, where A1(y) is defined by (15).
Proof Let (X '(x), T? (x)) be another y t-spectral pair of L.,(;.), x e Q. consisting of matrix functions with entries in Cx(fl). By Proposition 2.12 we have
X=(x) = X±(x)M(x),
r:, (x) = M-1(x)TT(x)M(x)
Further results on the Lopatinskii condition
589
for some matrix function M with entries in C'°(fl) such that det M(x) ' 0 for all x e 0. For short we write At = A. Then
At(y) = A f(y)M(y)
for all y e 00,
and it follows that
[arg det At(y)lm = [arg detA±(y)lm + [arg det M(y)],v. As in the proof of Proposition 15.11, C2 is contractible to a point xo a C2,
whence the function det M(x), x e a is homotopic to a constant. Its restriction to ail is, therefore, homotopic to a constant function, thus Corollary 15.9 implies
[arg det
0,
as was to be shown.
The following lemma is another version of Lemma 15.16 for which there is no special assumption on the orders of the rows of the boundary operator. Lemma 15.17 Let the yr-spectral pair (Xt(y). T?(y)) of 4(i.) be defined by the transformation (14), and then define Am(y) by (4). Then the quantity
[arg det A(y)];n is independent of the choice of the C" matrix ; t-spectral pair (X± (x), T,, (x))
of L.,(i), x e i2.
Proof The boundary operator has the form (I b). Now we operate on I with tangential derivatives to make the order the same in each row:
-4' = [eirrmb(),) 121 mk
, 1
(17 )
where _p = max Mk and tt/c'r = Let us write At = A' and A3 = Ate.. For the matrix polynomials associated with the
boundary operators .1 and 3, we clearly have By'().) = B.,()). Thus
A=(y) = A=(y), which has nonzero determinant for all y e c%], so 9' also satisfies the L-condition. Due to Propositions 15.12 and 15.5 we have [arg det
[arg det A±(y)],,n -,u[arg det(r,(y)1 + t2(y)T±(y)&n
Since the spectrum of T+ lies in the upper half-plane, the same is true of tii + (I - t) T+ for 0 < t 0 nsd Y, ' dt)) w(t) = 0, n-4
(Y,
,
1d i
dt)) w(t)I8 =o =
0
has the unique solution w = 0. Moreover, due to the one-to-one correspondence between solutions of the boundary value problems (19) and (20), it follows that ind(ddR, _VR) = ind(sd p .rd, -4 m -4)
= ind(dd, -4) + ind(.R,.4) = 2 - ind(dd, 9).
Boundary value problems for elliptic systems
594
The following proposition will be of use in §16.15. Proposition 15.20 Let the matrix polynomial L(A) and contours y+ and y- be
as in Theorem 3.20, and B(2) = Ei=o Bj.11 an r x p matrix polynomial of degree p. Let (X f, Tf) be yt-spectral pairs of L(2), respectively, and (XR, TR ) the y+-spectral pair of Lo(2) defined in Theorem 3.20. Also, let BR(1) = Fj¢o BB.i /
be the real 2r x 2p matrix polynomial corresponding to B(2). Then
e BR + --
(24)
where ABA = D=o B n jXR(TR )j and AB = I_j-o BjX±Tt, and 1, denotes the
r x r identity matrix.
Proof Write Bj = Mj + iNj where Mj and Nj are real r x p matrices,
j=0,...,1-l. Then
(Mj - Nj\ Mj
Nj
so with the formula for (XR, TR) in Theorem 3.20, it follows that w
is
j=o
BR.jXR(TRW
-N;( Iil
(Mj
j=o Nj
Mj
/\ -
I
X+T+
X_Tj
11
where I denotes the p x p identity matrix. As in Lemma 3.19, it is easily verified that M,
Mj,/\
iI
ilJ - lil, il,!\Bj
Bj
and now (24) follows immediately.
Remark Note that the matrix ASR is square (2r x 2r) exactly when 2r is equal to the degree of det L(A), which is the condition of proper ellipticity. Exercises
Let S2 be a bounded and simply connected region with C°° boundary. Show that for any 6 > 0 there exists a bounded region 12' n 91 which is also simply connected and such that 11' lies inside a 6-neighbourhood of 11. Hint: Let fl' = S2 v V where V = (y + tn(y); -e < t < c) is a tubular neighbourhood of it and a is sufficiently small. a <x'+y' 2. Q= bounded, simply connected region. Does the conclusion of Exercise 1 hold in this case? 1.
16 The index in the plane
The region 0 c R2 is bounded with smooth boundary, and for simplicity we assume that it is simply connected. Consider the 2 x 2 system 0
a22 - (1
1
0
) ax,
xeQ
+ A0(x) u = f(x),
u, -cos q,(y) - u2 sin qp(y) = 9(y),
(1)
y e ail,
where u = [u, u2]T is a real 2-vector function. Writing this problem in complex form with w = u, + iu2, z = x, + ix2, and 0/0± = ((a/ax, + i 0/ax2), we obtain the Riemann-Hilbert problem: Ow/0z + a(z)w + b(z)x' = f(z), Re e'a'(Ylw = g(y),
z E Q,
(1')
y e ail,
where a, b E C °°(S2) and (p e C °°(an). (Often this is referred to as the generalized Riemann-Hilbert problem and the "Riemann-Hilbert problem" is the case a = b = f = 0.) The R-linear operator W.(Q, C) -+ L2(C , C) X WWJ2(all, R) defined by w F- (aw/05 -aw - bw, Re elm wlm) is Fredholm, its solvability properties depend on the winding number, X = (1/2n) [tp(y)]0Q, and the index formula
index = I - 2X = I - [W(y)IM
(2)
is well known. (See [Wen] or [Ga] and also Exercises 7, 8 and 9 at the end of this chapter.) In this chapter we consider boundary value problems (,d, 9) where the principal part of the elliptic operator a/ is homogeneously elliptic (Definition 9.25) of order 1. We shall show that the index of (.d, 5B) is equal to the topological index defined in §15.3. First we prove an index formula due to A. Vol'pert for first order elliptic systems, by starting from the formula (2) for the index of the Riemann-Hilbert problem. Higher-order systems are
then handled by a standard device which involves a reformulation as a first-order system. It should be pointed out that we have already shown in §15.3 that the topological index is additive under composition of boundary value problems, but this result is not used in the proofs here. 595
596
Boundary value problems for elliptic systems
16.1 A simple form for first-order elliptic systems with real coefficients Consider a first-order elliptic boundary value problem (BVP) in the plane
+ A0(x)u = f(x),
aYz - A(x) ax
B(y)u = 9(Y),
x = (xl, x2) E (1,
(3)
y e 43C ,
where A(x) and A0(x) are real 2r x 2r matrix functions with entries in C'°(U), and B(y) is a real r x 2r matrix function with entries in The region S? c I8z is bounded and simply connected, with smooth boundary. Any elliptic operator 4 where the principal part is homogeneous of order
1 can be written in the form (3). Indeed, the principal part nsd(x, ) is invertible for all x e a 0 0 S E R2, and letting _ (0, 1) we see that the coefficient of 0/0xe is invertible for all x e f2. The BVP (3) is L-elliptic if det(I;. - A(x)) 0 0 for all x e
a R, and the
r x 2r Lopatinskii matrix B(Y)' J
(I). - A(Y))-' di.
has rank r for all y E dig, where the integral is taken along a contour y+ in the half-plane Im A > 0 containing the zeros of det(I). - A(y)) there. If (3) is L-elliptic then the linear operator W2(i2, R2r) -. L2(Q, I82r) x W21'2(OQ, Rr),
defined by u f- (Ou/Ox2 - A au/ax, + A0u, Bu), is Fredholm. Since the index
of Fredholm operators in Banach spaces is locally constant, ind(,d, -7) depends only on the principal part of V, that is, ind(d, 9) = ind(tt.pf, -4) because one can replace the coefficient A0(x) by tA0(x), 0 0 for all j = 1, ... , r. Also, Y+
Y+ = CX+ X+]-1
(see §3.1 (6))
so that the rows of Y+ (resp. Y+) form a linearly independent set of left eigenvectors of A corresponding to the eigenvalues in the upper (resp. lower) half-plane. Now let u be a real-valued solution of the equation au
au
ax+AaY
=f,
(15)
Boundary value problems for elliptic systems
604
and let w = Y+u and g = Y+ f. Then
(wJ-\Y+)u, (16)
and the equation (15) becomes
ax
(w) +
T+/ ay (w)
\T +
(17)
or OWj
ax
+Aj awl-9J,
j = 1,...,r.
(18)
ay
As an aside, note that (16) is inverted as follows: w\
_
U = [X+ X+]
w
/=
2 Re X+ w
In view of (18), we are left with the problem of finding a fundamental solution for a scalar operator of the form a/ax + AO a/ay, where AO = a + ib and b > 0; it is easily seen that a fundamental solution is given by -1
1
2niAox-y (Make a change of variables in the fundamental solution 1/nz for alai.) Hence the operator on the left side of (18) has fundamental solution given by
E+
, where E.
E+ = - 1diag 2xi
1
A,x - y
Then the original operator I a + A
r J=1
has fundamental solution ay
E=[X+ X+]
(E+
a
Y
2ReX+E+Y+.
E+X Y+) Y1
Now write X+ = [X1
X,] and Y =
:
,
where XX and Y are the
Y,
columns and rows of X+ and Y+, respectively. Also, let y+ = y1 + + y, where y, is a small circle in the upper half-plane containing Al but no other
605
The index in the plane eigenvalues of A. Then
X;Y X+E+Y+ = Y 1 J=1A.Jx-y
((IA
1
1
(1A -
('
A)
Ax - y
27ci;_1
2ni J,.
dA
A)-1
Ax - y
(the second equality holds since .1t is a simple pole of QA - A)-1, i.e. A. is an eigenvalue of multiplicity 1). This proves (14) in the case that A (ias distinct eigenvalues. Suppose now that A is any real 2r x 2r matrix such that det(1A - A) # 0 for all A E R. Note that the right side of (14) defines a function in C"(I82\0) that is locally integrable near the origin, and depends continuously on the matrix A. Lets > 0. By Lemma 16.10, there exists a matrix A. with distinct eigenvalues such that I! A, - All < e. By choosing a small enough we may assume that A, has real entries and det(L. - A,) 0 for all ). e R. Now let E. denote the right side of (14) when A is replaced by A,. Then we have
a
``+A, m`=b1, Y
and letting a - 0, we have E. -> E and DE -+A-=61, H
2x
ay
as required. The following proof is from [AMR]. Lemma 16.10 Let A be an n x n matrix. For any e > 0, there exists a matrix A, with distinct eigenvalues such that 11 A, - All < e.
Proof Let p(A) = det(IA - A) be the characteristic polynomial of A and let {U1, , p"-1 be the roots of the derivative p'(1). Then A has multiple eigenvalues if and only if a1U1)...a,t"-1) = 0 This last expression is a symmetric polynomial in p1, ... , p"-1, and so is a polynomial in the coefficients of p'; it is therefore a polynomial q in the n2 entries of A. Then q-1(0) is the set of complex n x n matrices which have multiple eigenvalues. Since q # 0, the complement of q- 1(0) is dense in C"2. then all its (Proof by contradiction: if q vanishes on an open set in derivatives also vanish, and hence all its coefficients are zero.) C"=,
16.4 Index formulas for higher-order systems with real coefficients In this section we suppose that s4 is an elliptic operator with principal part homogeneous of degree 1, and that the boundary operator 9 is a differential
Boundary value problems for elliptic systems
606
operator. In the first part of this section we also suppose that the principal part of 9 in each row is equal to the same number p; that is, with the notation of §15.1, (d, 9) e BE'-. For the moment we also assume that it < I - 1.
We will derive a formula for the index of (s?, 9) by reformulating (ns0', irs) as a first-order system and then applying Theorem 16.8. In view of §15.4 we may suppose that sal and 9 have matrix coefficients with real entries.
Let L.().) = 7r d(x, (1, A)) = U=o Aj(x)A', x e fl Since sal is elliptic, then A,(x) = itd (x, (0, 1)) is invertible for all x e i2 By considering the operator
A- 'd, we may assume that A, = I, that is, L.,(A) is monic. Also we let (1, A)) = _-o Bj(y)aj, y e ail. Recall from §15.1 that (d, R) satisfies the L-condition if and only if det A '(y) 0 0, where
B,(...) =
1-1
A '(y) =
j=o
Bj(Y)X+(Y)T+(Y)
(19)
As usual, we let (X+(x), T+(x)) be a y+-spectral pair of L.,Q) consisting of smooth matrix functions on G. To reformulate (rrd, its) as a first-order system, let
j1,
u
vj=axi ,-jax2,
(20)
so that the equation R.s?(x, a/ax)u = f(x) takes the form avi-1 ax2
+ Ai-1(x) ax 1 + ... + Ao(x) 1
ax,1
= f(x),
(21)
and there are the "compatibility conditions" avj
=
avj+1 1 = 0,...,1 - 2.
axe
(22)
ax-1
Thus, we obtain the first-order system .?(x, a/ax)v =
49V
- C1(x)
z
-
= F(x),
xeQ
(23a)
1
where v = [vo . . . v1_ J' and F = [0 . . . 0 f] T are pl-vector functions and C1(x) is the companion matrix for L.,(A):
0 1
1
0
1
Cl =
I
-Ao -A, ... ... -A,-1I Note that sl is elliptic if and only if ? is elliptic, since det Lx(A) det(11. - C1(x)).
The index in the plane
607
If p = I - 1 the boundary conditions a.V(y, c/aax)u = g(y) take the form
i(Y)v = g(y),
y e oil,
(23b)
_ [Bo Bl - -B_1], an r x pl matrix function. Then (19) can be written in the form where
A.4+(y) _
T+(y))j=o.
(24)
Now we are ready to prove the first version of the index formula (for real coefficients). At first sight it might appear that (25) implies that the index of (d, R) depends only on the values of the coefficients of d on the boundary
if, but in fact the choice of (X+, T+), and hence A+, depends on the coefficients throughout the region fl.
Theorem 16.11 Let the hypotheses be as stated above: (.Qt, 9) a BE'-" (for arbitrary µ) and the boundary value problem operators sat and . have real matrix coefficients. Then ind(&,/, -d) _
-1 [arg det A+(y)]en + r(2µ + 2 - 1).
(25)
tr
Proof Without loss of generality (.91, -4) = (anat. irs), that is, Qt contains no
derivatives of order 0
where M(c) is defined as usual and F(c) _ [cj bk jI ] k.j 1 o If 9 is a differential operator we also have (with G_ defined by the integral f _)
G+(y, -c') = M(- 1)G-(y,
')F(-1)(-1)'-I
(50)
For the Dirichlet problem the matrix G+ takes the form
M.* (y, '') =
1 ft 2n
I I \ L,-"(A)U ...As-'I] dl s-lI
Note that M+ = -M,; because the integrand is O(2- 2) as JAI - * co. An application of Cauchy's theorem then gives f,,+ = IT. and f,_ = - j As
usual, y+ and y- are simple, closed contours in the upper and lower
half-planes containing the eigenvalues of L,,, 4. (A) there, oriented in the counterclockwise direction. The following theorem is a particular case of the formula of AgranoviLDynin (Theorem 10.19).
Theorem 16.20 Let d a Ell" and let -4 be a boundary operator satisfying the L-condition. If the Dirichlet problem (d, 9) satisfies the L-condition then
ind(d, R) = ind(.W, 2) - I [arg det G+(y, T(y))]an + Zn [arg det G+(y, -T(y))]an
Hence ind(si, 2) = ind(d, 2) + ind c+, where W+ is a p.d.o. on 3( with principal symbol G+.
Proof Let (x+ (y, c'), T+(y, c')) be the y+-spectral pair of L .(A) defined as in (48). Let A+ = 4 and 8+ = col(X+ T+)i=o. By Tf eorem 16.19 we have
ind(sal, 2) = 2 [arg det A+(y, r(y))]an
+ 2n [arg det A+(y, -T(y))] , + r(2 - 1) and
Boundary value problems for elliptic systems
618
ind(d, 2) = 2a [arg det 3+(y, t(Y))]aa + Za [arg det c+(y, -T(Y))]an + r(2 - 1)
Hence ind(., 2) = ind(d, 2) + K, where K
= 2n [arg det R(y, r(y))]an + Z [arg det R(y, -r(y))]an n
and R = A+ 3+ 1. Now, observe that
G. = A+.[Y+... T+ 1Y+]
-lp
1
= R M+ ,
As mentioned above, Mu = -M-; thus (50) implies that det M ,+,,(y, - r(y)) and det M+(y, r(y)) are equal except for a constant factor ± 1. Hence they have the same winding number and K
2n
[arg det G+(Y, t(Y))]an + - [arg det G+(Y, -T(Y))]an
The last statement of the theorem now follows from Noether's formula for the index of p.d.o.'s on Oil (Theorem 8.83). Remark Due to §3.6, L7(A) has a monic y+-spectral right divisor, L, (A), of degree s. A right division by L, (A) gives B,(A) = Q,(A)L. ()) + Ry(A) where R,(A) _ E;=o RJA' and [R0(Y)...R:-1(Y)]
=
This is just the equation R = A+ "+1 in the proof above. Corollary 16.21 In addition to the hypotheses of Theorem 16.20, suppose that
2 is a differential operator and has order < s - 1. Then
ind(d, dl) = ind(.d, 2). Proof We define 5:t (Y)
= 27ti
ft
B,(A)Lq
'(A)[I... As- 11] dA
and note that G+(y, t(y)) = G+(y). If.4 is a differential operator we also have G+(y, Thus Theorem 16.20 can also be stated as follows: 1
1
n ind(.a7, 9) = ind(.a1, 2)-! [arg det G+(y)]01 + 2[arg det G-(y)]O.
The index in the plane Since the order of . and we have
619
is 5 s - 1, the terms in the integrand of G't are O(A-2)
G+(Y) - G-(Y) = tai J By())Lr 1(;.)[I... A- 11] dA = 0
r
where r is a simple closed contour containing all the roots of det L,(A) in its interior. Hence G+(y) = G_(y) which implies ind(sd1, 9) = ind(d, -9). Exercises 1.
Let T. X -+ Y be a Fredholm operator, where X and Y are Banach
spaces over R. Show that
ind(Tc) = ind T where TC: X ® C -+ Y® C denotes the complexification of T defined by
v®zi--+Tv®z, VEX,zEC. Let X and Y be Banach spaces over R, and let S: X ®C - Y ®C be a (C-linear) Fredholm operator. Show that ind(SR) = 2 ind S 2.
where SR: X ® X - Y® Y denotes the realization of S. Recall from §5.1 that the realization of S is defined as follows: If we write S(v (3 1)=(Slv) ®1 +
(S2r) ® i, where S;: X -, Y, then S1
S2l
' S2
S1/
SR =
Let B(A) = Fi=o BMA be an r x p matrix polynomial of degree p and let B'(A) = (cA + d)4B(A) where c, d e C and q is any nonnegative integer. Let 3.
X be a p x r matrix and Tan r x r matrix. Show that je+Q
k
Y B;XT'= Y i=o
I=o
where B'(A) _ Ey_$ B. 4.
Consider a system of p Laplace operators S1 = 1 a2/ax; + 1 a2/ax2,
with boundary operator - = B1(y) a/ax1 + B2(y) a/ax2.
(a) Show that the L-condition for (d, 2) is det(B1(y) + iB2(y)) # 0 for all y 6 arl (b) Using the fact that the Dirichlet problem for the Laplace operator has index 0, show that ind(sd1, 9) _
-1 [arg det(B1(y) + iB2(y))],10 + 2p.
The following is a generalization of the preceding exercise.
620 5.
Boundary value problems for elliptic systems
Let sat be a real p (x p second order elliptic operator such that det J Lx- '(i.) d; 96 0
for all x e S2,
1
where Lx(.) = nsi(x, (1, 2)). Show that (a) There exists a'r+-spectral right divisor, 12 - S,,, of L,(;.) for all x e n such that S, depends smoothly on x.
(b) The pair (X+(x), T+(x)) = (I, S,) is a Y'-spectral pair of L,(1.) and the Dirichlet problem satisfies the L-condition and has index 0. (c) The boundary operator 9 = BI(y) t/dx, + B2(y) d/dx2 satisfies the Lcondition if and only det(B,(y) + B2(y) S,) : 0 for all y e dig. (d) If (sat, -4) satisfies the L-condition, then 1
ind(.ot, 9) =
I
[arg det(B,(y) +
2p
71
Let sat be a (properly) elliptic operator of order 2s. If the Dirichlet boundary operator -9 = {I, I 010n,..., 1(8/in)'-'} satisfies the L-condition, show that 6.
ind(sl,I) =
1
2
[arg det E+(y)]e, +
1
2a
[arg det E_(y)]
where 23 =col(XtTt)=o, and (X+(x), T+(x)) is any smooth matrix j+1 spectral pair of L,(2), x e Q. Addendum
Let U denote the unit disc 1:1 < 1 and T the boundary Izi = 1. Let g e C(W) be a continuous real-valued function on T. Consider the problem of finding an analytic function w = u - iv in U such that its (harmonic) real part, it, has prescribed values, Ow
az
Re w = g on T.
= 0 in U,
(51)
It is well known that all solutions of this problem are given by
w(:) =
l J1eil+z -- f02. e'` - z
,q(e") dt + 1k,
(52)
where k is a real constant. Formally, this result can be obtained by looking at the Fourier expansion of g on the unit circle T,
g(e") _ Y 4(n) e"", Re z
The index in the plane
621
where the Fourier coefficients are defined by 2R 1
9(n) =
g(e1') adt
0
Since g has real values then g(n) = 4(-n) and the Fourier series for g takes the form g(ee) = Rel 4(0) + 2
O(n) em : I.
J
n=o
Thus we let 1
w(z) =
2R
ao
S(z, e")&") dt,
Izj '< 1,
(53)
where the Schwarz kernel S is z"e- in' =
S(z, e") _ -1 + 2 M=O
e"+z e" _ z
,
{z1 < 1.
(54)
Exercises (continued) 7.
Let g e C(GU). Show that all solutions of the problem (51) have the form
(52). Hint: First of all, verify that the function defined by (53) solves the problem (51). See [Ru 1], pp. 233-235. As usual let f2 be a bounded and simply connected region in R2 with C ' boundary. Show that the elliptic boundary problem operator 8.
Q0: W21.(f2) - L2(Q) x WZr2(aR, 88)
(55)
defined by w - (Ow/O!, Re wj,.fl)
is surjective and has a kernel of dimension 1. Conclude that
ind20=1. Hint: By the Riemann mapping theorem (see [Ru 1], p. 283) there exists a one-to-one conformal mapping 4 of Cl onto the unit disc U. It can be shown that li extends to a homeomorphism of S2 onto U, and is in fact smooth up to the boundary; see [Ru 1] and [War].
The formula (2) for the index of the Riemann-Hilbert problem (1') can now be obtained from the known index of (55) and Theorem 10.19 (Agranovic-Dynin).
Let B = [cos V -sin cp] and qp e C'(00). Show that the index of the operator 9.
2: WZ(C, R2) -+ L2(C, 882) x W2r2(8f1;, R)
622
Boundary value problems for elliptic systems
defined by the left-hand side of the equations in (1), i.e. u t-+ (Au, Bu) =
-
-11 au
du
(0
ox2
1
0) OX,
+ A o(x) u, Bu
/
is given by the formula
ind 2 = I - 2X = 1 - 1 [c'(Y)]an
(56)
n
0
1l
i
0I
Hint: The matrix polynomial L(J) = li. -
associated to the elliptic
operator A has a y '-spectral_pair X+ = [i - i ] r, T+ = i. By virtue of §15(14) the matrix polynomial L(A) = rrA(y, t + An) also has y+-spectral pair
X+ = f
\-i
,
T+ = i.
Then AB = BX+ = e". Similarly, with an appropriate choice of y- spectral pair we have A. = e-'1. By virtue of the discussion at the beginning of §15.1 we obtain a y+-spectral pair (X,., T+) defined on ST*(M) such that
e"", Ag (Y, ) =
1e-'),
if S'=t(Y) if -t(Y)
Now conclude from Theorem 10.19 that
ind2=ind2o+indQ, where Q is an elliptic p.d.o. on 8i2 with principal symbol A8 +. It follows by Noether's formula (Theorem 8.82) that ind Q = - X - X where X =
2n
C 0, it is easily verified that x(A) has degree 1, and we let rj (j = 0,... , 1) denote the 4's such that the equation a(A) = 0 has exactly j roots in the upper half-plane, and 1- j in the lower half-plane. 623
624
Boundary value problems for elliptic systems
Similarly, for operators d e 8 such that det L(i.) < 0, there are I + I classes 8; (j = 0,..., 1) determined by the roots of fl(i.) = 0. Note that case
(b) can be transformed to (a) by interchanging the rows of L(i.); this interchanges the roles of ot(;.) and P(;.).
Example 17.1 Let d be the real 2 x 2 operator corresponding to the Bitsadze operator a2/0x; + 2i a2/ax, axe - a2/axZ (see Chapter 9, Exercise 6). Then we have
;2-1
-2i. 12
2i.
-1
and det L(i) = ().2 + 1)2 > 0 for real i.. Also a(.) = 2(i. - i)2 which has two roots in the upper half-plane, so that d e 82. If we take the conjugate of the Bitsadze operator we get an operator in 80. The next theorem will show that 8 and 8; are components (i.e. maximal connected subsets) of if. Bojarski stated this result in [Bo 1] without proof. Theorem 17.2 The set of has 2(l + 1) homotopy classes, namely, 8o, ... , B, and
Proof It suffices to consider only those elliptic operators d for which det L(i.) > 0. Also there is the homotopy d + t(.d - 7r.4), 0 < t 4 det L(2) > 0
for all 0 < t < 1. When t = 1, we have
I a(A) -#(A) 2 p(A)
a(2)
Hence, without loss of generality, we may assume that L(A) and L(2) have the form A) -q(A) P(A) (4) -4(A) L(A) = L(A) = q(A) P(A) ' 4(A) P(A)) Since a = 2(p + iq) and a" = p" + iq, then p(2) + iq(2) # 0 and P(A) + 14(A) # 0 for all A e R. By Lemma 3.19, we also have p(2) + iq(A)
1
1
(-i
1
i 1
"'
P(2) - iq(2))( -i with an analogous expression for E(A) where the polynomials p, q are L(2) =
i/
(5)
replaced by p', q". By assumption, the equations a(A) = 0 and a(2) = 0 have the same number of roots in the upper half-plane. It is then clear that there exists a homotopy p` + iq' connecting p + iq and P + i4. In view of (5), this gives us a homotopy L`(2) connecting L(A) and L(2): L`(2) =
1
1
(-i
i
(p`(2) + iq`(A)
\
I -z l( AA) - iq`(A)/ \ - i i ) 1
-q`(A)) ' 0 < t < 1, which completes the proof of the second step. Cpt(A) q`(A)
P`(A)
For second-order elliptic operators d with det d(1, 2) > 0, there are three homotopy classes eo, Bl and 82. The next theorem shows that for any sad a J'j,
the Dirichlet problem (d, 2) satisfies the L-condition. Since the 2 x 2 Laplace operator (e A) also belongs to 91, it follows that the Dirichlet problem has index 0 for any operator in 8'1.
Boundary value problems for elliptic systems
626
Theorem 17.3 Let d be a second-order 2 x 2 elliptic operator (1= 2). If sad a 1 (or ti) then the Dirichlet problem for si satisfies the L-condition and has index 0. Proof To verify that the L-condition holds it is sufficient (in view of Theorem 10.5 it is also necessary) to show that det
If
A)]
(6)
dA jA 0
For 2 x 2 systems, the condition (6) can be reformulated in terms of the scalar functions a(A) and fl(A). In view of (1)
[nd 1, AA -I
-
a22(A)
-al2(A)
D(A) \-a21(A)
a11(A)
I
'
a, z(A)a21(A), so that (6) is equiv-
where D(A) = det aA(1, A) =
alent to a22(2)
dA
D(A)
a, '(A)
a12(2) a21(A) Y. D(A) dA jr+ D(A) dA
dA #
r+ D(A)
which, in turn, is equivalent to dAlz
D()
dAI
#' fr+ D(A)
or
0
dA
D(A)
11+
(7)
for all p e C such that IµI = 1. Since .4 a e,, the polynomial a(A) has one root above the real axis and one below. In view of (3) and Rouche's theorem, the same is true of the A-polynomial a(A) + µfl(A); let p, and µz denote these
roots, where Imp, > 0 and Im uz < 0. Also the real polynomial D(A) has roots A,, A,, A2 and A2, where Im A > 0 and Im A2 > 0. Then the condition (7) may be written in the form
5
I 27ri
f
(A-k1)(A-u2)
-d2#0.
(A - A1)(2 - A1)(A - A2)(A - A)
We will show that J :A 0 for any µ,, u21 A, and A2 such that Imp, > 0,
Imp20and ImA2>0.
Evaluating the residues of .0 in the upper half-plane and assuming that At # Az so that /the poles are simple, one obtains
._
(AI - i1)(Al _102) (A1 - Al)(AI - A2KA1 - A2)
+
(A2 - P1)(A2 - P2) (A2 - 21)(A2 - A1X22 - A2)
After taking a common denominator and cancelling the factor A, - A2, one
627
Elliptic systems with 2 x 2 real coefficients
obtains -101112 Im(2, + 22) - 12212 IM t, -12,12 Im 22] - 2i((p, + 112) IM(2122) 41m2,Im12.121 -.1212
By a limiting process, it is clear that the latter formula is true even if Al = 22.
Replacing 2 with 2 + m in the integral 5, where m is a suitably chosen real number, we may assume that Im(2122) = 0. Thus, to prove .0 = 0, it is necessary to show that -PIP2 'm(21 + 22) 96 12212 Im Al + 12112 Im 22 or
-µl#kµ2,
where
k=12212 Im21+I2112 Im22
IP212 Im(21 + 22)
But the fact that -,u, # kµ2 is clear since k > 0, Im(- µ,) < 0 and Im02) > 0. This proves that the Dirichlet problem for sad satisfies the L-condition. Now, since both sad and the 2 x 2 Laplace operator are in 4f,, there is a homotopy sad' in &I, such that d° = sd, sa71 = (
A). The L-condition holds
for the Dirichlet problem (sz(`, I) for all 0 < t < 1. Hence by Theorem 10.25 ind(d, 91) = ind(sV1, -9) = 0. Remarks (1) Theorem 17.3 remains true even if the coefficients of sW are not assumed to be constant; the same proof shows that det
fr+ [a.4(x, (1, A))]-' d2 # 0
(8)
for all x e i2. The idea of the proof of Theorem 17.3 was sketched in the paper [Bo 2]. (2) We can also prove that ind(sl, -9) = 0 using the formula in §16.5. Let (X+(x), T+(x),Y+(x)) be a y+-spectral triple of L,(2) = rrsd(x, (1, 2)), where X+, T+ and Y. are 2 x 2 smooth matrix functions. Since the matrix (8) is equal to X+(x)- Y+(x), it follows that det X+(x) # 0 for all x e Cl. Thus
ind(sV, 2) =
-1a [arg det X+(y)]an = 0.
17.2 An example: the Neumann BVP for second-order elliptic operators Recall that if 2
d(x, 8/8x) _
Ajk(x) 8218x; 8xk + lower order terms, J.k=1
x e Cl,
Boundary value problems for elliptic systems
628
then the Neumann boundary operator is W(y,
2
0/ax) = Y Ajk(Y)vk(Y) a/axj,
y e an,
j.k=1
where v = (v1, v2) _ (-t2, t1)
Lemma 17.4 For the second-order elliptic operator a2
d(x, a/ax) = A(x) axe + 2B(x) 2
02
a2
ax2 ax
+ C(x) 1
8x2
+
let (Xt(x), Tt(x)) be a yt-spectral pair of Lx(d) = nsjf(x, (1, A)). If '(y, a/ax) _ C2(y) a/axe + C1(y) a/0x1 denotes the Neumann boundary operator then C2(Y)Xt(Y)Tt(Y) + C1(Y)X±(Y) = (A(Y)X t (Y)TT (Y) + B(Y)X t (Y)) - (r1(Y)I + r2(Y)TT (Y))
(9)
for all y e aS2.
Proof We have W(y, a/ax) = C2(y) a/axe + C, (y) a/ax 1 where C. = s1 A - t2B
and C, =t1B-t2C.Thus C2XtTT + C1Xt = (t1A - t2B)XtTt + (r1B - r2C)Xt = r1(AXtTT + BXt) - r2(BX1TT + CXt) Since L(A) = A22 + 2BA + C, then AX ± T t + 2BX t Tt + CX t
=
0. Hence
(AXtTT + BXt)TT = -(BX±TT + CXI), and then (9) follows.
Corollary 17.5 Let L',(A) = n.4(y, t(y) + Av(y)) = A(y)A' + 2,9(y) A + C(y)
be the transform of L,(A) under P(2) = (-t2 + ).t1)/(t1 +.ir2) with ytspectral pair (X t (y), T± (y)) defined by §15.3 (14). Then A(y)f(± (Y)T± (Y) + B(Y)X x (Y) = A(y)X± (y)T± (Y) + B(y)X± (Y)
for all y e aQ.
Proof Note that s'(y, a/ax) = c2(y) a/av + c1(y) a/at, where C2(y) = A(y) and C3(y) = B(y), and the associated matrix polynomial is C, Q) ='(y, r(y) + Av(y)) = A(Y) 2 + B(Y) By Proposition 15.12, we then have A(Y)X t (Y)T? (Y) + :9(Y)g±(Y) = (C2(Y)X t (Y)TT (Y) + C1(Y)X t (Y))
X (t,(Y)I +
t2(Y)TT(Y))-1
= A(y)X±(y)T±(y) + B(Y)Xt(Y), the last equality holding because of (9).
629
Elliptic systems with 2 x 2 real coefficients
As an example, let us consider the 2 x 2 operator Ax, a/ax) = A(x) axz + 2B(x) axaax z
2
where ((P2VI(X(X)
l),
(PA(xx (
1
l + C(x)
axz + .. . l
B(x) = -I, C(x) = A(x)'
A(x) = and (pl, (P2 are real-valued functions in C°°(i2) such that [Wl(x)]2 + [4)2(x)]2 # 1 for all x e i2. It is clear that d is elliptic, since LX(A.) = xd(x, (1, 2))
= A(x)12 + 2B(x)2 + C(x) 4p2(x)(A2 + 1) + ()2 - 1)
W, (x)(22 + 1) - 22 4Pz(xxZ2 + 1) -
().2
- 1)
-4Pl(x)(22 + 1) - 22
and det Lx(A) = (1 + 22)2(1 - 1(p(x)I2) for all A e R, where V (x) = 4p,(x) + In view of Example 3.17,
X+(x)
=
r -`
- i4P2(x))
r \
T+(x) = (0
1)
4PlO , is a y+-spectral pair of L.,,(2), depending smoothly on x e f2. The Dirichlet boundary value problem for sad satisfies the L-condition provided that the to C" is invertible for all y e ail (see Remark 15.1 map u i - u(0) from and Corollary 15.13). Since det X+ (y) = ikp(y) # 0, the Dirichlet problem satisfies the L-condition if and only if for all y e dig,
4p(y) # 0
and by Theorem 16.15 the index is
ind(d, q) = -1 [arg det X.. (y)]aa Ir
= 1 Carg 4P(y)]es.
(10)
n
Consider now the Neumann boundary value problem for sad. By Corollary 17.5 we have A(Y)X+(Y)T+(Y) + B(Y)X+(Y)
= A(Y)X+(Y)T+(Y) + B(Y X+(Y)
r
(Pi(Y)
- \4P2(Y) -1
r
!4P(Y)
= \ -w(Y)
(Pz(Y) + 1
_91(Y)
-i I(P(Y)IZ -
1)'
1
-i
-i4pz(y) (i 14P1(Y) \0
1) i
1
-i
-14P2(Y)
iwl(Y) (11)
630
Boundary value problems for elliptic systems
Hence
det(A(y)X+(y)T+(y) + B(y)X+(y)) = ip(y)(Iq(y)12 - 2),
so the Neumann boundary value problem for .4 satisfies the L-condition if and only if cp(y) # 0, Igp(y)I # 0, (y)I 96 f for all y e Q. By Theorem 16.11, the index of the Neumann problem is 1
ind(sl, W) = 4 - 1 [arg det(C2X+ T+ + C1X+)]an (here it = 1, 1 = 2 and r = p1/2 = 2). By Corollary 17.4 we have C2X+T+ + C1X+ _ (AX+T+ + BX+)(t11 + t2T+)
and by virtue of (11) we have det(AX+T+ + BX+) = io(IT12 - 2). Since 1912 - 2 is real-valued, it has winding number equal to 0, so the winding number of det(AX+T+ + BX+) is equal to that of 1 on 6. The formula (10) shows that in general the Dirichiet problem has non-zero index. However,
note that ind(.d, 2) = 0 if sat e Bi. 17.3 An example: the elliptic system for plane elastic deformations Consider an infinitely long, cylindrical, elastic solid with plane cross-section () and assume that all displacements and forces lie in this plane (and depend
only on the coordinates in the plane). For an isotropic material the equilibrium equations for the displacement vector u = [u1 u21T take the form
d(8/ax)u = (A + u) grad div u + it Au = f(x)
(12)
where 2, u are Lame's elastic constants and f = [f1 f2]T is the body force. Let us write (12) in another form using Hooke's Law. This law, connecting stress to strain, states that
stress = 2u(strain) + 2(trace of strain)!.
(13)
Elliptic systems with 2 x 2 real coefficients
631
Let the stress tensor be denoted by a = [ai,] and the strain tensor by e = [e j]. By definition,
e,;(u) =
i, j = 1, 2
so that aij(u)
\-ut +
= µ ax,
au,l + A div u&,, axij
i, j = 1, 2
and then 2
(div a), = E
aaij
j=1 ax;
=k
a
(div u)bii ` a2u` J+ j=, ax, ax, ax;/ j=, \axj + 02uj
Aui+(A+t)
a
a
div u.
Thus, .emu = div a(u). We now derive a Green formula for sad. Let n denote the outward-pointing
unit normal vector field along 011. By the divergence theorem we have
I
1 div(a v) dx = (a v)Tn ds, (14) J n an where v = [v1 v21T is any 2-vector function and a = [at,] is au 2 x 2 matrix function (not necessarily a stress matrix) with entries in C'(S ). Also, div(o-v) = E
8
i, j ax,
(aijvj)
ot, =Ea,jz L+ ,.j
axt
t,j axt
vi,
and if a is a symmetric matrix
_ E et;(v)ai; + E v;(div a)j. i.j j Thus, we obtain div a,
where tr(eTa) denotes the matrix inner product D,, ei,ai j. By (14) we then have the formula
f f (tr(e(v)Ta) + vT div a) dx = n
fan vTaTn ds
(15)
where v is any 2-vector function, e = e(v) is the associated strain tensor and a is any 2 x 2 symmetric matrix function. Substituting or = a(u) in (15) gives
us one equation, then reversing the roles of u and v gives us another; subtracting the two equations and using the fact that tr(e(v)Tc(u)) =
632
Boundary value problems for elliptic systems
tr(e(u)Ta(v)), we obtain the Green formula
fan (v'-Vu - UTMv) dS,
n
for u, v e C2((1), where du = div a(u) and Ru = a(u) - n. This formula suggests
a boundary condition for d, namely, that the surface traction, 3u = a(u) n, be prescribed on a(. For future reference note that the stress matrix is µ(dui/axe + auZ/axl) 1 a(u) _ ((A + 2µ) aul/ax, + A au2/40xe (16) Mu,/ax, + (A + 2µ) auZ/ax2J µ(au,/ax2 + 0u2/axl)
and then R u =a(u)-n, where n = [r2
-ri]r
Physically, there are certain restrictions on the Lame constants A, µ which can be determined by consideration of the strain energy. Hooke's law (13) may be written in the form or = Ce, where C = [cikl] is a symmetric tensor of rank 4, and then the strain energy 'z tr(eTCe) equals
µ
et2j+ZA-(tre)2. r.;
On physical grounds the strain energy should be positive definite: tr(eTCe) 3 0
and tr(eTCe) = 0 if and only if e = 0. Given that attention is restricted to plane elastic deformations, it is not hard to show that the strain energy is positive definite and only if µ > 0 and A + µ > 0. However, the strain energy should be positive definite with respect to all three-dimensional elastic deformations, where the displacement vector u is [u, u2 u3]r and the u; depend on x,, x2 and x3. (All the definitions and calculations in this section remain valid in space with the obvious modifications.) This leads to a stronger restriction on A, A. Exercise 1 Show that the strain energy is positive definite with respect to the class of all three-dimensional elastic deformations if and only if µ > 0 and
3A+2µ>0.
Under the assumption that the strain energy is positive definite, we can use the formula (15) to determine explicitly the kernel and cokernel of (.sad, 9).
Let u e ker(d, -4). Then div a(u) = 0 in f and a(u) - n = 0 on aft, so that (15) implies
0 = ffn tr(era) dx =
JJn
tr(eTCe) dx
whence tr[eTCe] = 0. Then e = e(u) = 0, or
+-=0 ax; ax, au1
aU1
which has solutions u = [a, a2]' + b[x2 Hence a = dim ker(sif, 9) = 3.
i,j=1,2 -x,]T, where a a2 and b e R.
Elliptic systems with 2 x 2 real coefficients
633
To determine the dimension of the cokernel, we consider solvability conditions for the BVP saf (a/ax) u = 0,
x e (2
My, a/ax) u = g(y),
y e at,
(17)
Suppose we let v = [a, a2]T in the formula (15). Then e = e(v) = 0, and since a, and a2 are arbitrary constants, we obtain
ddx = Jan ff(Olax)u
n
R(y, a/ax)u ds.
Hence for the BVP (17) the conditions on g = [g, g2]T are
f
f
gl(Y)ds=0,
an
92(Y)ds=0.
an
Another solvability condition is obtained by letting v = b[x2
-x,]T in (15):
(Y2g1(Y) - Y1g2(Y)) ds = 0.
fan These are the only solvability conditions, that is, ig = 3. Since the equation sI(a/ax)u = f(x) has a solution u e C'(S1, f82) for any f e C°°(f, R2) it follows that ind(d, 9) = ind(.Ik« r) = 3 - 3 = 0 (see Lemma 16.13). Exercise 2 Within the class of three-dimensional elastic deformations, determine the kernel and cokernel of (sad, 9).
We end this chapter by applying the results of §17.1 to the operator (12), with no a priori assumptions on A and µ (i.e. with no requirement that the strain energy be positive definite). We have (L (z) = .V(l, z) _ (a. + u)
1
z z2)+(zz+
z2 +
z
µz2 + (A + 2p)
(2 + µ)z
(A + 2µ)z2 +
(2. + µ)z
1
JI
so that det L(z) = µ(2 + 2µ)(z2 + 1)2. Hence sat is elliptic provided p # 0 and d + 2p # 0 which we assume from now on. Also note that a(z) = (I + 3µ)(z2 + 1),
#(z) = _(A+ µ)(z - i)2.
Thus, ./ a c8, if p(2 + 2p) > 0 and a e e2 if p(A + 2p) < 0. Since det L(z) has a single root i of multiplicity 2 above the real axis, it is easily verified that a y+-spectral pair for L(z) is given by
X+=
-i 1
J.+3µ
2.+k 0
i
,
1
T+-(0 i)
Boundary value problems for elliptic systems
634
provided A + p # 0 (if A +,u = 0 then (12) is a system of two Laplace equations which can be dealt with separately). Since det X+ = (2 + 3p)/(2 +.U),
we find that the Dirichlet problem for sad satisfies the L-condition provided A + 3p # 0. If this condition holds then Theorem 16.15 implies 1
ind(d, 2)
[arg det X+ ]an = 0,
since det X+ is constant. Note: When 2 = - 3p, the operator sad is essentially the Bitsadze operator (Example 17.1).
Remark It is not hard to show that the condition p() + 2p) > 0 is equivalent to sad being strongly elliptic.
Now consider the boundary operator Mu = o(u) n. In view of (16) By(z) = V(y, (1, z)) = B1(y)z + B0(y), where
B1(y) = t
\
µt1 µT2
222
-(2 + 2µ)r1/
,
Bo(y) =
((2 + 2µ)t2 _ t1) `
-2T1
AT2
After some calculation it follows that AR+ (y) = B1(y)X+T+ + B0(y)X+ 2µ(t1
+ ire)
2µi(21 + iT2)
1
2p) + (4p2 + 6µZ)22] 2 + p [-2iµ(2 + [2µZ21 + 2ipt2(2 + 2p)]
+p and then det A+ = -4µ2(t1 + it2)2 960. Hence (sd, R) satisfies the L2
condition. By Theorem 16.11,
ind(st, 9) = 4 - [arg det AN(y)]8 n
=4-4=0, where we have used the fact that (1/2n)[arg(t1 + ir2)]M = 1.
References
[AB] M.F. Atiyah, R. Bott, The index problem for manifolds with boundary, Bombay Colloquium on Differential Analysis, Oxford Univ. Press, Oxford (1964), 175-186. [Ad] R.A. Adams, Sobolev Spaces, Academic Press, New York 1975. [ADN] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math. 12 (1959), 623-727; II, Comm. Pure Appl. Math. 17 (1964), 35-92. [Ag] S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Reinhold, Princeton, N.J. 1965. [Agr] M.S. Agranovi6, Elliptic singular integro-differential operators, Usp. Mat. Nauk 20, 5 (1965), 3-120 (Russian; English translation in Russian Math. Surveys 20 (1965), 1-122). [AkG] N.I. Akheizer, I.M. Glazman, Theory of Linear Operators in Hilbert Space, Pitman, London 1981. [AMR] R. Abraham, J.E. Marsden, T. Ratiu, Manifolds, Tensor Analysis, and Applications, Second Edition, Springer-Verlag, New York-Berlin 1988. [Atiy] M.F. Atiyah, Algebraic topology and elliptic operators, Comm. Pure Appl. Math. 20 (1967) 237-249. [Be] Yu.M. Berezanskii, Expansions in Elgenfunctions of Selfadjoint Operators, Translations of Mathematical Monographs, Vol. 17, Amer. Math. Soc., Providence, R.I. 1968. [BGR] J.A. Ball, Gohberg, L. Rodman, Interpolation of Rational Matrix Functions, Operator Theory: Advances and Applications, Vol. 45, Birkhi user Verlag, Boston-Basel 1990. [Bi 1] A.V. Bitsadze, Boundary Value Problems for Second-Order Elliptic Equations. North-Holland, Amsterdam 1968. [Bi 2] A.V. Bitsadze, Some Classes of Partial Differential Equations, Gordon & Breach, London 1988. [Bo 1] B. Bojarski, On the first boundary value problem for elliptic systems of second order in the plane, Pol. Akad. Nauk 7, 9 (1959), 565-570. [Bo 2) B. Bojarski, On the Dirichlet problem for a system of elliptic equations in space, Pol. Akad. Nauk 8, 1 (1960), 19-23. [BT] R. Bott, L.W.Tu, Differential Forms in Algebraic Topology, Springer-Verlag, New York-Berlin 1982. [Ca] H. Cartan, Calcul D4erentiel, Hermann, Paris 1967. 635
636
References
[CZ] A.P. Calderon, A. Zygmund, On the existence of certain singular integrals, Acta Math. 88, 1-2 (1952), 85-139. [DN] A. Douglis, L. Nirenberg, Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math. 8 (1955), 503-538. [Do] A. Dold, Lectures on Algebraic Topology, Springer-Verlag, New York-Berlin 1972. [Don] W.F. Donoghue, Distributions and Fourier Transforms, Academic Press, New York 1970. [DS 1] N. Dunford, J.T. Schwartz, Linear Operators, Vol. I, Wiley, New York 1958.
[DS 2] N. Dunford, J. T. Schwartz, Linear Operators, Vol. II, Wiley, New York 1963. [Eg] Yu.V. Egorov, Linear Differential Equations of the Main Types, Nauka, Moscow 1984 (Russian; English translation in Contemp. Sov. Math.: New York 1986). [Es] G.I. Eskin, Boundary Value Problems for Elliptic Pseudod fferential Equations, Translations of Mathematical Monographs, Vol. 52, Amer. Math. Soc., Providence, R.I. 1981. [Fe] B.V. Fedosov, An analytic formula for the index of an elliptic boundary-value problem, Mat. Sbornik 93, 135 (1974), No. 1 (Russian; English translation in Math. USSR Sbornik 22 (1974), No. 1, 61-90). [Ga] F.D. Gakhov, Boundary Value Problems, Dover Publications, New York 1990. [GLR] I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York 1982. [GK] I.C. Gohberg, M.G. Krein, The basic propositions on defect numbers, root numbers and indices of linear operators, Usp. Mat. Nauk 12, 2 (1957), 43-118 (Russian; English translation in Amer. Math. Soc. Transl., Ser. 2, 13 (1960), 185-264). [GP] V. Guillemin, A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, N.J. 1974.
[Gr] G. Grubb, Functional Calculus of Pseudo-Differential Boundary Problems, Birkhauser Verlag, Boston-Basel 1986. [Hi] M.W. Hirsch, Differential Topology, Springer-Verlag, New York-Berlin 1976. [Ho 1] L. Hormander, The Analysis of Linear Partial Differential Operators, Vol.1, Second Edition, Springer-Verlag, New York-Berlin 1990. [Ho 2] L. Hormander, The Analysis of Linear Partial Differential Operators, Vol. II, Springer-Verlag, New York-Berlin 1983. [Ho 3] L. Hormander, The Analysis of Linear Partial Differential Operators, Vol. III, Springer-Verlag, New York-Berlin 1985. [HR] H. Holmann, H. Rummler, Alternierende Differentia formen, B.I., Mannheim, 1972. [HW] W. Hurewicz, W. Wallman, Dimension Theory, Princeton University Press, Princeton 1941. [Hus] D. Husemoller, Fibre Bundles, McGraw-Hill, New York 1966. [Hu] Sze-Tsen Hu, Homotopy Theory, Academic Press, New York 1959. [KN] J.J. Kohn, L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 265-305. [Ku] H. Kumano-go, Pseudo-Differential Operators, M.I.T. Press, Cambridge, Mass. 1974.
[La 1] S. Lang, Algebra, Second Edition, Addison-Wesley, Reading, Mass. 1984.
References
637
[La 2] S. Lang, Differential Manifolds, Springer-Verlag, New York-Berlin 1985. [LM] J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, Berlin 1972. [Lo] Ya. B. Lopatinskii, On a method of reducing boundary value problems for a system of differential equations of elliptic type to regular integral equations, Ukr. Mat. Zur. 5, 2 (1953), 123-152 (Russian; English translation in Amer. Math. Soc. Transl., Ser. 2, 89 (1970), 149-183). [LS] L.H. Loomis, S. Sternberg, Advanced Calculus, Addison-Wesley, Reading 1968. [Mi] S.G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, Oxford 1965. [MP] S.G. Mikhlin, S. Pri Bdorfl; Singular Integral Operators, Springer-Verlag, New York-Berlin 1986. [Mir] C. Miranda, Partial Differential Equations of Elliptic Type, Springer-Verlag, New York-Berlin 1970. [Ne] J. Les methodes directes en theorie des equations elliptiques, Academia, Prague 1967. [Pa] R.S. Palais, Seminar on the Atiyah-Singer Index Theorem, Princeton University Press, Princeton, N.J. 1965. [Ru 1] W. Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill, New York 1987. [Ru 2] W. Rudin, Functional Analysis, McGraw-Hill, New York 1973. [Sch] M. Schechter, Modern Methods in Partial Differential Equations, McGraw-Hill, New York 1977. [Se 1] R. Seeley, Extension of C' functions defined in a half space, Proc. Amer. Math. Soc. 15, 625-626 (1964). [Se 2] R. Seeley, Integro-differential operators on vector bundles, Trans. Amer. Math. Soc. 117 (1965), 167-204. [Se 3] R. Seeley, Singular integrals and boundary value problems, Amer. J. Math. 88 (1966), 781-809. [Shu] M.S. Shubin, Pseudo-Differential Operators and Spectral Theory, Springer-Verlag, New York-Berlin 1987. [Sp 1] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. I, Publish or Perish, Inc., Boston, Mass. 1970. [Sp 2] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. V, Publish or Perish, Inc., Boston, Mass. 1975. [Str] G. Strang, Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, Mass. 1986. [Tay] M.E. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ. 1981. [TL] A.E. Taylor, D.C. Lay, Introduction to Functional Analysis, Second Edition, R.E. Krieger Pub[. Co., Malabar 1986. [Tre 1] F. Treves, Introduction to Pseudo-Differential and Fourier Integral Operators, Vol. 1, Plenum Press, New York-London 1982. [Tre 2] F. Treves, Introduction to Pseudo-Differential and Fourier Integral Operators, Vol. U. Plenum Press, New York-London 1982. [Tri] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam 1978. [Ve] I.N. Vekua, New Methods of Solution for Elliptic Equations, OGIZ, Moscow-Leningrad 1948. [Vo 1] L.R. Volevil, Solvability of boundary problems for general elliptic systems, Mat. Sbor. T. 68 (110), 3 (1965) 373-416 (Russian; English translation in Amer. Math. Soc. Transl., Ser. 2, 67 (1968), 182-225).
638
References
[Vo 2] L.R. Volevic, A problem in linear programming coming from differential equations, Usp. Mat. Nauk 18 (1963), no 3 (111), 155-162 (Russian). [Vol'p] A.I. Vol'pert, On the index and the normal solvability of boundary value problems for elliptic systems of differential equations in the plane, Trudy Mos. Mat. Obs. 10 (1961), 41-87 (Russian). [Wa] F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott-Foresman, London 1971. [War] W. Warschawski, On differentiability at the boundary in conformal mapping, Proc. Am. Math. Soc. 12 (1961), 614-620. [Wei] J. Weidmann, Linear Operators in Hilbert Spaces, Springer-Verlag, New York-Berlin 1980. [Wen] W.L. Wendland, Elliptic Systems in the Plane, Pitman, London 1979. [WI] J. Wloka, Partial Differential Equations, Cambridge University Press 1987. [Wo] M.W. Wong, An Introduction to Pseudo-Differential Operators, World Scientific, Singapore 1991.
Index
admissible pair 12 alternating 108 amplitude 239 asymptotic expansion 243, 297 base space 12
C' atlas 114 Calderon operator 545
Calderon projector 29
double-layer 526 double of a manifold 177 Ehrling's lemma 286 elliptic 270, 324 embedding 129 extension of C°0 functions 506 exterior algebra 108 exterior differentiation 184 exterior product 108
category 156
chart 114 classic operator 307, 321
collar 175 companion matrix 48 companion triple 49 complete symbol 113, 301 complexification 110 conormal bundle 152 contractible 163 coordinate map 113 cotangent bundle 181 covector field 181 covering condition 422 cut-off function 257 degree 582, 588 A-condition 422 density 203 diffeomorphism 107, 116 differentiable 101 differential 120 differential form 181 closed 189
exact 189 directional derivative 102, 131 Dirichlet problem 423, 613 DN-elliptic 333, 379 DN numbers 378
finite spectral triple 24 formal adjoint 289 Fredholm operator 366 freezing lemma 275, 276 functor contravariant 156 covariant 156
smooth 156 fundamental solution 602 y-spectral pair 24
partial 30 y-spectral triple 24 Gdrding's inequality 271 Gauss' theorem 209 general symbol 239 Green operator 499 Hermitian bundle 166 Hermitian metric 166 Hodge decomposition theorem 336 homogeneous function 256 homogeneous polynomial map 106 homogeneously elliptic 382 homotopy 162, 443, 451, 583 hypersurface 124 639
640
immersion 129 index 366 stability 371 index formula 607, 615 index-lowering operator 165 index-raising operator 165 infinite spectral triple 75 inverse function theorem 107
Jordan chain 8 canonical set 17 Jordan pair 18 finite 18 y-spectral 23, 35
partial 12 L-condition 389 L-elliptic 392 Laplace-Beltrami operator 228
Laplace-de Rham operator 225 left divisor 57 y-spectral 58 Leibniz rule 112 Hi rmander's generalized form 113 line bundle 147 linear differential operator 113, 210 linearization 47 local operator 209
locally Euclidean space 113 locally finite 139 Lopatinskii condition 422 main symbol 302, 316 manifold with boundary 170 monic 43 multilinear 107
Neumann boundary operator 628 Noether's formula 349 normal bundle 152 normal derivative 283 orientation of a manifold 197 orientation of a vector bundle 192 oscillatory integral 235 paracompact 140 parametrix 268, 327 partial derivative 104 partial spectral pair 12 partition of unity 139 patch function 257 Petrovskii elliptic 383 plane elastic deformations 630 Green formula 632 Poincare lemma 189
Index
Poisson operator 497 principal part 308, 321 properly elliptic 381 pseudo-differential operator 237, 293, 313
adjoint 248, 304, 319 composition 249, 304 transformation 252 pull-back 131, 185, 292 push-forward 131, 291
quotient manifold 128 quotient topology 118 realization 110 regularizer 367 smoothing 374 Riemann-Hilbert problem 595 generalized 600 index 595, 601 real form 601 Riemannian manifold 165 canonical volume form 198 codifferential 224 divergence 199 gradient 199 Hodge star operator 221, 224 Laplacian 199 right divisor 57 y-spectral 58
a-compact 139 single-layer 517 singular support 296 Smith canonical form 6 smooth map 116 smooth Tietze extension theorem 161 smoothing operator 374 Sobolev space 278, 288, 310 compact embedding 286 extension operator 281 restriction operator 281 trace operator 282 Sobolev's lemma 286 spectrum 1, 8 standard pair 43 standard triple 44 stereographic projection 115 Stokes' theorem 207 strongly elliptic 425 sub-bundle 149 submanifold 124 submersion 127 support 137, 139, 201, 291 symbol index 357
tangent bundle 120 tangent map 122
Index
tangent space 120 target space 12 Taylor's formula 105 topological index 588 trace operator 282 transfer 291, 312 transmission property 512 trivial elliptic b.v.p. operator 468 trivial on the boundary 469 trivialization-chart 312 tubular map 172 tubular neighbourhood 172
projection 143 pull-back 153 section 146 tensor product 155 total space 143 transition matrices 144 trivial 143 trivializing maps 142 vector field 130 complete 135 internal curve 132 Volevi4elliptic 379
vector bundle 142 cocycle conditions 144 direct sum 154 dual 155 local basis 146
weakly continuous 352 weakly convergent 352 wedge product 182 Weyl's lemma 326, 375, 394 winding number 582
641
This book examines the theory of boundary value problems for elliptic systems of partial differential equations, a theory which has many applications in mathematics and the physical sciences. The aim is to simplify and to algebraize the index
theory by means of pseudo-differential operators and new methods in the spectral theory of matrix polynomials. This latter theory provides important tools that will enable the reader to work efficiently with the principal symbols of the elliptic and boundary operators. It also leads to important simplifications and unifications in the proofs of basic theorems such as the reformulation of the Lopatinskii condition in various equivalent forms, homotopy lifting theorems, the reduction of a system with boundary conditions to a system on the boundary, and the index formula for systems in the plane. This book is suitable for use in graduate level courses on partial differential equations, elliptic systems, pseudo-differ-
ential operators, and matrix analysis. All the theorems are proved in detail, and the methods are well illustrated through numerous examples and exercises.
Cover design by Brian H. Crede/bc graphics
ISBN 0-521-43011-9
CAMBRIDGE UNIVERSITY PRESS