Rabier,rabier@imap.pitt.edu Received 24 March 2009; Revised 4 June 2009; Accepted 11 June 2009 Recommended by Peter Bates We fill a gap in the L p theory of elliptic systems on bounded d
Trang 1Volume 2009, Article ID 637243, 9 pages
doi:10.1155/2009/637243
Research Article
A Complement to the Fredholm Theory of
Elliptic Systems on Bounded Domains
Patrick J Rabier
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
Correspondence should be addressed to Patrick J Rabier,rabier@imap.pitt.edu
Received 24 March 2009; Revised 4 June 2009; Accepted 11 June 2009
Recommended by Peter Bates
We fill a gap in the L p theory of elliptic systems on bounded domains, by proving the
p-independence of the index and null-space under “minimal” smoothness assumptions This result has been known for long if additional regularity is assumed and in various other special cases,
possibly for a limited range of values of p Here, p-independence is proved in full generality.
Copyrightq 2009 Patrick J Rabier This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
Although important issues are still being investigated today, the bulk of the Fredholm theory
of linear elliptic boundary value problems on bounded domains was completed during the 1960s For pseudodifferential operators, the literature is more recent and begins with the work of Boutet de Monvel1; see also 2 for a more complete exposition. While this was
the result of the work and ideas of many, the most extensive treatment in the L pframework is arguably contained in the 1965 work of Geymonat3 This note answers a question explicitly left open in Geymonat’s paper which seems to have remained unresolved
We begin with a brief partial summary of3 in the case of a single scalar equation Let
Ω be a bounded connected open subset of RN , N≥ 2, and let P denote a differential operator
onΩ of order 2m, m ≥ 1, with complex coefficients,
P
|α|≤2m
Next, letB B1, ,Bmbe a system of boundary differential operators on ∂Ω with B of
order μ ≥ 0 also with complex coefficients,
B
|β|≤μ
Trang 2With M : max{2m, μ1 m
following regularity hypotheses:
H1; κ Ω is a C M ∂-submanifold ofRN i.e., ∂Ω is a C M submanifold ofRNandΩ lies
on one side of ∂Ω;
H2; κ the coefficients a α are in C M Ω if |α| 2m and in W M Ω otherwise;
H3; κ the coefficients b β are of class C M −μ ∂Ω if |β| μ and in W M −μ ∂Ω
otherwise
Then, for k ∈ {0, , κ}, the operator P maps continuously W M Ω into
W M Ω and B maps continuously W M Ω into W M −μ ∂Ω for every p ∈
1, ∞
Tp,k: P, B : W M Ω −→ W M Ω ×m
1
W M −μ ∂Ω 1.3
is a well-defined bounded linear operator Geymonat’s main result 3, Teorema 3.4 and Teorema 3.5 reads as follows
Theorem 1.1 Suppose that (H1; κ), (H2; κ), and (H3; κ) hold for some κ ≥ 0 Then,
i if p ∈ 1, ∞ and k ∈ {0, , κ}, the operator T p,k is Fredholm if and only if P is uniformly
ii if also κ ≥ 1 and T p,k is Fredholm for some p ∈ 1, ∞ and some k ∈ {0, , κ} (and hence
and k.
The assumptions made inTheorem 1.1are nearly optimal In fact, most subsequent expositions retain more smoothness of the boundary and leading coefficients to make the parametrix calculation a little less technical
The best known version of the Lopatinskii-Schapiro LS condition is probably the combination of proper ellipticity and of the so-called “complementing condition.” Since we will not use it explicitly, we simply refer to the standard literaturee.g., 3 5 for details
We will fill the obvious “gap” betweeni and ii ofTheorem 1.1by proving what follows
Theorem 1.2. Theorem 1.1 (ii) remains true if κ 0.
Note that k 0 corresponds to the most general hypotheses about the boundary and the coefficients, which is often important in practice
From now on, we setTp : Tp,0 for simplicity of notation The reason why κ ≥ 1 is required in partii ofTheorem 1.1is that the proof uses parti with κ replaced by κ − 1 By
a different argument, a weaker form ofTheorem 1.2was proved in3, Proposizione 4.2
p-independence for p in some bounded open interval around the value p 2, under additional
technical conditions
straightforward by-product of the Sobolev embedding theorems and, in fact, indexTp 0 in
Trang 3this case However, this invertibility can only be obtained under more restrictive ellipticity hypothesessuch as strong ellipticity and/or less general boundary conditions Agmon 6, Browder7, Denk et al 8, Theorem 8.2, page 102.
The idea of the proof ofTheorem 1.2is to derive the case κ 0 from the case κ ≥ 1 by
regularization of the coefficients and stability of the Fredholm index The major obstacle in
doing so is the mere C M regularity of ∂Ω, sinceTheorem 1.1with κ ≥ 1 can only be used if ∂Ω
is C M or better This will be overcome in a somewhat nonstandard way in these matters, by artificially increasing the smoothness of the boundary with the help of the following lemma
Lemma 1.3 Suppose that Ω is a bounded open subset of R N and that Ω is a ∂-submanifold of R N of
The next section is devoted to thesimple proof ofTheorem 1.2based onLemma 1.3
and to a useful equivalent formulation Corollary 2.1 Surprisingly, we have been unable
to find any direct or indirect reference toLemma 1.3in the classical differential topology or
PDE literature It does not follow from the related and well-known fact that every ∂-manifold
does not ensure that both can always be embedded in the same euclidian space It is also clearly different from the results just stating that Ω can be approximated by open subsets with a smooth boundary as in 9, which in fact need not even be homeomorphic to Ω.
Accordingly, a proof ofLemma 1.3is given inSection 3
Based on the method of proof and the validity ofTheorem 1.1for systems after suitable modifications of the definition ofTp,kin1.3 and of the hypotheses H1; κ, H2; κ, and H3;
κ, there is no difficulty in checking thatTheorem 1.2remains valid for most systems as well, but a brief discussion is given inSection 4to make this task easier
may be replaced by a collection of such systems, one for each connected component of ∂Ω.
Theorems1.1and1.2remain of course true in that setting, with the obvious modification of the target space in1.3
2 Proof of Theorem 1.2
As noted in3, page 241, the p-independence of ker Tprecall Tp: Tp,0 follows from that
of indexTp , so that it will suffice to focus on the latter
The problem can be reduced to the case when the lower-order coefficients in P and
B vanish since the operator they account for is compact from the source space to the target space in 1.3, irrespective of p ∈ 1, ∞ Thus, the lower-order terms have no impact on
the existence of indexTp or on its value It is actually more convenient to deal with the intermediate case when all the coefficients aα are in C M −2m Ω and all the coefficients b β
are in C M −μ ∂Ω, which is henceforth assumed.
First, M ≥ 2 since M ≥ 2m and m ≥ 1, so that by H1; 0 andLemma 1.3, there are a bounded open subset Ω of RN such that Ω is a ∂-submanifold of R N of class C∞and a C M
diffeomorphism Φ : Ω → Ω mapping ∂ Ω onto ∂Ω
The pull-backΦ∗u : u ◦ Φ is a linear isomorphism of W j,p Ω onto W j,p Ω for every
j ∈ {0, , M} and of W M −μ −1/p,p ∂Ω onto W M −μ −1/p,p ∂ Ω for every 1 ≤ ≤ m Meanwhile,
Trang 4Pu Φ−1∗PΦ ∗u where P is a differential operator of order 2m with coefficients a αof class
C M −2mon Ω and B u Φ−1∗BΦ∗u where B is a differential operator of order μ with coefficients bβ of class C M −μ on ∂ Ω.
From the above remarks, the operatorwhere B : B1, , Bm
Tp: P, B : W M,p
Ω−→ W M −2m,p
Ω×m
1
W M −μ −1/p,p
∂ Ω 2.1
has the form Tp UpTpVpwhereUpandVpare isomorphisms As a result, Tpis Fredholm with the same index asTp Since the coefficients of P and P and of B and B have the same smoothness, respectively, we may, upon replacingΩ by Ω and Tpby Tp , continue the proof
under the assumption that ∂Ω is a C∞submanifold ofRN but the a α are still C M −2mΩ and
the b β still C M −μ ∂Ω.
The coefficients aα can be approximated in C M −2m Ω by coefficients a∞
α ∈ C∞Ω and the coefficients bβ can be approximated in C M −μ ∂Ω by C∞functions b∞β on ∂Ω since ∂Ω
is C∞; see, e.g.,10, Theorem 2.6, page 49, which yields operators P∞andB∞
, 1≤ ≤ m, of order 2m and μ , respectively, in the obvious way.
Let p, q ∈ 1, ∞ be fixed The corresponding operators T∞
p and T∞
q are arbitrarily norm-close toTpandTqif the approximation of the coefficients is close enough If so, by the openness of the set of Fredholm operators and the local constancy of the index, it follows that
T∞
p andT∞
q are Fredholm with indexT∞
p index Tpand indexT∞
q index Tq But since ∂Ω
is now C∞and the coefficients a∞
α and b β∞ are C∞, the hypotheses H1; κ, H2; κ, and H3;
κ are satisfied by Ω, P∞andB∞and any κ ≥ 1 Thus, index T∞
p index T∞
q by partii of
Theorem 1.1, so that indexTp index Tq This completes the proof ofTheorem 1.2
Corollary 2.1 Suppose that (H1; 0), (H2; 0), and (H3; 0) hold, that P is uniformly elliptic in Ω,
W M −2m,qΩ ×m
1W M −μ −1/q,q ∂Ω, then u ∈ W M,q Ω.
andTpis Fredholm byTheorem 1.1i Let Z denote a finite-dimensional complement of
rgeTp in W M −2m,pΩ × m
1W M −μ −1/p,p ∂Ω Since W M −2m,qΩ × m
1W M −μ −1/q,q ∂Ω
is dense in W M −2m,pΩ ×m
1W M −μ −1/p,p ∂Ω and rge T p is closed, we may assume that
Z ⊂ W M −2m,qΩ ×m
1W M −μ −1/q,q ∂Ω If not, approximate a basis of Z by elements of
W M −2m,qΩ ×m
1W M −μ −1/q,q ∂Ω If the approximation is close enough, the approximate basis is linearly independent and its span Z of dimension dim Z intersects rge T ponly at {0} by the closedness of rge Tp Thus, Z may be replaced by Z as a complement of rgeTp SinceTp and Tq have the same index and null-space byTheorem 1.2, their ranges
have the same codimension Now, Z∩ rge Tq {0} because Z is a complement of rge T pand rgeTq⊂ rge Tp This shows that Z is also a complement of rgeTq
Therefore, sincePu, Bu ∈ W M −2m,qΩ ×m
1W M −μ −1/q,q ∂Ω, there is z ∈ Z such
thatPu, Bu − z : w ∈ rge T q⊂ rge Tp This yields z Pu, Bu − w ∈ rge T p , whence z 0 and soPu, Bu w ∈ rge T q This means that Pu, Bu Pv, Bv for some v ∈ W M,qΩ ⊂
W M,p Ω Thus, T p v − u 0, that is, v − u ∈ ker T p Since kerTp ker Tq ⊂ W M,qΩ by
Theorem 1.2, it follows that u ∈ W M,q Ω.
Trang 5It is not hard to check thatCorollary 2.1 is actually equivalent toTheorem 1.2 This was noted by Geymonat, along with the fact thatCorollary 2.1was only known to be true in special cases3, page 242
3 Proof of Lemma 1.3
Under the assumptions ofLemma 1.3,Ω has a finite number of connected components, each
of which satisfies the same assumptions asΩ itself Thus, with no loss of generality, we will assume thatΩ is connected
If X and Y are ∂-manifolds of class C k with k ≥ 1 and X and Y are C1 diffeomorphic,
they are also C k diffeomorphic 10, Theorem 3.5, page 57 Thus, since Ω is of class C M
with M ≥ 2, it suffices to find a bounded open subset Ω of R Nsuch that Ω is C∞and C M−1 diffeomorphic to Ω
In a first step, we find a C M function g : RN → R such that ∂Ω g−10 and ∇g / 0
on ∂Ω while g < 0 in Ω, g > 0 in R N \ ∂Ω and lim |x| → ∞ g x ∞ This can be done in various ways and even when M 1 However, since M ≥ 2, the most convenient argument is to rely
on the fact that the signed distance function
d x :
⎧
⎨
⎩
distx, ∂Ω, if x / ∈ Ω,
−dist x, ∂Ω, if x ∈ Ω 3.1
is C M in U a , where a > 0, and
U a: x∈ RN :|dx| dist x, ∂Ω < a 3.2
is an open neighborhood of ∂Ω in R N This is shown in Gilbarg and Trudinger11, page 355 and also in Krantz and Parks12 Both proofs reveal that ∇dx / 0 when x ∈ ∂Ω, that is, when dx 0 Without further assumptions, the C M regularity of d breaks down when
M 1.
Let χ ∈ C∞R be nondecreasing and such that χs s if |s| ≤ b/2 and χs sign sb
if|s| ≥ b, where 0 < b < a is given Then, g : χ ◦ d is C M in U a , vanishes only on ∂Ω, and
∇g / 0 on ∂Ω Furthermore, since g b on a neighborhood of ∂Ω ∪ U a {x ∈ R N : dx
a } in U a and g −b on a neighborhood of ∂Ω \ U a {x ∈ R N : dx −a} in U a , g remains
C Mafter being extended toRN by setting gx b if x ∈ R N \ Ω ∪ U a , and gx −b if
x ∈ Ω \ U a
This g satisfies all the required conditions except lim |x| → ∞ g x ∞ Since gx b > 0
∂ Ω, it follows from a classical theorem of Whitney 13, Theorem III
that theorem that there is a CM function h onRN , of class C ωinRN \∂Ω such that, if |γ| ≤ M, then ∂ γ h x ∂ γ g x if x ∈ ∂Ω and |∂ γ h x − ∂ γ g x| < |gx|/2 if x ∈ R N \ ∂Ω.
Evidently, h does not vanish on RN \ ∂Ω and h has the same sign as g off ∂Ω, that
is, hx < 0 in Ω and hx > 0 in R N \ Ω Furthermore, ∇hx ∇gx / 0 for every x ∈
∂ Ω, so that ∇hx / 0 for x ∈ U2c for some c > 0 Upon shrinking c, we may assume that
Trang 6Ω \ U2c / ∅ Also, lim|x| → ∞ h x lim|x| → ∞ g x ∞ For convenience, we summarize the relevant properties of h below:
i h is C MonRN and C ωoff ∂Ω,
ii ∇hx / 0 for x ∈ U2c,
iii Ω {x ∈ R N : hx < 0},
iv ∂Ω h−10,
v lim|x| → ∞ h x ∞.
Choose ε > 0 It follows from v that K ε : {x ∈ RN : hx ≤ ε} is compact and,
fromiii and iv, that K ε ⊂ Ω ∪ U c if ε is small enough argue by contradiction Since
h−1ε ∩ Ω ∅ by iii and iv and since h−1ε ⊂ K ε , this implies h−1ε ⊂ U c \ ∂Ω Thus, by
i and ii, h−1ε is a C ωsubmanifold ofRN and the boundary of the open setΩε : {x ∈
RN : hx < ε} ⊃ Ω In fact, Ω ε K ε is a ∂-manifold of class C ωsince, once again byii, Ωε
lies on one side of its boundary
We now proceed to show thatΩε is C M−1diffeomorphic to Ω This will be done by a variant of the procedure used to prove that nearby noncritical level sets on compact manifolds are diffeomorphic However, since we are dealing with sublevel sets and since critical points will abound, the details are significantly different
Let θ ∈ C∞
0 U2c be such that θ ≥ 0 and θ 1 on U c Since ∇h / 0 on U2cbyii, the
function θ∇h/|∇h|2 extended by 0 outside Supp θ is a bounded C M−1vector field onRN
Since M − 1 ≥ 1, the function ϕ : R × R N → RNdefined by
∂ϕ
∂t t, x −θ ϕ t, x ∇h ϕ t, x
∇hϕt,x2,
ϕ 0, x x,
3.3
is well defined and of class C M−1 and ϕt, · is an orientation-preserving C M−1 diffeomor-phism ofRN for every t ∈ R We claim that ϕε, · produces the desired diffeomorphism from
ΩεtoΩ.
It follows at once from3.3 that d/dth ◦ ϕ −θ ◦ ϕ ≤ 0, so that h is decreasing along the flow lines and hence that ϕt, · maps Ω ε into itself for every t ≥ 0 Also, if x ∈ Ω, then hϕt, x ≤ hx < 0 for every t ≥ 0, so that ϕt, x ∈ Ω by iii If now x ∈ ∂Ω ⊂ U c ,
then hx 0 and hϕt, x is strictly decreasing for t > 0 small enough It follows that
h ϕt, x < 0, that is, ϕt, x ∈ Ω for t > 0 Altogether, this yields ϕε, Ω ⊂ Ω.
Suppose now that x ∈ Ωε \ Ω K ε \ Ω Then, x ∈ U c and hence θx 1 For
obvious that θϕt, x 1 until t is large enough that ϕt, x /∈ U c But since ϕ t, x ∈ Ω ε
and h ◦ ϕ·, x is decreasing along the flow lines, ϕt, x /∈ U c implies ϕt, x ∈ Ω Since x /∈ Ω, this means that ϕτx, x ∈ ∂Ω for some τx ∈ 0, t Call τ∗x > 0 the first and, in fact,
only, but this is unimportant time when ϕτ∗x, x ∈ ∂Ω From the above, ϕt, x ∈ U c for
t ∈ 0, τ∗x and hence for t ∈ 0, τ∗x since ∂Ω ⊂ U c Then, θ ϕt, x 1 for t ∈ 0, τ∗x,
so that hϕt, x hx − t for t ∈ 0, τ∗x In particular, since ϕτ∗x, x ∈ ∂Ω and hence
h ϕτ∗x, x 0, it follows that hx − τ∗x 0 In other words, τ∗x hx ≤ ε Thus,
Trang 7h ϕε, x ≤ hϕτ∗x, x 0, that is, ϕε, x ∈ Ω If x ∈ ∂Ω ε so that hx ε and hence
τ∗x ε, this yields ϕε, x ∈ ∂Ω On the other hand, if x ∈ Ω ε \ Ω, then τ∗x hx < ε Since ϕτ∗x, x ∈ ∂Ω ⊂ U c , hϕt, x is strictly decreasing for t near τ∗x and so hϕε, x <
h ϕτ∗x, x 0, whence ϕε, x ∈ Ω.
The above shows that ϕε, · maps Ω ε into Ω, ∂Ω ε into ∂Ω, and Ω ε into Ω That
it actually maps Ωε ontoΩ follows from a Brouwer’s degree argument: Ω is connected and no point of Ω is in ϕε, ∂Ω ε since, as just noted, ϕε, ∂Ω ε ⊂ ∂Ω Thus, for y ∈
Ω, degϕε, ·, Ω ε , y is defined and independent of y Now, choose y0 ∈ Ω \ U2c / ∅, so that
θ y0 0 Then, ϕt, y0 y0 for every t ≥ 0 and so ϕε, y0 y0 Since ϕ ε, · is one to one and
orientation-preserving, it follows that degϕε, ·, Ωε , y0 1 and so degϕε, ·, Ωε , y 1 for
every y ∈ Ω Thus, there is x ∈ Ω ε such that ϕε, x y, which proves the claimed surjectivity.
At this stage, we have shown that ϕε, · is a C M−1diffeomorphism of RN mapping
Ωε intoΩ, ∂Ω ε into ∂Ω, and Ω εinto and onto Ω It is straightforward to check that such a
diffeomorphism also maps ∂Ωε onto ∂Ω approximate x ∈ ∂Ω by a sequence from Ω and
hence it is a boundary-preserving diffeomorphism of ΩεontoΩ This completes the proof of
Lemma 1.3
but this does not mean that the same thing is true of the C Mdiffeomorphism ofLemma 1.3
4 Systems
Suppose now that P : Pij , 1 ≤ i, j ≤ n, is a system of n2 differential operators on Ω, which is properly elliptic in the sense of Douglis and Nirenberg14 We henceforth assume some familiarity with the nomenclature and basic assumptions of4,14 Recall that Douglis-Nirenberg ellipticity is equivalent to a more readily usable condition due to Voleviˇc15 See
5 for a statement and simple proof
so that orderPij ≤ s i j , that have been normalized so that max {s1 , , s n} 0 and min{t1, , t n } ≥ 0.
It is well known that since N ≥ 2, proper ellipticity implies Σ n
i 1s i i 2m with
m ≥ 0 We assume that a system B : B j , 1 ≤ ≤ m, 1 ≤ j ≤ n of boundary differential
operators is given, with orderBj ≤ r jfor some{r1 , , r m } ⊂ Z.
Let
R : max{0, r1 m M : 1, , t n }, 4.1
and call a ijα and b jβthecomplex valued coefficients of PijandBj , respectively Given an
integer κ ≥ 0, introduce the following hypotheses generalizing those for a single equation in
the Introduction
H1; κ Ω is a C M ∂-submanifold ofRN
H2; κ The coefficients a ijα are in C R −s i Ω if |α| s i j and in W R −s i Ω otherwise
H3; κ The coefficients b jβ are in C R −r ∂Ω if |β| r j and in W R −r ∂Ω otherwise.
Trang 8For p ∈ 1, ∞ and k ∈ {0, , κ}, define
Tp,k: P, B :n
j 1
W R j Ω −→n
i 1
W R −s i Ω ×m
1
W R −r ∂Ω. 4.2
Thenas proved in 3,Theorem 1.1holdsonce again, the LS condition amounts to proper ellipticity plus complementing condition and proper ellipticity is equivalent to ellipticity if
to this case if M ≥ 2 If so,Corollary 2.1is also valid, with a similar proof and an obvious modification of the function spaces
is vacuous and the system Pu f can be solved explicitly for u in terms of f and its derivatives This is explained in14, page 506 If so, the smoothness of ∂Ω i.e., H1; κ is irrelevant, andTheorem 1.2is trivially true regardless of MTpis an isomorphism A special
case when m 0 arises if t1 · · · t n 0 in particular, if M 0, for then s1 · · · s n 0
from the conditions 2m Σn
i 1s i i ≥ 0 and s i ≤ 0.
From the above,Theorem 1.2may only fail if m ≥ 1, R 0, and M 1 The author
was recently informed by H Koch16 that he could proveLemma 1.3when M 1, so that
Theorem 1.2remains true in this case as well.
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... Douglis, and L Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II,” Communications on Pure and AppliedMathematics,... class="text_page_counter">Trang 9
14 A Douglis and L Nirenberg, “Interior estimates for elliptic systems of partial differential equations,”
Communications on. .. a classical theorem of Whitney 13, Theorem III
that theorem that there is a CM function h on< /i>RN , of class C ωinRN