A Complement to the Fredholm Theory of Elliptic Systems on Bounded Domains

Boundary Value Problems

Volume 2009, Article ID 637243,9pages 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,[email protected]

Received 24 March 2009; Revised 4 June 2009; Accepted 11 June 2009

Recommended by Peter Bates

We fill a gap in the Lp _{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 ofp. Here,p-independence is proved in full generality.

Copyrightq2009 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 also2for a more complete exposition.While this was the result of the work and ideas of many, the most extensive treatment in theLp_{framework 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 of3in the case of a single scalar equation. Let

Ωbe a bounded connected open subset ofRN,N≥2, and letPdenote a differential operator onΩof order 2m,m≥1,with complex coefficients,

P

|α|≤2m

aαx∂α. 1.1

Next, letB B1, . . . ,Bmbe a system of boundary differential operators on∂ΩwithB of

orderμ≥0 also with complex coefficients,

B

|β|≤μ

WithM : max{2m, μ11, . . . , μm1}andκ ≥ 0 denoting a chosen integer, introduce the following regularity hypotheses:

H1;κ Ωis aCMκ∂-submanifold ofRNi.e.,∂Ωis aCMκsubmanifold ofRNandΩlies on one side of∂Ω;

H2;κthe coefficientsaαare inCM−2mκΩif|α| 2mand inWM−2mκ,∞Ωotherwise;

H3;κthe coefficients bβ are of class CM−μκ_{∂Ω if |β| μ and in W}M−μκ,∞_∂Ω

otherwise.

Then, for k ∈ {0, . . . , κ}, the operator P maps continuously WMk,p_{Ω into} WM−2mk,p_ΩandB

maps continuouslyWMk,pΩintoWM−μk−1/p,p∂Ωfor everyp ∈

1,∞

Tp,k: P,B:WMk,pΩ−→WM−2mk,pΩ× m

1

WM−μk−1/p,p∂Ω 1.3

is a well-defined bounded linear operator. Geymonat’s main result 3, Teorema 3.4 and Teorema 3.5reads as follows.

Theorem 1.1. Suppose that (H1;κ), (H2;κ), and (H3;κ) hold for someκ≥0.Then,

iifp∈1,∞andk∈ {0, . . . , κ},the operatorTp,kis Fredholm if and only ifPis uniformly

elliptic inΩandP,Bsatisfies the Lopatinskii-Schapiro condition (see below);

iiif alsoκ≥1andTp,kis Fredholm for somep∈1,∞and somek∈ {0, . . . , κ}(and hence

for every suchpandk by (i)), both the index and null-space ofTp,k are independent ofp

andk.

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.,3–5for details.

We will fill the obvious “gap” betweeniand iiofTheorem 1.1by proving what follows.

Theorem 1.2. Theorem 1.1(ii) remains true ifκ 0.

Note thatk 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 whyκ ≥ 1 is

required in partiiofTheorem 1.1is that the proof uses partiwithκreplaced byκ−1.By a different argument, a weaker form ofTheorem 1.2was proved in3, Proposizione 4.2 p -independence forpin some bounded open interval around the valuep 2,under additional technical conditions.

IfTp λ,0is invertible for someλ ∈Cand everyp∈ 1,∞,thenTheorem 1.2is a

this case. However, this invertibility can only be obtained under more restrictive ellipticity hypothesessuch as strong ellipticityand/or less general boundary conditionsAgmon6, Browder7, Denk et al.8, Theorem 8.2, page 102.

The idea of the proof ofTheorem 1.2is to derive the caseκ 0 from the caseκ≥1 by regularization of the coefficients and stability of the Fredholm index. The major obstacle in doing so is the mereCM_{regularity of∂Ω,sinceTheorem 1.1withκ≥1 can only be used if∂Ω}

isCM1_{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 thatΩis a bounded open subset ofRN_{and thatΩis a∂-submanifold ofR}N_of

classCM_{withM≥2.Then, there is a bounded open subsetΩ ofR}N_{such thatΩ is a∂-submanifold}

ofRN_{of classC}∞_(evenCω_{) and thatΩandΩ areC}M_{diffeomorphic (as∂-manifolds).}

The next section is devoted to thesimpleproof 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

X of classCM _{with M ≥ 1 isC}M _{diffeomorphic to a∂-manifold Y of class C}∞ _{since this} 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 Ωcan 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,kin1.3and of the hypothesesH1;κ,H2;κ, andH3; 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.

Remark 1.4. When the boundary∂Ωis not connected, the systemBof boundary conditions 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, thep-independence of kerTprecallTp: Tp,0follows 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 inPand

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 inCM−2mΩ and all the coefficients bβ

are inCM−μ_∂Ω_{,which is henceforth assumed.}

First,M≥2 sinceM≥2mandm≥1,so that byH1; 0andLemma 1.3, there are a bounded open subsetΩ ofRN _{such thatΩ is a∂-submanifold ofR}N _{of classC}∞_{and aC}M

diffeomorphismΦ:Ω → Ωmapping∂Ω onto∂Ω.

Pu Φ−1∗_PΦ ∗_{uwherePis a differential operator of order 2mwith coefficientsaα of class} CM−2m _{onΩ andB}

u Φ−1∗BΦ∗uwhereB is a differential operator of orderμ with

coefficientsbβof classCM−μon∂Ω.

From the above remarks, the operatorwhereB: B1, . . . ,Bm

Tp:

P,B:WM,pΩ−→WM−2m,pΩ× m

1

WM−μ−1/p,p

∂Ω 2.1

has the formTp UpTpVpwhereUpandVpare isomorphisms. As a result,Tpis Fredholm

with the same index asTp.Since the coefficients ofPandPand ofBandBhave the same

smoothness, respectively, we may, upon replacingΩbyΩ andTpbyTp,continue the proof

under the assumption that∂Ωis aC∞submanifold ofRN_{but thea}

αare stillCM−2mΩand

thebβstillCM−μ_∂Ω_.

The coefficientsaαcan be approximated inCM−2mΩby coefficientsa∞α ∈C∞Ωand

the coefficientsbβ can be approximated inCM−μ_∂Ω_byC∞_functionsb∞

βon∂Ω since∂Ω

isC∞; see, e.g.,10, Theorem 2.6, page 49, which yields operatorsP∞andB∞ , 1≤≤m,of order 2mandμ,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 indexTpand indexT∞q indexTq.But since∂Ω

is nowC∞and the coefficientsa∞_α andb_β∞ areC∞,the hypothesesH1;κ,H2;κ, andH3;

κare satisfied byΩ,P∞andB∞and anyκ≥ 1.Thus, indexT∞_p indexT∞_q by partiiof

Theorem 1.1, so that indexTp indexTq.This completes the proof ofTheorem 1.2.

Corollary 2.1. Suppose that (H1;0), (H2;0), and (H3;0) hold, thatP is uniformly elliptic inΩ, and that P,B satisfies the LS condition. Let p, q ∈ 1,∞. If u ∈ WM,p_{Ω and Pu,Bu ∈} WM−2m,q_Ω×m

1WM−μ−1/q,q∂Ω, thenu∈WM,qΩ.

Proof. Since the result is trivial if p ≥ q,we assumep < q. Obviously, Pu,Bu ∈ rgeTp

andTpis Fredholm byTheorem 1.1i. LetZ denote afinite-dimensionalcomplement of

rgeTp in WM−2m,pΩ ×m 1WM−μ−1/p,p∂Ω. Since WM−2m,qΩ ×

m

1WM−μ−1/q,q∂Ω

is dense in WM−2m,p_Ω×m

1WM−μ−1/p,p∂Ωand rgeTp is closed, we may assume that Z ⊂ WM−2m,q_Ω×m

1WM−μ−1/q,q∂Ω.If not, approximate a basis ofZ by elements of WM−2m,q_Ω×m

1WM−μ−1/q,q∂Ω.If the approximation is close enough, the approximate

basis is linearly independent and its spanZof dimension dimZintersects rgeTponly at

{0}by the closedness of rgeTp. Thus,Zmay be replaced byZas 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∩rgeTq {0}becauseZis a complement of rgeTpand

rgeTq⊂rgeTp.This shows thatZis also a complement of rgeTq.

Therefore, sincePu,Bu∈ WM−2m,q_Ω×m

1WM−μ−1/q,q∂Ω,there isz ∈Zsuch

thatPu,Bu−z: w∈rgeTq⊂rgeTp.This yieldsz Pu, Bu−w∈rgeTp,whencez 0

and soPu,Bu w∈rgeTq.This means thatPu,Bu Pv,Bvfor somev∈WM,qΩ⊂ WM,p_Ω.Thus,T

pv−u 0,that is,v−u ∈kerTp.Since kerTp kerTq ⊂ WM,qΩby

It 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,Ωhas a finite number of connected components, each of which satisfies the same assumptions asΩitself. Thus, with no loss of generality, we will assume thatΩis connected.

IfXandY are∂-manifolds of classCk_{withk≥1 andX andY areC}1 _{diffeomorphic,}

they are also Ck _{diffeomorphic 10, Theorem 3.5, page 57. Thus, sinceΩ is of classC}M

withM≥2,it suffices to find a bounded open subsetΩ ofRN_{such thatΩ isC}∞_andCM−1

diffeomorphic toΩ.

In a first step, we find aCM_{functiong :R}N _{→ Rsuch that∂Ω g}−1_{0and∇g /0}

on∂Ωwhileg <0 inΩ,g >0 inRN_{\∂Ωand lim}

|x| → ∞gx ∞.This can be done in various ways and even whenM 1.However, sinceM≥2,the most convenient argument is to rely on the fact that the signed distance function

dx:

⎧ ⎨ ⎩

distx, ∂Ω, if x /∈Ω,

−distx, ∂Ω, if x∈Ω 3.1

isCM_inU

a, wherea >0, and

Ua:

x∈RN :|dx| distx, ∂Ω< a 3.2

is an open neighborhood of∂ΩinRN.This is shown in Gilbarg and Trudinger11, page 355 and also in Krantz and Parks12. Both proofs reveal that ∇dx/0 whenx ∈ ∂Ω,that is, whendx 0.Without further assumptions, theCM _{regularity ofdbreaks down when} M 1.

Letχ∈C∞Rbe nondecreasing and such thatχs sif|s| ≤b/2 andχs signsb

if|s| ≥b,where 0 < b < ais given. Then,g : χ◦disCM_{inUa,vanishes only on∂Ω, and}

∇g /0 on∂Ω.Furthermore, sinceg bon a neighborhood of∂Ω ∪ Ua {x∈RN_:dx a}inUaandg −bon a neighborhood of∂Ω\Ua {x∈RN:dx −a}inUa,gremains CM_{after being extended toR}N _{by settinggx bifx ∈ R}N_{\Ω∪Ua,andgx −bif} x∈Ω\Ua.

Thisgsatisfies all the required conditions except lim|x| → ∞gx ∞.Sincegx b >0

for|x|large enough, this can be achieved by replacinggxby1|x|2gx.Sinceg /0 off

∂Ω,it follows from a classical theorem of Whitney13, Theorem III withx: |gx|/2 in that theoremthat there is aCM_functionhonRN_{,of classC}ω_inRN_{\∂Ωsuch that, if|γ| ≤M,}

then∂γ_{hx ∂}γ_{gxifx∈∂Ωand|∂}γ_hx−∂γ_{gx|<|gx|/2 ifx∈R}N_\∂Ω.

Evidently, hdoes not vanish on RN _{\∂Ωand hhas the same sign asg off∂Ω, that}

Ω\U2c/∅.Also, lim|x| → ∞hx lim|x| → ∞gx ∞.For convenience, we summarize the relevant properties ofhbelow:

ihisCM_onRN_andCω_off∂Ω,

ii∇hx/0 forx∈U2c,

iii Ω {x∈RN_:hx<0},

iv∂Ω h−1_0,

vlim_|x| → ∞hx ∞.

Chooseε > 0.It follows fromv thatKε : {x ∈ RN _{: hx ≤ ε} is compact and,}

fromiii and iv, thatKε ⊂ Ω∪Uc if ε is small enough argue by contradiction. Since h−1_{ε∩Ω ∅byiiiandivand sinceh}−1_ε⊂K

ε,this impliesh−1ε⊂Uc\∂Ω.Thus, by

iandii,h−1_{εis aC}ω_{submanifold ofR}N _{and the boundary of the open setΩ}

ε : {x ∈ RN _{:hx< ε} ⊃Ω.In fact,Ω}

ε Kεis a∂-manifold of classCωsince, once again byii,Ωε

lies on one side of its boundary.

We now proceed to show thatΩε isCM−1diffeomorphic toΩ.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.

Letθ ∈ C₀∞U2cbe such thatθ ≥0 andθ 1 onUc.Since∇h /0 onU2cbyii, the

functionθ∇h/|∇h|2extended by 0 outside Suppθis a boundedCM−1_{vector field onR}N_.

SinceM−1≥1,the functionϕ:R×RN _{→ R}N_{defined by}

∂ϕ

∂tt, x −θ

ϕt, x ∇h

ϕt, x

_{∇hϕt, x}2,

ϕ0, x x,

3.3

is well defined and of class CM−1 _{and ϕt,·is an orientation-preservingC}M−1 _diff

eomor-phism ofRN_{for everyt∈R.We claim thatϕε,·produces the desired diffeomorphism from}

ΩεtoΩ.

It follows at once from3.3thatd/dth◦ϕ −θ◦ϕ ≤ 0,so thathis decreasing along the flow lines and hence thatϕt,·mapsΩεinto itself for everyt≥ 0.Also, ifx∈Ω,

thenhϕt, x ≤ hx < 0 for everyt ≥ 0,so thatϕt, x ∈Ωbyiii. If nowx ∈ ∂Ω ⊂ Uc,

then hx 0 and hϕt, xis strictly decreasing for t > 0 small enough. It follows that

hϕt, x<0,that is,ϕt, x∈Ωfort >0.Altogether, this yieldsϕε,Ω⊂Ω.

Suppose now thatx ∈ Ωε \ Ω Kε \Ω.Then, x ∈ Uc and henceθx 1.For t > 0 small enough,ϕt, x ∈ Uc and soθϕt, x 1 for t > 0 small enough. In fact, it is

obvious that θϕt, x 1 until tis large enough that ϕt, x/∈Uc.But sinceϕt, x ∈ Ωε

andh◦ϕ·, xis decreasing along the flow lines,ϕt, x/∈Ucimpliesϕt, x∈Ω.Sincex /∈Ω,

this means thatϕτx, x ∈∂Ωfor someτx ∈0, t.Callτ∗x > 0 the firstand, in fact, only, but this is unimportanttime whenϕτ_∗x, x ∈∂Ω.From the above,ϕt, x ∈Uc for t∈0, τ∗xand hence fort∈0, τ∗xsince∂Ω⊂Uc.Then,θϕt, x 1 fort∈0, τ∗x,

so thathϕt, x hx−tfort∈0, τ∗x.In particular, sinceϕτ∗x, x∈∂Ωand hence

hϕε, x ≤ hϕτ∗x, x 0,that is,ϕε, x ∈ Ω.Ifx ∈ ∂Ωε so thathx εand hence τ_∗x ε, this yieldsϕε, x∈∂Ω.On the other hand, ifx∈Ωε\Ω,thenτ∗x hx < ε. Sinceϕτ∗x, x∈∂Ω⊂Uc,hϕt, xis strictly decreasing fortnearτ∗xand sohϕε, x< hϕτ∗x, x 0,whenceϕε, x∈Ω.

The above shows that ϕε,· maps Ωε into Ω, ∂Ωε into ∂Ω, and Ωε 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, ∂Ωε since, as just noted, ϕε, ∂Ωε ⊂ ∂Ω. Thus, for y ∈

Ω,degϕε,·,Ωε, yis defined and independent ofy.Now, choosey0 ∈Ω\U2c/∅,so that θy0 0.Then,ϕt, y0 y0for everyt≥0 and soϕε, y0 y0.Sinceϕε,·is one to one and orientation-preserving, it follows that degϕε,·,Ωε, y0 1 and so degϕε,·,Ωε, y 1 for

everyy∈Ω.Thus, there isx∈Ωεsuch thatϕε, x y,which proves the claimed surjectivity.

At this stage, we have shown thatϕε,·is aCM−1 _{diffeomorphism of R}N _mapping

Ωε intoΩ,∂Ωεinto ∂Ω, andΩεinto and onto Ω.It is straightforward to check that such a

diffeomorphism also maps∂Ωεonto∂Ω approximatex ∈ ∂Ωby a sequence fromΩand

hence it is a boundary-preserving diffeomorphism ofΩεontoΩ.This completes the proof of

Lemma 1.3.

Remark 3.1. TheCM−1_{diffeomorphismϕε,·above is induced by a diffeomorphism ofR}N_,

but this does not mean that the same thing is true of theCM_{diffeomorphism 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

5for a statement and simple proof.

Let{s1, . . . , sn} ⊂ Z and{t1, . . . , tn} ⊂ Zbe two sets of Douglis-Nirenberg numbers,

so that orderPij ≤ si tj, that have been normalized so that max{s1, . . . , sn} 0 and

min{t1, . . . , tn} ≥0.

It is well known that sinceN ≥ 2,proper ellipticity implies Σn

i 1si ti 2mwith m ≥ 0.We assume that a systemB : Bj, 1 ≤ ≤ m, 1 ≤ j ≤ nof boundary differential

operators is given, with orderBj ≤rtjfor some{r1, . . . , rm} ⊂Z.

Let

R: max{0, r11, . . . , rm1}, M: Rmax{t1, . . . , tn}, 4.1

and callaijαandbjβthecomplex valuedcoefficients ofPijandBj,respectively. Given an

integerκ≥0,introduce the following hypothesesgeneralizing those for a single equation in the Introduction.

H1;κ Ωis aCMκ_{∂-submanifold ofR}N_.

H2;κThe coefficientsaijαare inCR−siκΩif|α| sitjand inWR−siκ,∞Ωotherwise.

Forp∈1,∞andk∈ {0, . . . , κ},define

Tp,k: P,B: n

j 1

WRtjk,pΩ−→ n

i 1

WR−sik,pΩ× m

1

WR−rk−1/p,p∂Ω. 4.2

Thenas proved in3,Theorem 1.1holdsonce again, the LS condition amounts to proper ellipticity plus complementing condition and proper ellipticity is equivalent to ellipticity if

m >0 andN≥3and it is straightforward to check that the proof ofTheorem 1.2carries over to this case ifM ≥ 2.If so,Corollary 2.1is also valid, with a similar proof and an obvious modification of the function spaces.

Remark 4.1. If m 0, there is no boundary condition in particular,R 0, and H3; κ

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;κis irrelevant, andTheorem 1.2is trivially true regardless ofMTpis an isomorphism. A special

case whenm 0 arises ift1 · · · tn 0in particular, ifM 0, for thens1 · · · sn 0

from the conditions 2m Σn

i 1siti≥0 andsi≤0.

From the above,Theorem 1.2may only fail ifm ≥1,R 0, andM 1.The author was recently informed by H. Koch16that he could proveLemma 1.3whenM 1,so that

Theorem 1.2remains true in this case as well.

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