Eigenvalues of Elliptic Systems for the Mixed Problem in Perturbations of Lipschitz Domains with Nonhomogeneous Neumann Boundary Conditions

MIYAZAKI Kohei and TAYLOR Justin L.

Department of Mathematics&Statistics,Murray State University,Murray,Kentucky,U.S.A.

Abstract.We study eigenvalues of an elliptic operator with mixed boundary conditions on very general decompositions of the boundary.We impose nonhomogeneous conditions on the part of the boundary where the Neumann term lies in a certain Sobolev or Lp space. Our work compares the behavior of and gives a relationship between the eigenvalues and eigenfunctions on the unperturbed and perturbed domains,respectively.

Key Words:Eigenvalues;elliptic systems;mixed problem;perturbed domains.

1 Introduction

The literature contains much analysis on the study of eigenvalues for elliptic equations,but work on systems of equations is much more scarce.Many of the estimates used for equations do not hold for systems and thus analysis for systems requires something different.Moreover,we use the reverse H?lder technique frequently to achieve our estimates.In this paper,we look at the behavior of eigenvalues and eigenfunctions on perturbed domains and compare them to ones on the unperturbed domain.We give a simple characterization of families of perturbed domains,which include dumbbell shaped domains,but these families may be quite general.We work in Lipschitz domains and assume that the coefficients of the operator are bounded and symmetric. Furthermore,we assume that the Dirichlet setDsatisfies a corkscrew condition,which allows for a rather general decomposition of the boundary.A term which lies in a Sobolev space orLpspace is imposed on the Neumann setN.

There are many results on the study of eigenvalues for equations on perturbed domains when we have Dirichlet boundary conditions. A classic paper by Babuska and Vyborny[1]shows continuity of Dirichlet eigenvalues for elliptic equations under a regular variation of the domain.Work by Davies[2]and Pang[3]studies the relationship between Dirichlet eigenvalues and corresponding eigenfunctions in a domain Ω and eigenvalues and eigenfunctions in sets of the formR(ε)={x ∈Ω:dist(x,?Ω)≥ε}. Two papers by Chavel and Feldman[4]and Anné and Colbois[5]examine eigenvalues on compact manifolds with a small handle and Dirichlet conditions on the ends of the handle.More recent work for Dirichlet conditions includes work by Daners[6],which shows convergence of solutions to elliptic equations on sequences of domains.Burenkov and Lamberti[7]prove spectral estimates for higher-order elliptic operators on domains in certain H?lder classes.Kozlov[8]gives asymptotics of Dirichlet eigenvalues for domains in Rnand Grieser and Jerison[9]also give asymptotics for Dirichlet eigenvalues and eigenfunctions on plane domains.

When a Neumann condition is placed on part of the boundary,the eigenvalue problem is much more difficult to analyze. A classic example by Courant and Hilbert[10]shows that continuity of eigenvalues is not generally obtained for Neumann eigenvalues if the domain is onlyC0.In fact,Arrieta,Hale,and Han[11]show that if the domain is not sufficiently smooth,none of the Neumann eigenvaluesconverge form ≥3.However,if one places more regularity on the domains,rates of convergence are achievable.This is illustrated in work by Jimbo[12],Jimbo and Kosugi[13],and Brown,Hislop,and Martinez[14].

As mentioned earlier,there seems to be a lot less work on the study of perturbed domains with systems of equations.Fang[15]studied the behavior of the second eigenvalue in a perturbed domain for a system of equations in R2.Taylor[16]provided rates of convergence for Dirichlet eigenvalues involving elliptic systems on domains with low regularity.More recent work by Collins and Taylor[17]showed convergence of eigenvalues for the mixed problem with homogeneous Dirichlet and Neumann boundary conditions.The contribution in this paper continues the study of eigenvalues for these types of operators for the mixed problem when the Neumann term is nontrivial.

2 Preliminaries and main results

We will be working on a bounded Lipschitz domain Ω.This means that locally on the boundary,Ω is a domain which lies above the graph of a Lipschitz function.In order to give a formal definition,we introduce coordinate cylinders.Given a constantM ＞0,x ∈?Ω,andr ＞0,we define a coordinate cylinderZr(x)={y:|y′-x′|＜r,|yn-xn|＜(2+4M)r}.Here,we use coordinates(x′,xn)∈Rn-1×R and assume that this coordinate system is a translation and rotation of the standard coordinates.Ω is said to be a Lipschitz domain if for eachx∈?Ω,there exists a coordinate cylinderZr(x)and a Lipschitz functionφ:Rn-1→R with Lipschitz constantMso that

We letandbe two nonempty,open,disjoint,connected,and bounded Lipschitz domains in Rnand form the domainDecompose the boundary?Ω0=D∪NandD∩N=?whereDis nonempty and relatively open with respect to?Ω0.We assume the corkscrew condition onD.To define this condition,let Λ denote the boundary ofDin?Ω0.

Definition 2.1.We say that D satisfies the corkscrew condition if for all x∈Λand r∈(0,100r0),there exists xr ∈D so that|xr-x|＜r and dist(xr,Λ)＞M-1r.

The corkscrew condition allows for a quite general decomposition of the boundary.It follows that ifDsatisfies the corkscrew condition,then we have the following lemma shown in Taylor et al.[18]:

Lemma 2.1.If x ∈D and r ∈(0,100r0),then there exists xr ∈D and a constant c=c(M),only depending on M,so that|x-xr|＜r andΔcr(xr)?D.Furthermore,

In the above lemma,σdenotes the surface measure.We again note that the local domains mentioned earlier are star-shaped domains.Thus with the aid of Lemma 2.1,we may apply Sobolev and Poincaré estimates later.

We define the family of perturbations for a suitableε0,{Tε}0＜ε＜ε0to be a family of open sets with Lipschitz boundary such that

and if|Tε|denotes the RnLebesgue measure ofTε,then

whereCand 0＜d ≤nare independent ofε.This allows the perturbations to be quite general.We fix two pointsand define?Ω0. We require the connections fromTεtoandto be contained inandrespectively. That is,andWe also require that for eachε ＜ε0,we haveIn other words,the connection is always touchingDin some sense,but may also touchN. We define a similar condition forandThen for anyε,define Ωεto be the interior of the setSo,you may think ofTεas a“tube”connecting each of the two domains.We now have the family of domainsLetandWe letbe the vector-valued Sobolev space taking values in Rmand defineto be the closure ofinin a neighborhood ofD~}.We also define

the gradient?ushould be interpreted as a matrixandis the Frobenius norm of?u.

SinceDsatisfies the corkscrew condition for anyε,and Ωεis a bounded Lipschitz Domain,Lemma 2.1 implies the Poincaré inequality for any 1≤p＜∞,

Consequently,we have the trace inequality as given in Cianchi et al.[19],

Define

We also have Sobolev-Poincaré inequalities for our local domains,taken from Ott and Brown[20].

Lemma 2.2.Suppose1≤p＜n andThen we have

Lemma 2.3.Let1≤p＜n and choose q so thatThen we have

The previous lemmas hold withSr(x)as the domain of integration on the right side ifdist(Sr(x),D)＞0.Otherwise,we must expandSr(x)in order to use the corkscrew condition.The inequalities(2.5)and(2.6)imply the Poincaré inequalities for any 1≤p＜∞,

Now we define the eigenvalue problem.Letwhere theAijare coefficientm×mmatrices which have measurable and bounded entries,which also satisfy the symmetry conditionfori,j=1,...,nandα,β=1,...,m.We consider the mixed eigenvalue problem

Above in(2.9)and throughout the rest of this paper we use the convention of summing over repeated indices,whereiandjwill sum from 1 tonandα,β,andγwill sum from 1 tom.Lettingdefine the bilinear form on

We say that the numberλis aneigenvalueofLwithand

wheredenotes the pairing of duality onWe define theRayleigh quotient Rεon(W1,2(Ωε))mas

for.It easily follows that ifλis an eigenvalue with eigenfunctionu,then

This inequality will be true ifLsatisfies certain ellipticity conditions and if we assume certain smoothness conditions on the coefficients,such as uniform boundedness.In particular,we say thatLsatisfies astrong Legendre conditionor astrong ellipticity conditionif there existsθ ＞0 so that We also say thatLsatisfies theLegendre-Hadamard conditionif there existsθ ＞0 so that

IfLsatisfies(2.14),then it is clear thats inequality holds for any

It is well-known that ifLsatisfies theLegendre-Hadamard conditionwith continuous coefficients intheninequality holds for anyPlease see Treves[21]for details.

The Lamé system is defined asLu=-divζ(u),whereζ(u)denotes the stress tensor

Here,the Lamé moduliυ(x)andμ(x)are given by

whereδijis the Kronecker delta. The functionsυ(x)andμ(x)are both assumed to be bounded and measurable and satisfy the conditions

Using Korn’s 1st Inequality as given in[22],it is easy to see that for the Lamé system,inequality is satisfied for all

The aim of this paper is to relate eigenvalues and eigenfunctions on the unperturbed domain to eigenvalues and approximated eigenfunctions on the perturbed domain.In particular,we show that if an eigenfunction is in a certain class,the relationship to its corresponding eigenvalue on Ωεis similar to the relationship between an eigenfunction and corresponding eigenvalue on Ω0.That is,an eigenvalue on Ωεacts as an eigenvalue on Ω0and the approximated eigenfunction on Ωεacts as an eigenfunction on Ω0.These relationships are illustrated in the main results of our work,namely Theorems 4.1 and 4.2.

3 A reverse H?lder estimate

A key ingredient in our work will involve the reverse H?lder argument introduced by Gehring[23]and refined by Giaquinta and Modica[24,25].The proof of the following Caccioppoli estimate is similar to the one given in Collins and Taylor[17],but some modifications are needed since we have a boundary term.For anyandr＞0,define a local version of the Hardy-Littlewood maximal function

Theorem 3.1.Let u be an eigenfunction with eigenvalue λ associated to the operator L.If fN is a function in(L2(?Ωε))m,there existssuch that whenx∈Ωε,and Sr=Sr(x),we have

Proof.Since there are 3 ellipticity conditions,there are 3 cases to consider.We note that in Collins and Taylor[17,Theorem 3.1],all cases are proven with the exception of the boundary term.Following along their proof by using the same procedure,for any fixed constantwe may arrive at the inequality

To deal with the boundary term,we choose anyρ＞0 so that

Next,apply the Poincaré inequality(2.8)to get

Now divide both sides of(3.3)by|Sr|and use(3.4)to obtain

If we now choosewe have that

where we have used[26,Lemma A.2.]on the last line.Now if we divide(3.2)by|Sr|,apply the Sobolev-Poincaré inequality(2.5)to the first term on the right side,and use(3.6),we obtain the result(3.1).

The next lemma taken from Ott and Brown[27]gives anLqestimate onPr f.

Lemma 3.1.Let p＞1and choose q so thatΩand any0＜r＜r0,we have

where C is a constant only depending on M and the dimension n.

To prove the reverse H?lder estimate given in Theorem 3.3,we introduce the maximal functions below and a theorem taken from[24].

Definition 3.1.

Theorem 3.2.Let r ＞q ＞1,and QR be a cube inRn with sidelength R centered at 0.Also,define d(x)=dist(x,?QR).If f and g are measurable functions such that f ∈(Lr(QR))m,g ∈(Lq(QR))m,f=g=0outside QR,and with the added condition that

for almost every x in QR where b≥0and0≤a＜1

where ? and C depend on b,q,n,a and r.

We now state and prove the main theorem in this section.

Theorem 3.3.Let u be an eigenfunction with eigenvalue λ associated to the operator L.If fN is a function in(Lp′(?Ωε))m for p′＞2with fN=0onthen there existsso that

n,m,M,and the constants from(2.13)

Proof.Now ifuis an eigenfunction with eigenvalueλ,we haveand thus we may employ the Sobolev inequality to get that|u|∈(Lt(Ωε))mfor somet ＞2.We choose a cubeQR,centered at 0,with sidelengthRsuch thatuniformly inε.Using(3.1)and(3.7)by settingandu=0 outsideQR,we obtain by a standard covering argument that

Here,and?from Theorem 3.2 is independent ofεand any eigenvalue.Now setting,and applying Lemma 3.1 with another covering argument,we have the result.

4 Eigenvalue and eigenfunction analysis

In this section,we will look at estimates on the eigenvalues and corresponding eigenfunctions.Because of the nonhomogeneity of the Neumann boundary condition,there is no reason to expect that we have a countable set of eigenvalues nor do we expect that each eigenfunction is bounded or can even be integrated.However,if we only look at eigenvalues contained in an interval which have corresponding eigenfunctions with boundedL2norms,then we can say a lot about the relationship between the eigenvalues and eigenfunctions on different perturbations.For the remainder of the paper we assumefNis a function in(Lp′(?Ωε))mforp′＞2 withfN=0 onWe define the following space:

Definition 4.1.Let a,b∈Rsuch that a＜b.Given B＞0

i.There exists λ∈[a,b]such that Lφ=λφ inΩε in the weak sense for some ε≤ε0.

From this point,we denote an eigenvalue ofLwith respect to Ωεasλεandas a corresponding eigenfunction.The next lemma gives a uniform bound on thenorm of the gradient of an eigenfunction.The proof uses the reverse H?lder estimate(3.8).

Lemma 4.1.

(3.8)and C depends onn,m,M,max{|a|,|b|},B,and the constants from(2.13).Moreover,C is independent of ε.

Proof.The reverse H?lder estimate(3.8)implies

whereρ ＞0 is any constant.Therefore,applying the Poincaré estimate(2.4)on the last term and choosingρa(bǔ)ppropriately,we obtain the estimate

We next use the interpolation inequality

wheretsatisfies

andq*is the Sobolev conjugate ofq ≤2.It follows from(4.3)and Sobolev’s inequality that

The inequalities(4.2),(4.3),and(4.5)now give the result.

Recall that the connections fromTεto Ωεare contained inandLetandWe define a cutoff functionηε:Ωε →R to be such that 0≤η≤1,ηε=0 inoutsideandThe next lemma states that the eigenfunctionmay be approximated by the functioninL2.

Lemma 4.2.

(3.8)and C depends onn,m,M,max{|a|,|b|},B,and the constants from(2.13).Moreover,C is independent of ε.

Proof.inequality and Poincaré’s inequality(2.3)imply

The estimate(4.6)now follows from(4.1)and(2.2).

We also may approximate the scalar product of 2 eigenfunctions as in the next lemma.

Lemma 4.3.

(3.8)and C depends onn,m,M,max{|a|,|b|},B,and the constants from(2.13).Moreover,C is independent of ε.

Proof.inequality implies

Now Poincaré’s inequality(2.3),(4.1),and(2.2)give the result(4.7).

The next theorem states that eigenvalues,which are Rayleigh quotients of eigenfunctions,may be approximated with Rayleigh quotients of approximated eigenfunctions on Ω0.That is,λεacts as an eigenvalue on Ω0with corresponding eigenfunction

Theorem 4.1.0

Here,C depends onn,m,M,max{|a|,|b|},B,and theconstants from(2.13).Moreover,C is independent of ε.

Proof.We have

Following a similar argument from part of the proof of Lemma 4.2 in[17],we obtain

Next from H?lder’s inequality,we have

Therefore since

We also have that

It follows that(4.9)-(4.13)now give the estimate(4.8).

The next theorem says thatacts as an eigenfunction on Ω0with corresponding eigenvalueλε.

Theorem 4.2.

Proof.We first note that since Ω0is Lipschitz,we may extendwtoEwby even reflection into Ωεsuch thatSuch an extension is discussed in[20,Appendix A].With this in mind,we have

First,we may use(2.7),H?lder’s inequality,and(4.1)to get

Also,sinceφsolvesLφ=λεφ,

The estimate(4.14)now follows from(4.15)-(4.18).

5 Conclusion

This paper adds to study of eigenvalue problems for systems on perturbed domains.Some problems to consider in the future are below:

·Can we study problems with perturbations that have Neumann boundary data if we impose more regularity on the domains?

·If we allow the perturbation to be less general,can we achieve more results?

·Can we allow our domains to satisfy a Robin type boundary condition?

Acknowledgement

The authors thank the referees for their valuable comments and suggestions in the prepartion of this article.