Fanghua LIN

(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

1 Introduction

This write up covers the author’s mini-course consisting of three lectures at the 6th-symposium on Analysis and PDEs at the Purdue University,June1st–4th,2015.A somewhat detailed expositions on the relevant works were given in a special graduate course of the author(which was first given at the Courant Institute in the spring of 2014 and then in the fall 2014 at the NYU/Shanghai).My goal here is to give a rather brief survey of some extremum problems for Laplacian eigenvalues on bounded domains in Euclidean spaces with the zero Dirichlet boundary condition.I also want to explain how the solvability(existence and regularity)of these extremum problems is related to(a stronger version)a generalized version of the Polya conjecture.

It is with the deep respect and admiration that I write this article dedicating to Professor Haim Brezis.

2 Some Classical Results

2.1 Weyl’s asymptotic formula and Polya’s conjecture

Let us start with the following simplest examples of eigenvalues and eigenfunctions.

Example 2.1Given an ODE,?

One has a set of(properly normalized)eigenfunctions

and eigenvalues

If we denote

Here VNis the volume of the unit ballin RN,thenbe written as,for the dimension N=1,that

Example 2.2Let Q be the unit square in R2,and we consider

We again have a set of eigenfunctions:uk(x,y)=sin(mx)sin(ny),and corresponding eigenvaluesHere k=N(λ)is the number of lattice points in{(m,n) ∈One notices that for N=2,andThus one haswhen N=2.

What we have described in the above examples are nothing but special cases of the well known Weyl’s asymptotic formula(see,e.g.,[1,5,16,19,25–26]and references therein).

LetΩbe a bounded domain in RN.The eigenvalue problems

has a sequence of(normalized)eigenfunctions:{uk(x)}that it forms an orthonormal basis of L2(Ω).In particular,

and

The corresponding sequence of eigenvalues{λk}satisfies in addition that 0< λ1< λ2≤λ3≤ ···.Weyl’s asymptotic formula then implies that

Here|Ω|is the volume of Ω and N is the dimension of Ω.Weyl’s formula can be further improved(as conjectured by Weyl himself)(see[1,16,25]and references therein):

Here N(λ)denotes number of eigenvalues≤ λ (the minus sign in the formula is for the Dirichlet eigenvalues,and the plus sign corresponds to the eigenvalues with zero Neumann boundary conditions or we simply call them the Neumann eigenvalues).

Thus for a fixed domainΩof finite perimeter in RN,one has for the Dirichlet eigenvalues that

where D(N,|Ω|,|?Ω|)is a positive constant depending on N,|Ω|and|?Ω|.This leads also naturally to the well-known Polya’s conjecture(for the Dirichlet eigenvalues on a bounded domainΩ?RN).

Polya’s conjecture(see[27])is as follows:

for every bounded domain in RNand every positive integer k.

We note that,using the further improved expansions of Weyl’s formula,if k and|Ω|are fixed,then the optimal shape,that is,the right-hand side reaches the minimum value,of the domain Ω is a ball.Weak versions of Polya’s conjectured lower bound for Dirichlet eigenvalues are due to Berezin[4]and Li and Yau[23],with developments later by Laptev[22]and others using Riesz means and “universal” inequalities(see for example a survey article by Ashbaush[2]).Polya[27]proved the conjecture(??)for planar tiling domains(see also[22]).

On the other hand,if we fix a k(sufficient large)and if the Weyl asymptotic formula(?)is valid(uniformly inΩ),which is obviously an unknown issue,then one would expect,as k becomes larger and larger,that the “optimal” domain Ω of a fixed volume,which realizes the least value for the tight hand side of(?),would converge to a ball which solves the isoperimetric(inequality)problem.We shall come back to this interesting and difficult point later.

2.2 M in-M ax principle and nodal domains

Let us recall the classical min-max principle for Laplacian eigenvalues[14]:

(1)

(2)

Here

is the Rayleigh quotient,and Ej’s are j-dimensional subspaces.

One of the uses of(1)and(2)is to obtain estimates for both upper bounds(via(1))and lower bounds(via(2)).

In particular,λ1=inf{R(u):andfor k=2,3,···.

Next,we have the nodal domains theorem of Courant.

Theorem 2.1Let u be a k-th eigenfunction,i.e.,Δu+λku=0 inΩand u?=0(identically)inΩ withthen the number of nodal domains of u≤k.Here an open connected subset D ofΩis called a nodal domain of u if u never vanishes on D and if u≡0 on?D.

When N=1,then the above nodal domain theorem is an easy consequence of the Sturm-Liouville theory.Moreover,in the latter case,the number of nodal domains(intervals)of a k-th eigenfunction is exactly k.A simple proof of the Courant’s nodal domain theorem can be deduced from the min-max principle.

and hence,via min-max principle,one hasConsequently,inandTherefore v vanishes on a nonempty open subsetThe latter is impossible by the analyticity of v inΩ(see[14])or by the unique continuation theorem.

Corollary 2.1SupposeThen the number of nodal domains of u is exactly 2.

2.3 Partitions of a domain

In general,we let u ∈ E(λk){0}be a k-th eigenfunction of the Laplacian on Ω with Dirichlet boundary condition.Letμ(u)be the number of nodal domains of u.We say there is an m-nodal partition ofΩ if there is an eigenfunction u ∈ E(λk){0}(for some k≥ m)such that m= μ(u)(see also[17]for related discussion).A natural question is how largeμ(u)can be,for a u ∈ E(λk){0}.Courant’s nodal domain theorem says that,for any u ∈ E(λk){0},μ(u)≤ k.It turns out thatμk=max{μ(u):u ∈ E(λk){0}}is definitely smaller than k,for k large.The following elegant theorem was due to Pleijel.

Theorem 2.2(Pleijel)

for any bounded domainΩ in RN.Hereμkis defined above.

Next,we apply the Faber-Krahn inequality(see discussions below)toΩjto obtain(since Ωjis a nodal domain of u ∈ E(λk){0})thatwhere Bjis a ball in RNwithTo proceed,one examines,for example,the case N=2.Then one has

(here j0=2.4 is the first positive zero of the 0-th order Bessel function).

On the other hand,Weyl asymptotic formula implies thatwhen N=2.Henceorwhen N=2.

Finally,the cases N>2 can be handled in the same way,and the conclusion of the theorem follows.

We remark that

(a)Any l∞-minimal partition is a spectral equal partition(this may be viewed as an interesting exercise).

(b)The existence,regularity and regularity of free interfaces of minimal partition have been studied by many authors(see,for example,the survey article[17],and also[11–12]).

In[11],we also conjectured,in the case N=2,that

(i)exists,and it is independent ofΩ.

(ii)(Hexa).

3 Extremum Problems

3.1 Special cases

One of the main purposes of this article is to address the following extremum problem:

for a positive integer k and N ≥ 2.Hereλk(Ω)is the k-th Dirichlet eigenvalue of the Laplacian onΩ?RN.It is already unclear,at the first look,that what type of measurable subsetsΩin RNwith volume 1,so thatλk(Ω)would be well-defined.

When k=1,we have the following well-known Faber-Krahn inequality:

where B is a ball in RNwith|B|=1.Here infimum is taking among any measurable setsΩ in RNwith|Ω|=1,so that λ1(Ω)is defined.For example,Ω being a bounded,open subset of RNwill work.This inequality may be proved by many arguments including the classical symmetrization method(see[3,13,15,18,28]).

The case k=2 is already drastically different from the case k=1.In fact,there is no connected open setΩthat could solve the extremum problem(EP)when k=2.Indeed,let us assume thatsolves(EP)with k=2,and let u2be the corresponding second eigenfunction:

The above argument has showed also that the solution to(EP)when k=2 is given by a disjoint union of two balls of the equal volume

These preliminary observations lead to a couple rather basis questions:

(Q1)What type setsΩwould be admissible for the extremum problem(EP)?

(Q2)How one can handle the usual“concentration compactness”problems when a minimizing sequence{Ωj}of(EP)in RNmay be stretched and splitted to infinitely?

(Q3)Is it possible to find minimizersΩof(EP)such thatΩwould be open and connected?

There were numerous works addressing the first two questions.Much of discussion of these may be find in the excellent monograph by Henrot[18]and references therein.Here we shall be concentrated mainly on the third question and to discuss some recent progress on it.Before we do so,let us discuss a couple more specific cases.

The first special class of subsetsΩ?RNone would consider is the class of convex domains.We have the following.

Theorem 3.1There is a convex domain that solves the(EP):

for every positive integer k.

ProofLet{Ωn}be a minimizing sequence.Since Ωn’s are convex andone has that eitherΩnconverges in the Hausdorff distance to a bounded convex domainΩ?with|Ω?|=1,or there is a subsequence of{Ωn?}such that Ωn?would be contained in strips(after suitable rotations and translations)of formsuch thatwhileIf the latter is the case,then an easy calculation of the first eigenvalues of the regions of the formyields that(no matter what are the sized of Ln’s).In particular,

as n tends to infinite.This would contradict to the fact thatλk(Ωn?)converge to the value

and hence the latter is not possible.For the former case,an easy fact in the convex geometry implies that ifΩn’s converges to Ω?(convex)with|Ω?|=1 in the Hausdorff distance,then Ωn’s are uniformly Lipschitz domains.Moreover,λk(Ωn)converges to λk(Ω?)ascan be easily established.Hence the conclusion of the theorem follows.

The second class of subsetswe would discuss here are bounded sets.Here is one of the basis existence result(see[7–9,18]and references therein).

Theorem 3.2 There is a quasi-open setΩ?? B that solves the following constrained extremum eigenvalue problem:

Here B is a large ball(or any bounded,Lipschitz domain with|B|>1).

The proof of the above theorem is contained in the references[7–9,18],and it takes some pages to describe it.Here we shall discuss the relevant notion of quasi-open sets and some properties of such sets in the next section as these would be important to other parts of discussions in the paper.

3.2 Quasi-op en sets

Let f(x)be a real valued continuous function on RN.Then for any c∈R,O={x∈RN:f(x)>c}is an open set.The converse is also true,that is,if O is open in RN,then there is a(smooth)continuous function on RNsuch thatTo define quasi-open subsets of RN,we introduce the notion of quasi-continuous functions.A real valued function f(x)is called quasi-continuous if and only if,there is a subsetsuch that f is continuous onΩεand that the classical capacity ofis less thanε.A subsetΩ of RNis called quasi-open,if there is a quasi-continuous function f such thatΩ={x∈RN:f(x)>0}(which is defined upto zero capacity sets).One can check that a setΩ is quasi-open,if,?ε>0,? an open set Oεsuch thatEquivalently,a setΩ is quasi-open,if there is a sequence of open sets{Ωn},such that,and that

A theorem of Federer-Ziemer says that if f is an H1(RN)function,then f is quasi-continuous.It is then not hard to show that a subsetis quasi-open if and only if there is a nonnegative H1(RN)function f such that

Next,it is necessary to discuss also a few natural topologies on the space of quasi-open subsets in RNin order to solve(EP).For conveniences,let us assume that these quasi-open sets are contained in a fixed bounded domain.

Definition 3.1Let{Ωn}be a sequence of quasi-open sets in a bounded domain B.We say thatΩnisγ-convergent toΩ if and only if the associated potential functions of the domains are convergent,that is,in L2(B)as n→∞.Here

Theorem 3.3(see[29])Let{Ωn}andΩbe quasi-open sets in B.ThenΩnisγ-convergent toΩ if and only if,?f∈L2(B),the solution of

converges in L2(B)to v,the solution of

We note that v,Let us sketch a proof of the above theorem when ΩnandΩ are open and smooth domains.

ProofIt suffices to verify that ifΩnisγ-convergent toΩ,thenin L2(B).For this purpose,we consider first thatfor a large constant M.Then the maximum principle implies thatandA simple calculation yields

One notices that the same formula is valid also for uΩand uΩn.SinceΩnisγ-convergent to Ω,henceConsequently,Therefore,the followings are true:

Note the second item above,which follows from(i)and an integration,implies that uΩninNow(i)and the maximum principle imply also that

and that

Thusinin this case.

Now for anyone can find fM∈L2(B)with|fM|≤ M.Let the corresponding solutions beand vM,then the above arguments yieldsasOn the other hand,standard elliptic estimates imply thatasWe concludein L2(B).

Let us also introducea convergence of Hilbert spaces,in the sense defined by Mosco(see[7]),hereΩn,Ω ? B.A sequenceis called to be convergent toin the sense of Mosco,if the following two statements are held:

(a)there is a sequencesuch that

(b)Ifsuch that there is a subsequence{vnk}withthen

It is not hard to show,via min-max principle,that the statement(a)above implies that

On the other hand,the statement(b)would imply that

Though theγ-convergence(and convergence in the sense of Masco defined above)would imply the convergence of Laplacian eigenvalues with the zero Dirichlet boundary condition,they are in a way strong convergences.Consequently,it is not easy to work with for our extremum problems.The following weak-convergence of domains would be more suitable to solve the variational problems in shape optimizations.

Definition 3.2A sequence of quasi-open domainsΩn?B is said to converge toΩweakly withΩ?B,ifin L2(B),andΩ={x∈B:w(x)>0}.

We note that w in general is not equal to uΩ.One also notices that potential functions are continuous from the De Girogi elliptic regularity theory.The following proposition is trivial.

Proposition 3.1(Compactness)Let{Ωn}be a sequence of quasi-open subdomains in B.Then there is a subsequence{Ωnk}that converges weakly toΩ,a quasi-open subdomain in B.

and we would also obtain a contradiction.For general k-th,k>1,eigenvalues,it could be also verified in the same way using the min-max principle and an induction on k.

4 Connected Minimizers

The existence of minimizers of(EP)without boundedness constraint has been established recently in the work of Mazzoleni and Pratelli[24]and Bucur[6].In fact,in[24]a more general class of extremum problems for Laplacian-Dirichlet eigenvalues was considered,and existence of bounded minimizers was proven.As a consequence of their proofs,they also showed that for any quasi-open set A ? RN,one hasλk(A)≤ M(k,N)λ1(A).In[6],Bucur proved that minimizers of(EP)exists.Moreover,he showed every minimizer is bounded and has a finite perimeter.The last result will be discussed in the final section of this paper.The aim of this section is to study when such minimizers are connected domains.We should also point out that in another recent work,by Bucur-Mazzoleni-Pratelli-Velichkov[10],it was shown that minimizers are open sets.

4.1 Splitting(in)equality

Theorem 4.1Assume that there is a multiply connected domainthat solves,for given k,the following problem:

Denote the infimum value of above extremum problem byΛ(k,N).Then,for some 1≤m≤k,

Herecan be decomposed into mutually disjoint subdomainssuch thatand that the positive integers k1,k2,···,kmsatisfy k1+k2+···+km=k.And eachΩkjcan be scaled properly(so that its volume becomes 1)to solve(EP)with k=kj.

This result may be viewed as an extension of a theorem due to Keller-Wolf[20],and we shall see that it is an easy consequence of the min-max principle.On the other hand,using arguments from the proofs of concentration-compactness(see[8]),one may derive a similar statement for minimizing sequences.

Let us sketch a proof of Theorem 4.1.

Proof of Theorem 4.1Assume thatis not connected,and we writea union of two subdomains(which are not necessarily connected)such that|Ωi|>0(i=1,2)andLetbe an eigenfunction of the Dirichlet-Laplacian on.Thenare eigenfunctions of the Dirichlet-Laplacian on Ωialso.Assume thatis the j1-th eigenvalue on Ω1(with j1is the maximum so thatWe claim first that j1

Next,we claim that there are at least(k?j1)eigenvalues ofΩ2which are smaller than Λ(k,N).Otherwise,for(whilethe min-max principle(here we may choose the k ? 1 dimensional subspace ofto be spanned by j1eigenfunctions onΩ1and the first k?1?j1eigenfunctions onΩ2)is as follows:

The latter is not possible.ThusOn the other hand,ifΛ(k,N),then the other min-max principle

would implyagain impossible.

Finally,if we replace Ω1byandΩ2byThen we have(note thatandBy the minimality ofΛ(k,N),we thus conclude that the last two inequalities above are equalities.That is,

An easy induction leads to the conclusion of Theorem 4.1.

As a byproduct of the aboveproof,we have the following statement which again follow from the min-max principle.

Proposition 4.1LetΩbe a bounded open set in RN,and assume thatΩhas m connected components Ω1,Ω2,···,Ωm.Suppose that u is the k-th eigenfunction of the Laplacian on Ω with the Dirichlet boundary condition.Thenare eigenfunctions of Dirichlet-Laplacian on Ωj’s,j=1,2,···,m(unless that u vanished identically on some of Ωj’s).Let kjbe positive integers(or zero ifsuch thatis a kj-th eigenfunction.Then

There are a few simple consequence of the above proposition.

Corollary 4.1The minimization problem(EP)has a solutionwhich,in general,would have at most m connected components with m≤c0k for some c0<1.

ProofSuppose thathas m connected components.Then one of the component,saymust have its volumeIfis a k-th eigenfunction onwith the Dirichlet boundary condition,thenis a k1-th eigenfunction on Ωk1.Thus(by Faber-Krahn inequality),whereis a ball of volumeOn the other hand,for ball B of volume 1.Weyl’s asymptotic formula implies thatWe thus conclude that

We note that the above proof is very similar in the spirit to the proof of Pleijel’s theorem.

Corollary 4.2For every N≥2,(EP)has a solution for some k≥3,which is a connected open set.

ProofThe existence of a bounded(depending on N and k)minimizer which is also an open set in RNfor the(EP)was known(see[6,10,24]).To show,for some k,it is connected,we assume,to the contrary,thatfor all k ≥ 3 are disconnected.Then by discussions in this section,one would conclude thatmust consist of exactly k connected components.The latter is not possible by Corollary 4.1.

For k=3 and N=2 or 3,Keller-Wolf[20]observed earlier that solutions of(EP)are connected.Indeed,for N=2,k=3,ifis disconnected,then one would conclude that Λ(3,2)=3Λ(1,2),i.e.,is consisting of 3 disjoint equal balls of volumeeach.As Λ(1,2)=andΛ(3,2)≤ λ3(D)=46(here D is a disc in R2of area 1),one sees that it is not possible.

For N=3,k=3,one calculatesΛ(1,3)≤ 26,and(here B is a ball in R3of volume 1).We thus haveand again it is not possible.

4.2 Generalized Polya’s conjecture

The classical Polya’s conjecture can be stated as follows.

Conjecture 4.1

Polya proved for any planar tilling domain Ω of area 1,λk(Ω)≥ C2k ≡ 4πk for k=1,2,···.In fact,one can see,for any finite k ∈ N,the strict inequality is true from Polya’s proof.We believe the following conjecture may be also valid.

Conjecture 4.2(Generalized Polya’s Conjecture)

for all N>2 and k>1.Hereδk,Nare positive numbers depending only on N and k.

Proposition 4.2Assume that the generalized Polya’s conjecture is true,then there are infinitely many k’s,such that the extremum problem has a solutionwhich is a connected open set.

ProofSuppose that the conclusion of the above proposition is not true.Then for any k≥k0,one has

On the other hand,

Here mjis the number of times ofΛ(kj,N)appeared in the summation of the splitting equality,where δ0=min{δk,N:1 ≤ k ≤ k0}>0.We therefore obtain an contradiction,when k is sufficiently large.

4.3 Regularity of minimizers

The regularity of minimizersof the extremum problem(EP)is a challenging problem.We are going to describe a work(in progress)of the author with Dennis Kriventsov.Before we do so,let us describe a recent interesting work of Bucur[6]in which he proved thatare bounded and of finite perimeter.

We letbe a minimizer of the problem(EP).For quasi-open set Ω ? RN,we define uΩbe the potential function ofΩ:

For A1,A2quasi-open and bounded sets,we define

It is easy to see that a sequence{Ωn}of quasi-open sets(contained in a fixed ball),such thatΩnisγconvergent toΩ,hereΩ is a quasi-open subset(of the same ball)if and only ifas

Definition 4.1A quasi-open set A of finite Lebesgue measure is called a local shape subsolution for E(A),if there is anη>0 andΛ>0,such that,for allandquasi-open withone has

where

Remark 4.1Ifis a minimizer of(EP),thenis a minimizer ofwhere A quasi-open in RN,for some dilation constant t>0.The converse is also true.Indeed,for any quasi-open set A with 0<|A|< ∞,one haswhereis a homothety of A such thatHence if we letthen

Ifis a minimizer of(EP),thenis a minimizer of λk(A)+|A|,where t is the unique positive critical point ofConversely,if A quasi-open,0<|A|<∞ is a minimizer ofλk(A)+|A|,thenis a minimizer of(EP).Here?A is the homothety of A with

Theorem 4.3(Bucur)If A is a quasi-open set that minimizes{λk(B)+|B|:B quasi-open in RNwith 0<|B|< ∞},then A is a local shape subsolution of E(·).

The proof of this statement is based on the fact that

whereare resolvent operators of Laplacian on A and?A,respectively,with the zero Dirichlet boundary condition,and whereis a constant depending only on k and N.ThusOn the other hand,(see the remark above),thus one has proven thatis a local shape subsolution of the energy E(·).

Theorem 4.4(Bucur)If A is a local shape subsolution of the energy E(·),then A is bounded andχA∈BV.That is,A is a set of finite perimeter.

ProofLet u=uA,uε=(u?ε)+and

Note that asThus we obtain

Consequently,

Co-area formula implies that there is a sequencesuch thatfollows.Finally,a direct construction and comparison,using the property of“l(fā)ocal shape subsolution”of A,yields that forθ∈ (0,1),there is an r0>0,c0>0 such that for all x0∈RN,0

We claim the latter implies the boundedness of A.Indeed,if there is a sequence{yn}?A such thatSince yn∈A,one hasSinceis subharmonic in RN,

Bucur’s results described here provide a starting point for the regularity ofThe following is a statement that would be discussed in the forthcoming work(see[21]):Ifis a non-degenerate minimizer of(EP),thenis almost everywhere analytic.More precisely,away from an HN?1measure zero set,it is real analytic.A key point of the proof of this last result is to reduce it to the case of the study of certain extremum domains that are associated with their first Dirichlet eigenvalues for the Laplacian.One may ask that if in the 2D case,the boundary ofconsists of at most c(k)analytic arcs.In general,one obviously has to understand much better theproperty ofin order to study the asymptotics of these minimizers as k becomes very large.In particular,it may be closely related to both the generalized Polya conjecture and the optimal partition problems.

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