aa r X i v : . [ m a t h . SP ] J u l HIGHER DIMENSIONAL SURGERY AND STEKLOV EIGENVALUES
HAN HONG
Abstract.
We show that for compact Riemannian manifolds of dimension at least 3 withnonempty boundary, we can modify the manifold by performing surgeries of codimension 2or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certainchanges in the topology of a domain do not have an effect when considering shape optimiza-tion questions for Steklov eigenvalues in dimensions 3 and higher. Our result generalizesthe 1-dimensional surgery in [6] to higher dimensional surgeries and to higher eigenvalues.It is proved in [6] that the unit ball does not maximize the first nonzero normalized Stekloveigenvalue among contractible domains in R n , for n ≥
3. We show that this is also truefor higher Steklov eigenvalues. Using similar ideas we show that in R n , for n ≥
3, the j -thnormalized Steklov eigenvalue is not maximized in the limit by a sequence of contractible do-mains degenerating to the disjoint union of j unit balls, in contrast to the case in dimension2 [9]. Introduction
In this paper we study questions related to shape optimization for the Steklov eigen-value problem on compact Riemannian manifolds of dimension at least three. The classicaltheorem of R. Weinstock [13] states that among all simply connected domains in R withfixed boundary length 2 π , the unit disc uniquely maximizes the first Steklov eigenvalue. Itwas shown in [2] that the Weinstock inequality holds in any dimension, provided one re-stricts to the class of convex sets. Namely, for every bounded convex set Ω ⊂ R n we have¯ σ (Ω) < ¯ σ ( B n ), where ¯ σ j (Ω) = σ j (Ω) | ∂ Ω | / ( n − denotes the normalized eigenvalue. How-ever, in the wider class consisting of contractible domains, the higher dimensional analogueof Weinstock’s theorem fails since there exist contractible domains B nǫ,δ (see Figure 1) with¯ σ ( B nǫ,δ ) > ¯ σ ( B n ) when n ≥ n -dimensional Riemannian manifold M with boundary,a compact properly embedded m -dimensional submanifold Σ of M , and δ > Σ ,δ denote the Lipschitz domain obtained by removing the δ tubular neighbourhood of Σ Mathematics Subject Classification. from M . We call this procedure a surgery of codimension n − m (see Section 2 for moreprecise details). Our main result is the following. Theorem 1.1.
The Steklov spectrum of a compact Riemannian manifold with boundarychanges continuously under surgeries of codimension at least two, in the sense that, if n − m ≥ then lim δ → + ¯ σ j (Ω Σ ,δ ) = ¯ σ j (Ω) for j = 0 , , , . . . . One purpose of such surgeries is to simplify the topology of the manifold. When m = 1 and j = 1 this theorem was proved in [6], in which case the surgery can be applied to construct amanifold with connected boundary from any given manifold with boundary, while keeping thefirst normalized Steklov eigenvalue nearly unchanged. Theorem 1.1 implies more generallythat the supremum of the j -th normalized eigenvalue among all manifolds is the same asthe supremum among manifolds having relatively simple topology; that is, among manifoldswhich can be obtained by performing surgeries up to codimension two. Specifically, our resultimplies that given any compact Riemannian manifold Ω of dimension n ≥
3, and given any ǫ > k ∈ N , there exists a smooth subdomain ˜Ω of Ω such that the homomorphism ofthe m -th homology groups i ∗ : H m ( ∂ ˜Ω) → H m ( ˜Ω) induced by the inclusion i : ∂ ˜Ω ֒ → ˜Ω isinjective for m = 0 , , . . . min { n − , (cid:4) n (cid:5) } , and such that | ¯ σ j ( ˜Ω) − ¯ σ j ( ˜Ω) | < ǫ for j = 1 , . . . , k. This conclusion involves some standard but rather involved algebraic topology, and we omitthe proof since our results here are analytic in nature.An immediate consequence of Theorem 1 . n ≥
3, balls do not maximize highernormalized Steklov eigenvalues among contractible domains in R n . Corollary 1.2.
For n ≥ , there exists a contractible domain Ω ∗ in R n such that for any j ≥ , ¯ σ j (Ω ∗ ) > ¯ σ j ( B n ) . This is not surprising, especially in light of the fact that a disc does not maximize higherSteklov eigenvalues among simply connected domains in R . The classical result [11] givesthe upper bound ¯ σ j (Ω) ≤ πj for any simply connected domain Ω in R and all j ≥
0. When j = 1, equality is characterized in Weinstock’s theorem ([13]), but for j ≥ j identical discs. Byanalogy with the n = 2 case, it is natural to ask whether a similar result is true in higherdimensions. Using ideas from Theorem 1.1 we show that this is not the case. Theorem 1.3.
For n ≥ and j ≥ , the supremum of the j -th normalized Steklov eigenvalueamong contractible domains in R n is not achieved in the limit by a sequence of contractibledomains degenerating to the disjoint union of j identical round balls. This is a direct consequence of a more general theorem on the convergence of Stekloveigenvalues of overlapping domains as they are pulled apart, Theorem 3 .
2, discussed inSection 3.The paper is organized as follows. In section 2 we recall basic knowledge about theSteklov eigenvalue problem and prove our main result, Theorem 1 .
1, on higher dimensionalsurgeries and the Steklov spectrum. In section 3 we discuss applications and related resultsin connection with questions about shape optimization for the Steklov problem for domainsin R n , and prove Corollary 1.2, Theorem 3 .
2, and Theorem 1 . Acknowledgments.
The author would like to thank his advisors Jingyi Chen and AilanaFraser for helpful discussions.
IGHER DIMENSIONAL SURGERY AND STEKLOV EIGENVALUES 3 Continuity of Steklov eigenvalues under codimension 2 surgeries
In this section, we recall some basic facts about the Steklov eigenvalue problem, and proveour main theorem on the continuity of eigenvalues under certain higher dimensional surgeries,up to codimension 2. Let (Ω , g ) be a compact, connected n -dimensional smooth Riemannianmanifold with nonempty boundary. The Steklov eigenvalue problem is given by ( ∆ u = 0 in Ω ∂u∂ν = σu on ∂ Ωwhere ν is the outer unit normal to Ω. When the trace operator T : H (Ω) → L ( ∂ Ω) iscompact, in particular, if the domain has Lipschitz boundary, the spectrum of the Stekloveigenvalue problem is discrete 0 = σ < σ (Ω) ≤ σ (Ω) ≤ · · · and the eigenvalues have a standard Rayleigh quotient characterization(2.1) σ j (Ω) = inf (cid:26) R Ω |∇ u | R ∂ Ω u : 0 = u ∈ H (Ω) , Z ∂ Ω uφ i = 0 for i = 0 , . . . , j − (cid:27) where { φ , φ , φ , . . . } is a complete orthonormal basis of L ( ∂ Ω) such that φ i is an eigen-function with eigenvalue σ i (Ω) for each i = 1 , , . . . . Alternatively,(2.2) σ j (Ω) = inf E j +1 sup u ∈ E j +1 \{ } R Ω |∇ u | R ∂ Ω u where the infimum is taken over all ( j + 1)-dimensional subspaces E j +1 of the Sobolev space H (Ω).We consider a smooth m -dimensional submanifold Σ embedded in the manifold Ω, withboundary ∂ Σ embedded in ∂ Ω, and meeting ∂ Ω orthogonally along ∂ Σ. Let L δ = { x ∈ Ω : d ( x, Σ) < δ } and T δ = { x ∈ Ω : d ( x, Σ) = δ } . Denote Ω δ = Ω \ L δ . Definition 2.1.
Taking out the tubular neighborhood L δ of Σ gives a domain Ω δ . Through-out the paper, we call this procedure surgery of dimension m or surgery of codimensionn-m .We will apply the following “no concentration” lemma to show that if 2 ≤ m ≤ n −
2, thenthe interior and boundary L norms of a sequence of eigenfunctions don’t concentrate nearthe neck T δ as δ → + . When m = 1, that is when Σ is a smooth curve, this phenomenon wasverified in [6, Lemma 4.2]. We prove here that it is in fact still true provided 2 ≤ m ≤ n − Lemma 2.2.
Suppose ≤ m ≤ n − . There exists a constant r > such that if u δ ∈ W , (Ω δ ) and Z Ω r / \ Ω r u δ + Z Ω δ \ Ω r |∇ u δ | ≤ C for δ ∈ (0 , r / , with C independent of δ . Then (2.3) lim δ → k u δ k L ( T δ ) = 0 , HAN HONG (2.4) lim s → k u δ k L (Ω δ \ Ω s ) = 0 , (2.5) lim s → k u δ k L (( L s \ L δ ) ∩ ∂ Ω) = 0 for any < δ < s < r / .Proof. For simplicity of presentation we will assume that the normal bundle of Σ is trivial;otherwise, we can consider local trivializations and add up all of the corresponding estimates.Choose r > r neighbourhood of the zero section onto its image, and suchthat the metric on the r tubular neighbourhood L r of Σ in M is uniformly equivalent tothe product metric ˜ g + dr + r g S n − m − on Σ × D r , where ˜ g is the metric on Σ induced from g , g S n − m − is the standard metric on the sphere S n − m − and D t is the ball of radius t centredat the origin in R n − m . Since Σ intersects ∂ Ω orthogonally along ∂ Σ, we may further choose r sufficiently small such that the metric on L r ∩ ∂ Ω is uniformly equivalent to the productmetric g ∂ Σ + dr + r g S n − m − .We first localize the support of u δ to lie near Σ. Choose a smooth radial cutoff functionsuch that ϕ ( x, r, θ ) = ( r ≤ r r ≥ r and define v δ = ϕu δ . It follows from Schwarz and arithmetic geometric mean inequalitiesthat |∇ v δ | ≤ ϕ |∇ u δ | + u δ |∇ ϕ | ) . Since |∇ ϕ | can be bounded in terms of r which is a fixed number, we have(2.6) Z Ω δ |∇ v δ | ≤ Z Ω δ \ Ω r |∇ u δ | + 2 C Z Ω r / \ Ω r u δ ≤ C where C , C are constants depending only on C and r . We will choose δ much smallerthan r . From the construction we have that u δ = v δ on T δ , and thus to prove equation (2 . ε > Z T δ u δ = Z T δ v δ ≤ ε Z Ω δ |∇ v δ | for sufficiently small δ . Since the metric on Ω δ is uniformly equivalent to the product metricon the support of v δ , it suffices to prove the estimate for the product metric.For a fixed point p in Σ we denote the restriction of v δ ( x, r, θ ) to the annulus D r \ D δ in R n − m at this point p by v ( r, θ ). Choose a harmonic function h on D r \ D δ as follows ∆ h = 0 D r \ D δ h = v = 0 ∂D r h = v ∂D δ . Since harmonic functions minimize the Dirichlet energy we have(2.7) Z D r \ D δ |∇ h | ≤ Z D r \ D δ |∇ v | . IGHER DIMENSIONAL SURGERY AND STEKLOV EIGENVALUES 5
For any σ with δ ≤ σ ≤ r we have Z D r \ D σ ∆ h = Z ∂D r ∂h ∂r − Z ∂D σ ∂h ∂σ = − Z ∂D σ ∂h ∂σ = − σ n − m − ddσ (cid:20) σ − n + m +1 Z ∂D σ h (cid:21) where last equality follows since the volume measure on ∂D σ is σ n − m − times that on theunit sphere ∂D . Since ∆ h = 0, we have Z D r \ D σ ∆ h = 2 Z D r \ D σ |∇ h | , which together with (2 .
7) implies − σ n − m − ddσ (cid:20) σ − n + m +1 Z ∂D σ h (cid:21) = 2 Z D r \ D σ |∇ h | ≤ Z D r \ D δ |∇ h | ≤ Z D r \ D δ |∇ v | . Now dividing both sides by σ n − m − and integrating with respect to σ over the interval [ δ, r ]we obtain δ − n + m +1 Z ∂D δ v ≤ (cid:18)Z r δ σ − n + m +1 dσ (cid:19) Z D r \ D δ |∇ v | . If 1 ≤ m ≤ n −
2, this implies that Z ∂D δ v ≤ ε n ( δ ) Z D r \ D δ |∇ v | where ε m +2 ( δ ) = 2 δ ln( r δ ) and ε n ( δ ) = δn − m − for n ≥ m +3. Integrating the above inequalityover Σ we obtain(2.8) Z T δ v δ ≤ ε n ( δ ) Z Σ m Z D r \ D δ |∇ v | ≤ ε n ( δ ) Z Ω δ |∇ v δ | , since |∇ v | ≤ |∇ v δ | because ∇ v is the Euclidean gradient on the slice D r \ D δ at a point onΣ. Since lim δ → ε n ( δ ) = 0 for 1 ≤ m ≤ n −
2, this completes the proof of (2 . .
4) and (2 .
5) of the lemma. By the same argument usedto prove (2.8), we have that for any δ ≤ t < r / Z T t v δ ≤ ε n ( t ) Z Ω δ |∇ v δ | ≤ C ε n ( t ) HAN HONG where ε n ( t ) is defined as above, and C is as in (2 .
6) and is independent of δ . Integratingwith respect to t over [ δ, s ] for s < r / Z Ω δ \ Ω s v δ = Z L s \ L δ v δ ≤ ( C (cid:16) s ln( r s ) − δ ln( r δ ) + s − δ (cid:17) , n = m + 2 C ( s − δ ) / ( n − m − , n ≥ m + 3 . Since the right hand sides tend to zero as s → δ < s , this concludes the proof of(2 . t = { x ∈ Σ : dist Σ ( x, ∂ Σ) ≤ t } and denote ∂ Σ t = { x ∈ Σ : dist Σ ( x, ∂ Σ) = t } (note this is only part of the topological boundary of Σ t ). We choose t sufficiently small suchthat H ∂ Σ τ ≤ C for all τ ≤ t , where H ∂ Σ τ is the mean curvature of ∂ Σ τ with respect to outerunit normal. By the coarea formula, Z t Z ∂ Σ τ (cid:18)Z D s \ D δ ( x ) v δ (cid:19) dτ = Z Σ t (cid:18)Z D s \ D δ ( x ) v δ (cid:19) ≤ Z Σ Z D s \ D δ ( x ) v δ = Z L s \ L δ v δ . Denote F s,δ ( τ ) := Z ∂ Σ τ (cid:18)Z D s \ D δ ( x ) v δ (cid:19) . Then (2 .
9) yields lim s → Z t F s,δ ( τ ) dτ = 0 . Thus there exists τ ∈ (0 , t ) such that lim s → F s,δ ( τ ) = 0 . We differentiate F s,δ ( τ ) to get ddτ F s,δ ( τ ) = Z ∂ Σ τ (cid:18)Z D s \ D δ ( x ) v δ dv δ dτ (cid:19) − Z ∂ Σ τ (cid:18)Z D s \ D δ ( x ) v δ (cid:19) H ∂ Σ τ . Integrating over [0 , τ ], applying the coarea formula, and using Cauchy-Schwarz inequalityand arithmetic-geometric mean inequality result in that for any 0 < ε < | F s,δ ( τ ) − F s,δ (0) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Σ τ (cid:18)Z D s \ D δ ( x ) v δ dv δ dτ (cid:19) − Z τ Z ∂ Σ τ (cid:18)Z D s \ D δ ( x ) v δ (cid:19) H ∂ Σ τ dτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε Z Σ τ (cid:18)Z D s \ D δ ( x ) |∇ v δ | (cid:19) + (1 /ε + C ) Z Σ τ (cid:18)Z D s \ D δ ( x ) v δ (cid:19) ≤ C ε + (1 /ε + C ) Z L s \ L δ v δ Taking ε = s and letting s tend to zero, from (2 .
9) it follows that F s,δ (0) → s → . Hence we obtain the equation (2 . (cid:3) IGHER DIMENSIONAL SURGERY AND STEKLOV EIGENVALUES 7
Lemma 2.3.
Consider the following logarithmic cut-off function along the submanifold Σ m ,where ≤ m ≤ n − , ϕ δ = r ≤ δ
22 ln δ − ln r ln δ δ ≤ r ≤ δ δ ≤ r. Then R Ω |∇ ϕ δ | → as δ → + . Proof.
Integrating with respect to product metric gives Z Ω |∇ ϕ δ | ≤ Z Σ Z D δ \ D δ |∇ ϕ δ | = c ( n ) | Σ | (ln δ ) Z δδ r n − m − dr = c ( n ) | Σ | δ ( n )where δ ( n ) = − / ln δ when n = m + 2 and δ ( n ) = ( δ n − m − − δ n − m − ) / ( n − m − δ ) when n > m + 2. Since δ ( n ) → δ →
0, this completes the proof. (cid:3)
We now prove our main theorem, that given a compact Riemannian manifold with bound-ary (Ω n , g ), surgeries of dimension m (see Definition 2 .
1) can be performed while keepingthe Steklov eigenvalues nearly unchanged, provided 1 ≤ m ≤ n − Theorem 1.1. If ≤ m ≤ n − , then lim δ → + σ j (Ω δ ) = σ j (Ω) for j = 0 , , , . . . .Proof. Let u δ , u δ , u δ , . . . be L ( ∂ Ω δ )-orthonormal Steklov eigenfunctions such that u jδ is aSteklov eigenfunction of Ω δ with eigenvalue σ j (Ω δ ), ( ∆ u jδ = 0 in Ω δ∂u jδ ∂η = σ j (Ω δ ) u jδ on ∂ Ω δ . Claim 2.1.
For any j ∈ N , σ j (Ω δ ) is uniformly bounded from above for small δ. Proof.
Let { f , . . . , f j +1 } ⊂ H (Ω) be j + 1 functions on Ω with support in Ω r (where r isas in Lemma 2.2), that are linearly independent on ∂ Ω. Let E = span { f , · · · , f j +1 } . Then if δ < r , any function in E is a valid test function for the min-max variational characterization(2.2) of σ j (Ω δ ), and so σ j (Ω δ ) ≤ sup u ∈ E \{ } R Ω δ |∇ u | R ∂ Ω δ u = sup u ∈ E \{ } R Ω r |∇ u | R ∂ Ω r u ≤ Λ j where Λ j is independent of δ . (cid:3) Since u jδ is a Steklov eigenfunction of Ω δ with eigenvalue σ j (Ω δ ),(2.10) Z Ω δ |∇ u jδ | = σ j (Ω δ ) Z Ω δ ( u jδ ) = σ j (Ω δ ) ≤ Λ j . By (2.10) and since k u jδ k L ( ∂ Ω δ ) = 1, by standard theory,(2.11) k u jδ k L ( K ) ≤ C (Λ j , K )for any compact subset K of Ω \ Σ.Elliptic boundary estimates ([8, Theorem 6.29]) give bounds k u jδ k C ,α ( K ) ≤ C k u jδ k C ( K ) HAN HONG for any compact subset K of Ω \ Σ for all sufficiently small δ , where C = C ( j, α, Λ j , K ). BySobolev embedding and interpolation inequalities ([1, Theorem 5.2], [8, (7.10)]), k u jδ k C ( K ) ≤ C (cid:0) ε k u jδ k C ( K ) + ε − µ k u jδ k L ( K ) (cid:1) where ε > µ > n , and C depends on K .Hence k u jδ k C ,α ( K ) ≤ C with C independent of δ . By the Arzela-Ascoli theorem and adiagonal sequence argument, there exists a sequence δ i → j , u jδ i convergesin C ( K ) on compact subsets K ⊂ Ω \ Σ to a harmonic function u j on Ω \ Σ, satisfying ∂u j ∂η = σ j u j on ∂ Ω \ ∂ Σwith σ j = lim i →∞ σ j (Ω δ i ). Claim 2.2.
For each j ≥ ∈ N , u j can be extended to a Steklov eigenfunction of Ω witheigenvalue σ j .Proof. First observe that u j ∈ H (Ω \ Σ). Fix the compact subset Ω δ N for some large N .Then k u jδ i k L (Ω δi ) = k u jδ i k L (Ω δN ) + k u jδ i k L (Ω δi \ Ω δN ) is uniformly bounded independent of i .The first term is bounded by (2.11) and the second term is bounded by (2 .
4) of Lemma 2 . .
10) shows that k u jδ i k H (Ω δi ) is uniformly bounded. Thus u j ∈ H (Ω \ Σ).For any function ψ ∈ W , ∩ L ∞ (Ω), define ψ δ = ψϕ δ where ϕ δ is defined in Lemma 2 . u j is a harmonic function on Ω \ Σ and satisfies ∂u j ∂η = σ j u j on ∂ Ω \ ∂ Σ, and ψ δ vanishesnear Σ, we have Z Ω \ Σ ∇ u j · ∇ ψ δ = σ j Z ∂ Ω \ ∂ Σ u j ψ δ , and so(2.12) Z Ω ψ ∇ u j · ∇ ϕ δ + ϕ δ ∇ u j · ∇ ψ = σ j Z ∂ Ω u j ψ δ . From H¨older’s inequality it follows that (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ψ ∇ u j · ∇ ϕ δ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω | ψ ∇ u j ||∇ ϕ δ | ≤ (cid:18)Z Ω | ψ ∇ u j | (cid:19) (cid:18)Z Ω |∇ ϕ δ | (cid:19) . Therefore, by Lemma 2 . Z Ω ψ ∇ u j · ∇ ϕ δ → δ → . Since | ψ δ | ≤ | ψ | ∈ L ∞ (Ω) and u j ∈ H (Ω \ Σ), taking limits both sides of (2 .
12) and applyingthe dominated convergence theorem, we obtain Z Ω ∇ u j · ∇ ψ = σ j Z ∂ Ω u j ψ which implies the claim. (cid:3) Claim 2.3.
The set of functions { u j } is orthonormal on the boundary of Ω , i.e., Z ∂ Ω u j u l = δ jl for j, l ≥ . IGHER DIMENSIONAL SURGERY AND STEKLOV EIGENVALUES 9
Proof.
Let ǫ >
0. Since u j are Steklov eigenfunctions by Claim 2 .
2, and therefore smooth bythe regularity theory, there exists sufficiently small s > | R L s ∩ ∂ Ω u j u l | < ǫ/
4. Ac-cording to H¨older’s inequality and (2 .
3) and (2 .
5) of Lemma 2.2, we may furthermore assumethat s is chosen sufficiently small such that | R T δi u jδ i u lδ i | < ǫ/ | R ( L s \ L δi ) ∩ ∂ Ω u jδ i u lδ i | < ǫ/ i large enough. The fact that u jδ i uniformly converges to u j in any compact subsetimplies that | R ∂ Ω \ L s u jδ i u lδ i − R ∂ Ω \ L s u j u l | < ǫ/ i large enough. Hence it follows that (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ Ω u j u l − δ jl (cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ Ω \ L s u j u l − δ jl (cid:12)(cid:12)(cid:12)(cid:12) + ǫ/ < (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ Ω \ L s u jδ i u lδ i − δ jl (cid:12)(cid:12)(cid:12)(cid:12) + ǫ/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂ Ω δi u jδ i u lδ i − Z ( L s \ L δi ) ∩ ∂ Ω u jδ i u lδ i − Z T δi u jδ i u lδ i − δ jl (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ǫ/ < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ( L s \ L δi ) ∩ ∂ Ω u jδ i u lδ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T δi u jδ i u lδ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ǫ/ < ǫ/ ǫ/ ǫ/ ǫ Since ǫ > (cid:3)
Theorem 1 . σ j = σ j (Ω). This is what we shall do in thefollowing, by applying induction. It is easy to see that lim δ → σ (Ω δ ) = σ (Ω) since they arealways zero. Hereafter, we fix j and assume that for each 0 ≤ i ≤ j − u i is a Stekloveigenfunction of Ω with eigenvalue σ i (Ω). First, we show that σ j (Ω) ≤ σ j . To see this, fromClaim 2 . Z ∂ Ω u j = Z ∂ Ω u u j = · · · = Z ∂ Ω u j − u j = 0 , which guarantees that u j is an admissible test function for σ j (Ω), and thus σ j ≥ σ j (Ω) . Toprove the theorem, we need to prove the reverse inequality.Denote the j -th Steklov eigenfunction of Ω by v , and define f = v − j − X i =0 (cid:18)Z ∂ Ω δ vu iδ (cid:19) u iδ . In fact, f is the projection of v to the orthogonal complement of first j eigenspaces of Ω δ .Therefore, Z ∂ Ω δ f u iδ = 0 , for i = 0 , · · · , j − . Thus f is an admissible test function for σ j (Ω δ ) . On the one hand, let’s estimate the denominator of the Rayleigh quotient,(2.13) Z ∂ Ω δ f = Z ∂ Ω δ v − j − X i =0 (cid:18)Z ∂ Ω δ vu iδ (cid:19) . It’s clear that lim δ → Z ∂ Ω δ v = Z ∂ Ω v and lim δ → Z ∂ Ω δ vu iδ = Z ∂ Ω vu i = 0 for i = 0 , · · · , j − . On the other hand, we may estimate the numerator of the Rayleigh quotient as follows(2.14) Z Ω δ |∇ f | = Z Ω δ |∇ v | + j − X i =1 (cid:18)Z ∂ Ω δ vu iδ (cid:19) Z Ω δ |∇ u iδ | − j − X i =1 (cid:18)Z ∂ Ω δ vu iδ (cid:19) Z Ω δ h∇ v, ∇ u iδ i . Similarly, Z Ω δ |∇ u iδ | = σ (Ω δ ) < C and Z Ω δ h∇ v, ∇ u iδ i ≤ C (cid:18)Z Ω δ |∇ u iδ | (cid:19) ≤ C , where all the constant C , C , C are independent of δ for small δ. In the end, we combine the estimates (2 . .
14) in the characterization (2 .
1) and takethe limit to get lim sup δ → σ j (Ω δ ) ≤ lim δ → R Ω δ |∇ f | R ∂ Ω δ f = R Ω |∇ v | R ∂ Ω v = σ j (Ω) . This completes the proof of the theorem. (cid:3) Some results on shape optimization for Steklov eigenvalues in R n For any bounded domain Ω ⊂ R n there are upper bounds on all normalized Stekloveigenvalues, σ j (Ω) | ∂ Ω | n − ≤ C ( n ) j n ([4]). An explicit upper bound for the first normalizedSteklov eigenvalue was given in [6, Proposition 2.1], however the upper bound is not expectedto be sharp. It is an open question to determine the sharp upper bound. Question 3.1 ([10]) . On which domain (or in the limit of which sequence of domains) isthe supremum of ¯ σ j (Ω) over all bounded domains Ω ⊂ R n realized? A consequence of Theorem 1.1 is that the supremum of the j -th normalized eigenvalueamong all domains is the same as the supremum among domains having relatively simpletopology; that is, among domains which can be obtained by performing surgeries up tocodimension two (as discussed in Section 2).Another immediate consequence of our surgery result is that when n ≥ σ j (Ω), even among contractible domains. This was proved for j = 1in [6, Theorem 1.1]. Corollary 1.2.
For n ≥ , there exists a contractible domain Ω ∗ in R n such that for any j ≥ , ¯ σ j (Ω ∗ ) > ¯ σ j ( B n ) . IGHER DIMENSIONAL SURGERY AND STEKLOV EIGENVALUES 11
Proof.
Let B nǫ denote the ball of radius ǫ centred at the origin in R n . It was shown in [6,Proposition 3.1] that for ǫ > σ j ( B n \ B nǫ ) > ¯ σ j ( B n ). We may then performa 1-dimensional surgery on B n \ B nǫ to construct a contractible domain, while changing theSteklov eigenvalues by an arbitrarily small amount. For example, let B nǫ,δ denote the domainobtained by removing a δ -tube around a radial segment (see Figure 1). Then since n ≥ .
1, for sufficiently small ǫ > δ → ¯ σ j ( B nǫ,δ ) = ¯ σ j ( B n \ B nǫ ) > ¯ σ j ( B n ) . Therefore, we can find a small δ > σ j ( B nǫ,δ ) > ¯ σ j ( B n ) . We let Ω ∗ = B nǫ,δ . (cid:3) Figure 1.
The construction of the contractible domain B nǫ,δ in R n for n ≥ δ is radius of the neck and ǫ is the radius of inner ball, we delete the neck andinner ball from unit ballAs discussed in the introduction, this result is not surprising, especially in light of [9] whichshows that the supremum of the j -th normalized eigenvalue among simply connected domainsin R is achieved in the limit by a sequence of simply connected domains degenerating tothe disjoint union of j identical discs. We now consider contractible domains of this type inhigher dimensions.Consider the necklace-like contractible domain Ω nε,l in R n which is the union of l n -dimensional unit balls centred along a common axis and positioned such that adjacent ballsoverlap by a distance ε along the axis; see Figure 2. As ε tends to zero, Ω nε,l converges to l identical unit balls touching tangentially at l − p , · · · , p l − (see Figure 3). We denote l disjoint unit balls by ⊔ l B n . r = 1 Figure 2.
The construction of Ω nε,l in R n , n ≥ l overlapping unit balls.Using similar ideas as in the proof of Theorem 1.1, we show that the Steklov spectrum ofΩ nε,l converges to that of ⊔ l B n as ε → p p p l − r = 1 Figure 3.
Limit of Ω nε,l in R n , n ≥
2, as ε → Theorem 3.2.
For every j ≥ , the j -th normalized Steklov eigenvalue of Ω nε,l , where n ≥ ,converges to the j -th normalized Steklov eigenvalue of the disjoint union of l unit balls as ε tends to zero, i.e., (3.1) lim ε → + ¯ σ j (Ω nε,l ) = σ j ( ⊔ l B n ) · ( l · | S n − | ) n − . Remark . (1) The above theorem holds for more general overlapping domains. For ex-ample, in the construction we could replace the unit ball by the domain B nǫ,δ from the proofof Corollary 1.2 (Figure 1). More generally, the same proof works for domains with smoothboundary overlapping at elliptic points on the boundary, and such that other intersections donot occur; (2) Comparing (3 .
1) with Theorem 1.3.1 in [9], the index of the Steklov eigenvaluedoesn’t have to be same as the number l of balls we overlap. (3) This result was proved in[3, Example 3]. However, we use a different approach here. Proof of Theorem . . Since | ∂ Ω nε,l | converges to l | S n − | , it suffices to prove that(3.2) lim ε → + σ j (Ω nε,l ) = σ j ( ⊔ l B n ) . We fix l ≥ n ≥
2, and without ambiguity, we simply use the notation Ω ε for Ω nε,l inthe proof. We split the proof into four steps. Step 1: Uniform energy bound and uniform convergence in compact subsets.
We consider the Steklov eigenvalue problem on Ω ε . Let { u jε } ∞ j =0 be L ( ∂ Ω ε )-orthonormalSteklov eigenfunctions of Ω ε such that u jε is an eigenfunction with eigenvalue σ j (Ω ε ). By anargument similar to the proof of Claim 2 .
1, for each fixed j ∈ N , there exists ε > ε < ε , σ j (Ω ε ), and hence the energy, is uniformly bounded independent of ε . Asin the proof of Theorem 1.1, elliptic boundary estimates give uniform bounds on u jε and itsderivatives up to ∂ Ω ε , and there exists a sequence ε i → j , u jε i convergesin C ( K ) as i → ∞ on any compact subset K of ⊔ l B n \ { p , . . . , p l − } to a harmonic function u j on ⊔ l B n satisfying ∂u j ∂η = σ j u j on ⊔ l ∂ B n \ { p , . . . , p l − } with σ j = lim i →∞ σ j (Ω ε i ). Step 2: Nonconcentration of L norms of the eigenfunctions around p , · · · , p l − . For a point p on the boundary of the unit ball B n , we consider the geodesic ball B s ( p ) in B n centred at p with radius s . Notice that if s ≥ ε and p is chosen along the axis of thedomain Ω ε then B s ( p ) will contain the intersection of the adjacent overlapping balls. Theboundary of B s ( p ) consists of two parts: Γ s ( p ) = ∂ B s ( p ) ∩ B and Γ s ( p ) = ∂ B s ( p ) \ Γ s ( p ). IGHER DIMENSIONAL SURGERY AND STEKLOV EIGENVALUES 13
In the following we choose r small such that B s ( p ) is uniformly equivalent to a Euclideanhalf ball of radius s for s < r .We claim that for any smooth function u on B s ( p ) with u = 0 on Γ s ( p ),(3.3) Z B s ( p ) u ≤ C ( n ) s Z B s ( p ) |∇ u | and(3.4) Z Γ s ( p ) u ≤ C ( n ) s Z B s ( p ) |∇ u | . Here C ( n ) , C ( n ) are constants depending only on the dimension n . The above two Poincareinequalities follow from estimates of the first Dirichlet-Neumann eigenvalue and the firstDirichlet-Steklov eigenvalue of a Euclidean half ball (or see [7, Lemma 4.5 & Lemma 4.6]).According to [7, Lemma 4.3] there exists s and a smooth cut off function ζ , which is 0on B r \ B s and 1 on B s , such that for any smooth function u defined on B r ,(3.5) Z B r \ B s |∇ ζ | u ≤ C ( s ) Z B r \ B s u + Z B r |∇ u | ! . Here s < s = s ( s ) < r and s ( s ) = o (1) , C ( s ) = o (1). Now if u is any smooth functiondefined on the unit ball B n , applying ζ u to (3 .
3) we have Z B s ( p ) u ≤ Z B s ( p ) ( ζ u ) ≤ C ( n ) s Z B s ( p ) |∇ ζ u | ≤ C ( n ) s Z B s ( p ) |∇ u | + Z B s ( p ) \ B s ( p ) |∇ ζ | u ! ≤ C ( s ) Z B r ( p ) |∇ u | + Z B r ( p ) \ B s ( p ) u ! (3.6)where C ( s ) = o (1), and we used (3 .
5) in the last inequality. Similarly, applying ζ u to (3 . Z Γ s ( p ) ≤ C ( s ) Z B r ( p ) |∇ u | + Z B r ( p ) \ B s ( p ) u ! where C ( s ) = o (1) . Now we consider the eigenfunction u jε on domain Ω ε . In what follows we shall show thatinterior and boundary L norms of u jε don’t concentrate near p , . . . , p l − as ε →
0. Thecommon axis of the domain Ω ε intersects the boundaries of the overlapping balls at 2 l − p (1)1 ( ε ) , p (2)1 ( ε ) , · · · , p (1) l − ( ε ) , p (2) l − ( ε ), such that as ε → p (1) k ( ε ) and p (2) k ( ε ) converge to p k for k = 1 , · · · , l −
1. We apply the inequality (3 .
6) at these points to the restriction of the function u jε to each ball. Summing the inequalities yields that for any ε < s < s ( s ) < r , Z ∪ i =1 ∪ l − k =1 B s ( p ( i ) k ( ε )) ( u jε ) ≤ C ( s ) Z Ω ε |∇ u jε | + Z ∪ i =1 ∪ l − k =1 B r ( p ( i ) k ( ε )) \ B s ( p ( i ) k ( ε )) ( u jε ) ! ≤ C · C ( s )where the last inequality follows from the trace inequality and the uniform energy bound.Similarly, applying (3 .
7) we obtain that for any ε < s < s ( s ) < r , Z ∪ i =1 ∪ l − k =1 Γ s ( p ( i ) k ( ε )) \ Γ ε ( p ( i ) k ( ε )) ( u jε ) ≤ Z ∪ i =1 ∪ l − k =1 Γ s ( p ( i ) k ( ε )) ( u jε ) ≤ C ( s ) Z Ω ε |∇ u jε | + Z ∪ i =1 ∪ l − k =1 B r ( p ( i ) k ( ε )) \ B s ( p ( i ) k ( ε )) ( u jε ) ! ≤ C · C ( s ) . Hence following the ideas of the proofs of Claims 2.2 and 2 . k u jε k H (Ω ε ) is uniformlybounded and therefore u j ∈ H ( ⊔ l B n ), u j extends to a Steklov eigenfunction of ⊔ l B n , and { u j } are orthonormal on the boundary of ⊔ l B n . Step 3: Construction of cut off function.
Define the following cutoff function on ⊔ l B n , ϕ ε = , r < ε r − ln ε ln ε − ln ε ε ≤ r ≤ ε r > ε where r is the distance function to the nearest of the points p , . . . , p l − . Let T t = { x ∈⊔ l B n : r ( x ) ≤ t } . We have(3.8) Z ⊔ l B n |∇ ϕ ε | = 1(ln ε ) Z T ε \ T ε r ≤ (2 l − C ( n )(ln ε ) Z εε r n − dr = C ( n, l ) ǫ n ( ε )where ǫ ( ε ) = − / ln ε and ǫ n ( ε ) = ε n − (1 − ε n − ) / (ln ε ) for n ≥ Step 4: Proof of the convergence of eigenvalues.
We now use induction to provethat (3.2) holds: lim ε → σ j (Ω ε ) = σ j ( ⊔ l B n ). It is clear that σ (Ω ε ) = 0 for any ε and σ ( ⊔ l B n ) = 0, and so (3.2) holds for j = 0. For fixed j ∈ N , suppose that (3.2) holds for i = 0 , , · · · , j −
1; that is, u i is a Steklov eigenfunction of ⊔ l B n with eigenvalue σ i ( ⊔ l B n ) for i = 0 , , · · · , j −
1. According to Step 1, we have Z ∂ ( ⊔ l B n ) u j = Z ∂ ( ⊔ l B n ) u u j = · · · = Z ∂ ( ⊔ l B n ) u j − u j = 0 . Then u j is an admissible test function for the j -th eigenvalue of ⊔ l B n . Thus σ j ( ⊔ l B n ) ≤ σ j ,and we have(3.9) lim inf ε → + σ j (Ω ε ) ≥ σ j ( ⊔ l B n ) . In order to prove the theorem, we need to prove the reverse inequality. We will use a j -theigenfunction of ⊔ l B n , which we denote by v , to construct an admissible test function for the j -th eigenvalue of Ω ε , where ε is the square of ε . This part of the proof is slightly different IGHER DIMENSIONAL SURGERY AND STEKLOV EIGENVALUES 15 from the proof of Theorem 1 .
1, since we need to cut off and glue functions to construct theadmissible test function to approximate the j -th eigenvalue of Ω ε .Let f = vϕ ε − P j − i =0 ( R ∂ Ω ε vϕ ε u iε ) u iε , where we recall that { u iε } ∞ i =0 are orthonormaleigenfunctions of Ω ε . By construction, f is orthogonal to u ε , · · · , u j − ε on ∂ Ω ε , and so σ j (Ω ε ) ≤ R Ω ε |∇ f | R ∂ Ω ε f . A simple calculation turns the numerator to Z Ω ε |∇ f | = Z Ω ε |∇ ( vϕ ε ) | + j − X i =1 σ i (Ω ε ) Z ∂ Ω ε vϕ ε u iε ! − j − X i =1 Z ∂ Ω ε vϕ ε u iε Z Ω ε h∇ u iε , ∇ ( vϕ ε ) i . (3.10)The first term can be estimated as Z Ω ε |∇ ( vϕ ε ) | ≤ Z Ω ε |∇ v | + 2 Z T ε \ T ε ϕ ε |∇ v | + v |∇ ϕ ε | ≤ Z ⊔ l B n |∇ v | + C | T ε \ T ε | + C Z T ε \ T ε |∇ ϕ ε | = Z ⊔ l B n |∇ v | + C ( ε )(3.11)with C ( ε ) → ε → + by (3 . ≤ i ≤ j ,lim ε → + Z ∂ Ω ε vϕ ε u iε = Z ⊔ l B n vu i = 0and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Ω ε h∇ u iε , ∇ ( vϕ ε ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( σ i (Ω ε )) Z Ω ε |∇ ( vϕ ε ) | ! . Since σ i (Ω ε ) are uniformly bounded in ε , and by (3 . .
10) both tend to zero as ε → + . In all, we have an inequality for the numerator, Z Ω ε |∇ f | ≤ Z ⊔ l B n |∇ v | + C ( ε )with C ( ε ) → ε → + .On the other hand, we expand the denominator as follows Z ∂ Ω ε f = Z ∂ Ω ε ( vϕ ε ) − j − X i =0 Z ∂ Ω ε vϕ ε u iε ! → Z ∂ ( ⊔ l B n ) v as ε →
0, since the second term tends to zero as ε → Now we combine all the estimates to getlim sup ε → σ j (Ω ε ) ≤ lim ε → R Ω ε |∇ f | R ∂ Ω ε f ≤ R ⊔ l B n |∇ v | R ∂ ( ⊔ l B n ) v = σ j ( ⊔ l B n ) . Therefore together with (3 .
9) we have lim ε → σ j (Ω ε ) = σ j ( ⊔ l B n ). (cid:3) As a special case of Theorem 3.2, we obtain the following generalization of [9, Theorem1.3.1] to higher dimensions.
Theorem 3.4.
For the domains Ω nε,j ⊂ R n (see Figure ), n ≥ , we have lim ε → + ¯ σ j (Ω nε,j ) = ( j · | S n − | ) n − . Proof.
When l = j , σ j ( ⊔ j B n ) = 1, thus lim ε → + σ j (Ω nε,j ) = ( j · | S n − | ) n − . (cid:3) Following Remark 3.3, if we take the domains that we overlap to be B nǫ,δ , we see that when n ≥ nε,j do not attain the supremum of the j -th Steklov eigenvalue in thelimit as ε → j , in contrast to the case in dimension two [9]. Theorem 1.3.
For n ≥ the supremum of the j -th normalized Steklov eigenvalue amongcontractible domains in R n is not achieved in the limit by a sequence of contractible domainsdegenerating to the disjoint union of j identical round balls.Proof. Let ˜Ω nε,j be the domain obtained by overlapping j copies of the domain B nǫ,δ . Then bythe proof of Theorem 3 . ε → + ¯ σ j ( ˜Ω nε,j ) = j n − · σ j ( B nǫ,δ ) · | ∂ B nǫ,δ | n − . By Theorem 3.4, Corollary 1.2, and Theorem 1 . ε → + ¯ σ j ( ˜Ω nε,j ) > lim ε → + ¯ σ j (Ω nε,j ) . (cid:3) We end this section with following corollary.
Corollary 3.5.
For certain j and n , if ε > is sufficiently small, then ¯ σ j (Ω nε,j ) > ¯ σ j ( B n ) .Proof. By Theorem 3 .
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