Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is 8πk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$8\pi k$$\end{document}, which is the best upper bound for the kth\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k^{\text {th}}$$\end{document} area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realizing a weighted Neumann problem as a limit of Steklov problems on perforated domains. For k=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1$$\end{document}, the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.


Introduction
For a compact, connected Riemannian manifold (M, g) of dimension d, with or without C 1 boundary ∂M , the Laplace eigenvalue problem consists in determining all λ ∈ R for which the following eigenvalue problem admits a nontrivial solution: where ∂ n u is the outward normal derivative of u. Similarly, when ∂M = ∅ the Steklov problem consists in determining all σ ∈ R such that the following boundary value problem admits a nontrivial solution: Δ g u = 0 in M, ∂ n u = σu on ∂M. In the present paper we study the relation between σ k (M, g) and λ k (M, g). In order to do so, we introduce the unifying framework of variational eigenvalues associated with a Radon measure. Given a Radon measure μ on M , we define where F k+1 is a (k + 1)-dimensional subspace of C ∞ (M ) ∩ L 2 (M, μ). To the best of our knowledge, variational eigenvalues for Radon measures were first defined to describe Laplacians on fractal sets, see e.g. the survey of Triebel [Tri08]. In the context of spectral bounds and shape optimisation, they first appeared in the work of Kokarev [Kok14] as a relaxation of the optimisation constraint. One should also see the influential work of Korevaar [Kor93] and especially of Grigor'yan-Netrusov-Yau [GNY04] where the spectrum of energy forms is investigated. The variational eigenvalues admit a natural normalisation λ k (M, g, μ) := λ k (M, g, μ) μ(M ) One of the main interest of introducing these variational eigenvalues is that they unify the presentation of several eigenvalue problems. For instance, for μ = dv g the volume measure associated to metric g, λ k (M, g, μ) = λ k (M, g), while for μ = ι * dA g , the pushforward by inclusion of the boundary measure, λ k (M, g, μ) = σ k (M, g). We present results of two types. On one hand, we study variational eigenvalues on their own. In Sect. 3 we establish the necessary functional analysis preliminaries. We define p-admissible measures, which are essentially measures that can be viewed as elements of the dual space (W 1,p (M )) * with certain compactness properties, and give various examples of p-admissible measures. Section 4 is concerned with the properties of variational eigenvalues. For example, we show that the eigenvalues associated with a 2-admissible measure form a discrete unbounded sequence accumulating only GAFA CONTINUITY OF EIGENVALUES 517 The following notations allows us to clarify the statement of our results: where M is a compact surface and Ω is a C 1 domain. For these optimal eigenvalues the results of this section can be summarized as follows. In particular, using the known results on the exact values of Λ k (M ) obtained in [KNPP21,Kar21] respectively, Σ k (S 2 ) = 8πk, Σ k (RP 2 ) = 4π(2k + 1). (1.5)

Optimal isoperimetric inequalities for planar domains.
Because any domain Ω ⊂ R 2 is diffeomorphic to a domain in the sphere S 2 , it follows from (1.5) that σ k (Ω, g 0 ) ≤ 8πk. Following the ideas of [GL21] we show that this inequality remains sharp for planar domains.
Theorem 1.6. Let Ω ⊂ R 2 be a bounded simply-connected domain with C 1 boundary. There exists a sequence Ω ε ⊂ Ω of subdomains, with ∂Ω ⊂ ∂Ω ε , such that In particular, The domains Ω ε are obtained by removing small disks from Ω. In particular, this solves [GP17, Open problem 2] for d = 2.
1.3 Quantitative isoperimetric bounds for σ 1 . Following [FS16,Theorem 4.3], it was suggested in [GP17] that the number of boundary components in a maximizing sequence for Σ 1 (S 2 ) needs to be unbounded. Indeed, let M 0,b be a a compact orientable surface of genus 0 with b boundary components and define Σ 1 (0, b) = sup g σ 1 (M 0,b , g).
Theorem 1.8. For every ε > 0 there exists C > 0 such that for every b 1 and every metric g (1.6) This theorem follows from the more general Theorem 2.1 and the constant C is explicitly computable in terms of ε. It seems unlikely that this bound is sharp, yet there does not seem to be any obvious candidate for the sharp bound. We discuss in more details in Sect. 2 the parts of the proof where sharpness may be lost. For planar domains, Theorem 2.1 also leads to the following bound, which implies that any Σ 1 (R 2 )-maximising sequence of domains with fixed perimeter shrinks to a point. Theorem 1.9. For every ε > 0 there is C > 0 such that for every connected bounded domain Ω ⊂ R 2 with smooth boundary, Remark 1.10. It follows from the seminal work of Fraser-Schoen [FS16] that free boundary minimal surfaces in the unit ball are intimately related to the maximal Steklov eigenvalues. In particular, given an embedded free boundary minimal surface in the unit ball, its coordinates are Steklov eigenfunctions with eigenvalue σ = 1.
In [FL14,Conjecture 3.3], Fraser and Li conjectured that for each free boundary minimal surfaces M properly embedded in B ⊂ R 3 , this Steklov eigenvalue is always the first one, so that in such a case 2 Area g (M ) = H 1 (∂M ) = σ 1 (M, g). We can read Theorem 2.1 in this setting. Let M 0,b be a free boundary minimal surface of genus 0 with b boundary components properly embedded in B ⊂ R 3 by its first Steklov eigenfunctions. Then, for every ε > 0, with the constant C > 0 given by Theorem 1.8, Under the Fraser-Li conjecture, this holds for all free boundary minimal surfaces of genus 0 with b connected boundary component that are properly embedded in the unit ball B ⊂ R 3 . In other words, if the Fraser-Li conjecture is true properly embedded free boundary minimal surfaces of genus 0 with area close to 4π must have a large number of boundary components.
We assume that β ∈ L d/2 (Ω) if d 3, and β ∈ L 1 (log L) 1 (Ω) if d = 2 (see p. 9 for the definition of this space which contains L p , p > 1). If the flat metric on R d is denoted by g 0 , then the eigenvalues of this problem can be understood in the weak sense as in (1.1), as the variational eigenvalues λ k (Ω, g 0 , βdv g0 ).
Theorem 1.11. For any domain Ω ⊂ R d , and any nonnegative 0 For the same family Ω ε , We note that combining the methods of [GHL21, GL21], we could have proved a weaker form of this result, i.e. with β continuous, using an intermediate dynamical boundary value problem. The proof that we present here is more direct, allows for a more singular β, and gives more information on domains that are nearly maximizing λ k .
In order to prove this result, we realise the domains Ω ε by removing tiny balls from Ω whose centres are periodically distributed. The construction is in the spirit of homogenisation theory, with the distinction that the radius of the balls removed is not uniform, but rather varies according to the a continuous approximation of the function β, and is chosen so that the total boundary area tends to ∞ in a controlled way as ε → 0. Variation within periods in homogenisation theory has also been explored in [Pta15]. In our method of proof, the number of boundary components tends to ∞. By Theorem 1.8, this is unavoidable in dimension 2 since we can obtain planar domains with σ 1 as close to 8π as we want. In higher dimension, it is possible to achieve the same result with only one boundary component, see [FS13,GL21].
Finally, we remark that a straightforward modification of our method yields an analogous result for compact Riemannian manifolds, see Remark 6.1 and a similar result for β continuous on closed Riemannian manifolds in [GL21, Theorem 1.1].
Theorem 1.12. For any compact Riemannian manifold (M, g) of dimension d, and , there exists a family Ω ε ⊂ M of domains such that for each k ∈ N, Harmonic extensions of the associated eigenfunctions from Ω to M converge strongly to eigenfunctions of the limit problem in W 1,2 (M ).
For the same family Ω ε , Here, harmonic extensions of the associated eigenfunctions converge weakly to eigenfunctions of the limit problem in W 1,2 (M ).
1.5 Flexibility in the prescription of the Steklov spectrum. Bucur-Nahon [BN21] have recently shown that the Weinstock and Hersch-Payne-Schiffer inequalities are unstable, in the sense that there are simply-connected domains that are very far from the disk-or from a union of k identical disks-with their kth normalised eigenvalue arbitrarily close to 2πk. In fact, they prove the following result.
The domains Ω ε constructed in [BN21] are diffeomorphic to the original domains. They are obtained by a local perturbation of the boundary. We remark that a similar result can be obtained as an application of Theorem 1.11, see Remark 5.4 for details. However, the domains Ω ε obtained this way have many small holes concentrated near the boundary ∂Ω 1 . We further investigate flexibility results for the Steklov spectrum of domains in Euclidean space. In many ways, the Neumann and Steklov problems have similar features. This has led to an investigation of bounds for one eigenvalue problem in terms of the other, see e.g. [HS20,KS68]. It was previously thought that some universal inequalities between perimeter-normalised Steklov eigenvalues and area-normalised Neumann eigenvalues could exist. It is known from [GP10, Section 2.2] that normalised Steklov eigenvalues can be arbitrarily small while keeping the normalised Neumann eigenvalues bounded away from zero. We use Theorem 1.11 to prove that there are also domains with arbitrarily small area-normalised Neumann eigenvalues λ k (Ω, g 0 ), for which the Steklov eigenvalues are bounded away from zero. Theorem 1.14. There exists a sequence of planar domains Ω ε such that the nor-  1.6 Plan of the paper, heuristics, and strategies. The majority of the paper is centred around Theorems 1.2 and 1.11; we either discuss their applications, develop the theory towards their proof and justify some constraints that become apparent in the proof. We note that the proof of both of these theorems are very similar in nature under the scheme that we develop. In Sect. 2, we start by presenting applications of Theorem 1.11, including the proofs of Theorems 1.6 and 1.14. The proofs of Theorems 1.8, 1.9 are independent of the rest of the paper and are also presented there.
In Sect. 3, we present the general framework of variational eigenvalues associated to a Radon measure. This is a unifying framework which allows one to compare different, seemingly unrelated, eigenvalue problems on a manifold. We start with a general description of the setup and give conditions on and examples of measures giving rise to eigenvalues behaving like Laplace eigenvalues. Finally, we obtain continuity of the eigenvalues and eigenfunctions with respect to convergence of the measures in the dual of some appropriate Sobolev space.
An immediate application of the framework presented in Sect. 3, is given in Sect. 5. In particular, we prove that on any surface we can approximate Steklovtype eigenvalues with Laplace eigenvalues associated with a degenerating sequence of metrics, giving as an application a proof of Theorem 1.2.

Notation.
We make here a list of notation that is explicitly reserved throughout the paper.
Manifolds and their domains. Whenever we mention a manifold or a surface without qualification, it may have nonempty boundary, which is always assumed to be C 1 . In any PDE written in strong form, the boundary term may be ignored when the manifold has empty boundary. Closed manifolds and surfaces are compact and without boundary. We reserve the letter M for manifolds. When M has nonempty boundary, we denote by int(M ) the set M \ ∂M .
A domain in a manifold M is a bounded open connected subset of M with C 1 boundary if its boundary is nonempty. We reserve the letters Ω and Υ for domains.
Standard measures and metrics. Let (M, g) be a Riemannian manifold. We denote the volume measure dv g . If there is a canonical metric on M , it is denoted by g 0 . This could be the flat metric on R d or the round metric on the sphere. It is usually a constant curvature metric. If M has a boundary, we write dA g for the boundary measure induced by the metric. It is often useful to recall that dA g := H d−1 ∂M , where H d−1 is the (d − 1)-dimensional Hausdorff measure induced by the metric g on M . We abuse notation and make no distinction between dA g as a measure on ∂M , and the pushforward by inclusion ι * dA g which is a measure on M .
In cases where confusion may arise, if we want to explicitly distinguish the restriction of dv g to a domain Ω ⊂ M we write dv Ω g := (dv g ) Ω, Standard function spaces and capacity. Every vector space under consideration is defined over R. For X a topological vector space, ξ ∈ X * , x ∈ X we denote by ξ, x X := ξ(x) the duality pairing. Since all vector spaces are real, we use this notation to denote an inner product as well, without confusion. For p ∈ [1, ∞] we let p be its Hölder conjugate and for p ∈ [1, d) we let p be its Sobolev conjugate, given respectively by In order to characterise critical scenarios in dimension 2, we will require a generalisation of the usual Lebesgue L p and Sobolev W 1,p spaces. The first spaces we introduce are the Orlicz spaces L p (log L) a , for p 1 and a ∈ R and exp L a for a > 0. For a reference on Orlicz space, see e.g. [BS88, Chapters 4.6-4.8]. The space For p > 1 and a ∈ R, or p = 1, a 0, it can be endowed with the Luxemburg norm under which it is a Banach space. For a > 0, we also define the Orlicz spaces exp L a to be Just like the spaces L p (log L) a , they can be endowed with the Luxemburg norm under which it is also a Banach space. The space exp L 1 serves as a pairing space for L 1 (log L) 1 , see [BS88, Theorem 4.6.5], in the sense that there is C > 0 so that We identify exp L 1 with the dual of L 1 (log L) 1 . For every p 1 and a, ε > 0, we have the relations We also define for p 1 and a ∈ R the Orlicz-Sobolev spaces W 1,p,a (M ) as with the gradient being understood in the weak sense, see [Cia96, Section 2] for this definition. We note that for every p 1, a 0, ε > 0 we have the relations Finally, we will make use of the notion of p-capacity. Given two sets Υ ⊂⊂ Ω M , we write The p-capacity of Υ with respect to Ω is defined as and the p-capacity of Υ as We note that if Ω ∩ ∂M is not empty, we do not require in the definition of the capacity that f vanishes on that set.
Asymptotic notation. We make extensive use throughout the paper of the so-called Landau asymptotic notation. We write f2 → 0. The limits in the last two bullet points will either be as some parameter tends to 0 or ∞ and will be clear from context. The use of a subscript in the notation, e.g.
, indicates that the constant C, or the quantities involved in the definition of the limit may depend on the subscript.

Applications and Motivation.
In this section, we give application of Theorem 1.11 to shape optimisation for the Steklov problem in R 2 , and to spectral flexibility. We also provide the proofs of Theorems 1.8 and 1.9.

Approximation by Steklov eigenvalues.
We start by proving Theorem 1.6 from Theorems 1.2 and 1.11.
Proof of Theorem 1.6. Let Ω ⊂ R 2 be a simply-connected C 1 domain. We know from [Her70, Pet14, KNPP21] that Λ k (S 2 ) = 8πk. Let δ > 0, and g be a smooth metric on S 2 such that such that Let Υ be S 2 with a small disk removed. It is well known that as the radius of that disk goes to 0, the Neumann eigenvalues λ k (Υ, g) converge to λ k (S 2 , g), see [Ann86, Théorème 2]. Thus, removing a small enough disk , Let Φ : Ω → Υ be a conformal diffeomorphism. Since Dirichlet energy is a conformal invariant, the kth Neumann eigenvalue of Υ is equal to the variational eigenvalue λ k (Ω, g 0 , Φ * (dv g )). The homogenisation Theorem 1.11 guarantees the existence of Ω ε ⊂ Ω such that Putting this all back together yields the bound , and by Theorem 1.2 this is in fact an equality.
The exact same proof can be used to obtain the comparison between Steklov and Neumann eigenvalues.
Proof of Theorem 1.14. For δ > 0, proceed as in the proof of Theorem 1.6, but start with Ω ⊂ R 2 such that λ k (Ω) < δ 2 , for instance a very thin rectangle. By Theorem 1.11, one can choose ε in (2.1) small enough that λ k (Ω ε , g 0 ) < δ. This concludes the proof.

Geometric and topological properties of maximising sequences.
In the present section we prove Theorems 1.8 and 1.9 The domains Ω ε constructed in Theorem 1.11, are obtained by removing many tiny balls whose total boundary length tends to +∞. In particular, the length of each boundary component relative to the total length of the boundary tends to zero. We show that any maximizing sequence of domains for Σ 1 (S 2 ) or Σ 1 (R 2 ) exhibits this behaviour. Moreover, for any metric on M 0,b one has the following quantitative relation between the relative length of the longest boundary component and the Steklov spectral defect

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Theorem 2.1. Let (M, g) be a compact Riemannian surface of genus 0 with b boundary components and let L be the length of its longest boundary component.
The first inequality is the trivial statement that H 1 (∂M 0,b ) Lb, the main content of this theorem is the second inequality. One can interpret this result as a quantitative improvement of Kokarev's estimate (1.2). This is the essence of Theorems 1.8 and 1.9 which we now prove using Theorem 2.1.
Proof of Theorem 1.8. We may assume without loss of generality that σ 1 > 4π and therefore that b 3, and that ε is sufficiently small. Exponentiating the leftmost and rightmost side in (2.2) and rearranging yields .
To arrive at (1.6) we use the inequality (1+x) −1 < 1−x+x 2 and choose C depending on ε to be such that for ε small enough.
Proof of Theorem 1.9. Let Ω be a connected bounded domain in R 2 , and C ⊂ ∂Ω be the boundary of the unbounded connected component of R 2 \ Ω. Then, we have that 2 diam(Ω) H 1 (C ) L, where L is the length of the longest boundary component. The proof is completed in exactly the same way as above.
The proof of Theorem 2.1 is based on Hersch's renormalisation scheme [Her70], as well as on a quantitative version of Kokarev's no atom lemma [Kok14, Lemma 2.1].
Let B be the unit ball in R 3 . For ξ ∈ B, Hersch's conformal diffeomorphism Ψ ξ : S 2 → S 2 is defined as A. GIROUARD ET AL. GAFA Lemma 2.2. (Hersch's renormalisation scheme, see [GNP09,Lau21]). Let μ be a measure on S 2 such that for all x ∈ S 2 , μ({x}) 1 2 μ(S 2 ). Then, there exists a unique ξ ∈ B such that the pushforward measure (Ψ ξ ) * μ has its center of mass at the origin. In other words, for j ∈ {1, 2, 3}, the coordinate functions x j : Remark 2.3. In the classical formulation of Hersch's scheme as in e.g. [GNP09] the measure μ is precluded from having points of non-zero mass. In the form presented here the measure μ is allowed to have point masses. The proof is different from the classical topological arguments and can be found in [Lau21].
Given y ∈ S 2 , we define the closed hemisphere For Ω ⊂ S 2 y , recall that we define the capacity of Ω in S 2 y as Lemma 2.4. Let K a ⊂ S 2 y be a closed spherical cap of area a < 2π centred at y ∈ S 2 . The capacity of K a in S 2 y is given by .
Proof. Let Φ : D → S 2 be the stereographic parametrisation of S 2 y . By elementary trigonometry, Let χ a : D → R be the capacitary potential for B(0, r a ), i.e. the radial function defined in polar coordinates (t, θ) as for r a < t 1, It follows by invariance of the Dirichlet energy under conformal transformations that (Φ −1 ) * χ a is the capacitary potential of K a , and thus that .

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Proof of Theorem 2.1. The proof is based on Kokarev's proof of (1.2), keeping a precise track of all quantities involved. Note that the theorem is trivially true when def(M, g) 4π, so we assume that def(M, g) < 4π. Let C ⊂ ∂M be the longest connected component of the boundary and fix y ∈ S 2 . It follows from the Koebe uniformization theorem that there exists a diffeomorphism Φ : M → Ω ⊂ S 2 y , conformal in the interior of M , sending C to the equator, i.e Φ(C ) = ∂S 2 y . Let μ := Φ * ds. be the pushforward of the boundary measure by Φ. The equator carries the length of C : We apply the Hersch renormalisation scheme to the measure μ. By Lemma 2.2, there is a unique ξ ∈ B so that the measure ζ := (Ψ ξ ) * μ has its center of mass at the origin. In other words, we can read from (2.3) that for j ∈ {1, 2, 3}, the functions Using the pointwise identities 3 j=1 x 2 j = 1 and 3 j=1 |∇ g0 x j | 2 = 2, this leads to a strict form of Kokarev's bound from [Kok14]: Because the total area of S 2 is 4π, it follows that the opposite hemisphere S 2 −y is mapped by Ψ ξ to a spherical cap with small area: (2.4) Let z ∈ S 2 be the center of the spherical cap K a = Ψ ξ (S 2 −y ), where a = Area g0 (K a ). The center of the circle ∂K a is κz ∈ B, where 2π(1 − κ) = a < def(Ω, g)/2. The spectral defect is smaller than 4π by hypothesis. Hence, Let π z : R 3 → R correspond to the projection on the subspace Rz. That is, π z (x) := (x · z). Then the measure ρ := (π z ) * ζ = (π z • Ψ ξ ) * μ is supported in the interval (−1, 1) and has an atom of weight μ(∂S 2 y ) = H 1 (C ) located at κ ∈ (0, 1). Because the center of mass of ζ is the origin 0 ∈ B, we have In particular It follows that Let χ a ∈ W 1,2 (S 2 ) be the capacitary potential of K a ⊂ S 2 z , and m χa = 1 H 1 (∂M) S 2 χ a dζ. We can thus use χ a − m χa ∈ W 1,2 (S 2 ) as a trial function for σ 1 (M ) = λ 1 (Ω, g 0 , μ). By Lemma 2.4 . Now, using that χ a ≡ 1 on K a together with (2.5) we get where in the last step we have minimized the quadratic form. Putting all of this together leads to Recall that a = Area g0 (K a ) ≤ def(Ω, g)/2 to finish the proof.
Remark 2.5. As was mentioned in the introduction, it is unlikely that the inequality in Theorem 2.1 is sharp. In its proof, we identify two main arguments where a loss of sharpness may have occured. First, in (2.4), we bound the deficit from below by the area of a single disk, whereas it could be bounded from below by the total area of all b disks in the complement of Ψ ξ (Ω). This would lead to an improvement if all of those disks have comparable size. The arguments of [GL21] suggest that for sequences maximising the first Steklov eigenvalues all disks in the complement will have comparable size. However, we do not have a proof and it might not hold for all domains whose first normalised eigenvalue is close to the maximum. Another loss of sharpness is that the capacitary potential may not be the best trial function for σ 1 . Finding a better trial function would improve the bounds obtained in (2.6).

Admissible Measures and Associated Function Spaces
The goal of this section is to properly define which measures allow for the definition of variational eigenvalues, and to define associated Sobolev-type spaces appropriate for our purpose. At the end of this section, we will provide explicit examples of admissible measures.

Sobolev-type spaces.
Definition 3.1. For 1 p < ∞, M a compact Riemannian manifold and μ a Radon measure on M , we define W 1,p (M, μ) to be the completion of C ∞ (M ) with respect to the norm This completion (3.1) gives rise to an embedding τ μ p : In the classical setting where μ is the volume measure associated to g, the map τ μ p is the natural embedding of the Sobolev space W 1,p (M ) ⊂ L p (M ). If we want to make M explicit, we denote the embedding operator τ μ p,M . Since M is compact W 1,p (M, μ) ⊂ W 1,q (M, μ) whenever p q. For 1 < p < ∞, the closed unit ball in W 1,p (M, μ) is clearly weakly compact so that W 1,p (M, μ) is a reflexive Banach space.

Convention.
We adopt the following conventions in order to make the notation a bit lighter for spaces and operators that appear often. We write L p (M ) for L p (M, dv g ), L p (∂M ) for L p (M, dA g ) and W 1,p (M ) := W 1,p (M, dv g ). In general, the measure μ may be omitted from the notation when it is the natural volume measure given by the Riemannian metric, for instance as λ k (M, g) := λ k (M, g, dv g ).
Denote the average of a function f ∈ L 1 (M, μ) by Definition 3.2. We say that a Radon measure μ supports a p-Poincaré inequality if there is We denote by K p,μ the smallest such number K.
For general measures, the space W 1,p (M, μ) could be very different from the Sobolev space W 1,p (M ) and solutions to (weak) elliptic PDEs in those spaces could lack the natural properties one expects from them. For that reason we restrict ourselves to a particular class of admissible measures, first introduced in [KS20] for d = p = 2, see also [Kok14] for a similar definition. It is clear from that definition that dv g and the boundary measure dA g are padmissible for all p ∈ (1, ∞). The aim of the rest of this subsection is to prove the following two theorems. The first one gives a characterisation of p-admissible measures. The second one essentially says that when μ is a p-admissible measure there is an isomorphism between W 1,p (M, μ) and W 1,p (M ). Their proofs are intertwined but they are better stated separately for ease of reference. Theorem 3.5. Let p ∈ (1, ∞) and suppose that μ is not supported on a single point and supports a p-Poincaré inequality. There exists c p,μ , C p,μ > 0 so that for every In particular, the completions W 1,p (M, μ) and W 1,p (M ) of C ∞ (M ) are isomorphic.
We start by proving the first inequality in Theorem 3.5.
Proposition 3.6. Let p ∈ (1, ∞) and μ be a Radon measure on M supporting a p-Poincaré inequality. Suppose furthermore that μ is not supported on a single point. Then, there is c p,μ > 0 such that for all f ∈ C ∞ (M ) In particular, the identity map on C ∞ (M ) extends to a bounded operator T μ p : W 1,p (M ) → L p (M, μ).
Proof. If p > d, this follows from the boundedness of the map W 1,p (M ) → C(M ) → L p (M, μ). Suppose then that p d. Since μ is not supported on a single point, supports a p-Poincaré inequality and points have vanishing p-capacity, this means that μ has no point masses.
We proceed in a similar manner to the proof of [Kok14, Lemma 2.2] where d = p = 2 and μ is a probability measure. For any Ω ⊂ M with μ(Ω) > 0, define K p, * (Ω) via

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Let f be a smooth function supported on Ω and assume that μ(Ω) p/p μ(M ) −p 2 −p . From this assumption and Hölder's inequality, We therefore have that for such Ω .
Since μ has no point masses there is a finite covering of M with domains {Ω j } such that with associated smooth partition of unity ρ p j . Then, for all f ∈ C ∞ (M ), Summing up those inequalities proves as we claimed that As an immediate corollary, we get the necessity in Theorem 3.4. for measures ξ and μ. Note that μ − μ(M ) ξ(M ) ξ vanishes on constant functions, if they are shown to be in W 1,p (M ) * existence of a solution is easily guaranteed; we are specifically interested in estimating its norm in terms of trace operators and the Poincaré constants K p . We require a generalisation of the Lax-Milgram theorem to Banach spaces. It is clearly weakly nondegenerate if we restrict ourselves to functions of zero mean.
The following lemma establishes the Brezzi condition.
Lemma 3.9. Let M be a Riemannian manifold, p ∈ (1, ∞) with Hölder conjugate p = p/(p − 1) and μ a Radon measure supporting a p-Poincaré inequality and such that τ μ p is compact. Then, there exists κ > 0 such that for all ϕ ∈ W 1,p (M, μ), Proof. Towards a contradiction, we assume that such a κ does not exist. This implies the existence of a sequence ϕ n ∈ W 1,p (M, μ) such that We first prove that if (3.2) goes to 0, then |∇ϕ n | L p (M ) does as well. Since μ supports a p-Poincaré inequality, we have that (3.3) By density of smooth vector fields and duality, we have that where Γ(T M) is the set of smooth vector fields on M . By the L p -Helmholtz decomposition of vector fields on C 1 domains, see e.g.
Here ν is the normal vector to the boundary. By the divergence theorem As a result, one has Thus, from (3.3), we see that if (3.2) holds then |∇ϕ n | L p → 0. Therefore, we have that ϕ n is a sequence in W 1,p (M, μ) so that This means that ϕ is constant a.e., and since τ μ p extends the identity on C ∞ (M ), ϕ is also μ-a.e. constant. But then, ϕ − m ϕ,μ L p (M,μ) = 0, a contradiction.
Moreover, if μ supports a p-Poincaré inequality, ϕ ξ,μ satisfies Remark 3.11. The condition on the existence of T ξ,μ p is later shown to always be satisfied for p-admissible measures.

Proof. Let
and consider the bilinear form a : X p × X p → R given by It follows from Lemma 3.9 that a satisfies the Brezzi condition, and it is weakly nondegenerate on X p × X p . Furthermore, since μ has finite volume 1 ∈ L p (M, μ). This means that L := (T ξ,μ p ) * 1 ∈ X * p and for f ∈ X p By the Banach-Nečas-Babuška theorem there exists a unique ϕ μ,ξ ∈ X p so that for all f ∈ X p , a(ϕ ξ,μ , f) = L(f ). For f ∈ W 1,p (M, ξ), we obtain the identity (3.4) by noticing that formula (3.6) extends from X p to W 1,p (M, ξ) and computing We turn our attention to estimate (3.5). As in the proof of Lemma 3.9, we have that there exists C p > 0 so that From the weak characterisation of ϕ ξ,μ that for any f ∈ C ∞ (M ), Since the left-hand side is invariant under addition of a constant to f , we may assume that M f dξ = 0. By Hölder's inequality, if μ supports a p-Poincaré inequality we have that where T μ p is bounded from Proposition 3.6. Inserting this estimate into (3.8) and (3.7) completes the proof.
We can now prove that the spaces W 1,p (M ) and W 1,p (M, μ) are isomorphic.  (M, μ). Then, there is C p,μ such that If moreover μ supports a p-Poincaré inequality, then we can take Before carrying on with the proof, we note that we have proved in Proposition 3.6 that supporting a p-Poincaré inequality implies that T μ p is bounded, so that this proposition implies the second bound in Theorem 3.5.

Proof.
We have that From Lemma 3.10 with ξ = dv g , there is ϕ ∈ W 1,p (M ) such that The estimate on C p,μ can be then be read from the bound on ∇ϕ L p (M ) obtained in Lemma 3.10 under the p-Poincaré inequality condition.
We can now prove sufficiency in Theorem 3.4.  We finally write the two following propositions that allow us to rewrite Lemma 3.10 with weaker conditions, for ease of reference. The first proposition indicates that p-admissibility is a monotone condition.
Proposition 3.14. Suppose that T μ p : W 1,p (M ) → L p (M, μ) is bounded. Then for all q > p, T μ q is compact. In particular, μ is q-admissible and admissibility is a monotone condition.
Proof. If q > d, T μ q is compact since the embedding W 1,q (M ) → C(M ) is compact, so we suppose now that p < q d. Compactness of T μ q follows from general interpolation theory. Given two compatible normed vector spaces, i.e. spaces X 0 , X 1 that are both subspaces of a larger topological vector space V , Peetre's K-functional is defined on f ∈ X 0 + X 1 as For 0 < θ < 1 and 1 q < ∞, let (X 0 , X 1 ) θ,q be the interpolation space between X 0 and X 1 (see [BS88, Chapter 5]): We use the interpolation theorem found in [CF89, Theorem 2.1] which states the following. Given Y 0 , Y 1 compatible Banach spaces; X 0 , X 1 Banach spaces such that X 1 is continuously embedded in X 0 and T is a linear operator such that T : X 0 → Y 0 is bounded and the restriction T : X 1 → Y 1 is compact. Then, for 0 < θ < 1 and 1 q < ∞, the operator T : (X 0 , X 1 ) θ,q → (Y 0 , Y 1 ) θ,q is compact.
On the other hand, taking X 0 = W 1,p (M ) and X 1 = W 1,r (M ) it follows from [Bad09, Theorem 6.2, Corollary 1.3 and Remark 4.3], the later remark treating the case with C 1 boundary, that (X 0 , X 1 ) θ,q = W 1,q (M ). Therefore, the interpolation theorem tells us that T μ q : W 1,q (M ) → L q (M, μ) is indeed compact. We therefore rewrite the statement of Lemma 3.10 in the following way.

Examples and admissibility criteria.
Let us now give a few examples of admissible measures, as well as a local criterion that characterises them. We start with basic examples.
Example 3.17. On a smooth compact manifold M with C 1 boundary, the volume measure dv g is p-admissible for every p ∈ (1, ∞), as is the pushforward by inclusion of the boundary measure ι * dA g . Any linear combination of them is also p-admissible.

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A. GIROUARD ET AL. GAFA Example 3.18. It follows from the definition of the capacity that measures supported on a set of p-capacity zero do not support a p-Poincaré inequality, and as such are not admissible.
We now explore the edge cases of admissibility. We provide those examples for p = 2 since that is the context where they will be relevant. This last example allows us to obtain the weakest integrability condition on β in Theorem 1.11. We need to introduce the following characterisation of compactness beforehand. Maz'ya's compactness criterion [Maz11, Section 11.9.1] states that T μ p is compact if and only if The isocapacitary inequality [Maz11, Equations 2.2.11 and 2.2.12] states that for every Υ ⊂ M , with Vol g (Υ) Vol g (M )/2 that (3.10) Example 3.19. Let 0 β ∈ L 1 (log L) 1 (M ) (for d = 2) or 0 β ∈ L d/2 (M ) (for d > 2) be a positive density and μ = βdv g . Then, μ is admissible. For 1 p < d/2 (for d ≥ 3), and for p = 1 (when d = 2), there exists β ∈ L p (M ) such that βdv g is not an admissible measure. We split the proof of these claims in a few cases. It is easy to see that β ∈ L p and that the measure βdv g fails Maz'ya's compactness criterion, hence T μ is not compact and μ is not admissible. Case (ii): p = 1, d = 2. Similarly to the previous case, for some 0 < δ < 1 let This time, β ∈ L 1 (M ) but we can see that T μ is not even bounded on W 1,2 (M ), so certainly not admissible. Indeed, consider the function f (y) = − log(r y ) a . It is a standard computation to see that f ∈ W 1,2 (M ) when a < 1/2. On the other hand, choosing a = δ/2, for some ε > 0 M f 2 β dv g M ε 0 dr r log(1/r) = +∞.

CONTINUITY OF EIGENVALUES 539
Case (iii): d = 2 and β ∈ L 1 (log L) 1 (M ) or d 3 and p d/2. Suppose without loss of generality that |β| 1 a.e. For d = 2, it follows from Jensen's inequality with the convex function ϕ(x) = x log x that for any Υ ⊂ M , In other words, .
Taking m = r −1 , we have that for r small enough m m ε and we deduce that for every ε > 0, lim r→0 + sup β L 1 (log L) 1 (Υ) : diam Υ r ε so that by Maz'ya's compactness criterion T μ is compact and μ is admissible. For d > 2, it follows from Hölder's inequality that Therefore, inserting in Maz'ya's criterion along with the isocapacitary inequality The same argument as earlier but with the sets Υ m = Υ ∩ β d/2 < m shows that this limit converges to 0, so that μ is admissible.

Variational Eigenvalues
Eigenvalue convergence results are ubiquitous in the literature and the proofs of a large number of them follow similar steps. In the present section we formulate these steps explicitly in sufficient generality to allow direct application to many natural eigenvalue problems, including both the Steklov and Laplace problems.

4.1
Variational eigenvalues associated to a Radon measure. We generalise to higher dimension the definition of eigenvalues associated to a measure, introduced in [Kok14] for surfaces. Let (M, g) be a compact Riemannian manifold. For a Radon measure μ on M , we define the variational eigenvalues λ k (M, g, μ) in the following way. For any f ∈ C ∞ (M ) such that f ≡ 0 in L 2 (M, μ), we define the Rayleigh quotient R g (f, μ) by The eigenvalues λ k (M, g, μ) are then given by where the infimum is taken over all (k + 1)-dimensional subspaces F k+1 ⊂ C ∞ (M ) that remain (k + 1)-dimensional in L 2 (M, μ). A natural normalisation for these eigenvalues is  We revisit the examples from the previous section and how they give rise to natural eigenvalues.
Example 4.2. If M is closed and μ = dv g , the volume measure associated to g, then λ k (M, g, dv g ) are eigenvalue of the Laplace operator. In this case T μ is the usual embedding W 1,2 (M ) ⊂ L 2 (M ). If M is a compact manifold with boundary, then λ k (M, g, dv g ) are Neumann eigenvalues.  where ∂ n ± are normal derivatives in opposite directions on Σ.
Example 4.5. For β > 0, if μ = ι * dA g + β dv g , then λ k (M, g, μ) are eigenvalues associated with a dynamical boundary value problem, see [BF05,GHL21], given by The corresponding Laplace-type operator acts in L 2 (M ) ⊕ L 2 (∂M, dA g ) and is not densely defined. From the perspective of variational eigenvalues, this does not cause any problem. By Example 3.19 μ is admissible, so that the spectrum is indeed discrete. Proof. Let ε > 0 be arbitrary. Let F ⊂ C ∞ (M ) be a (k + 1)-dimensional subspace that remains (k + 1)-dimensional in L 2 (M, μ) and such that Convergence μ n * − μ implies that for large n the subspace F is (k + 1)-dimensional in L 2 (M, μ n ) and As a result, for large n one has For many applications it is important to establish continuity of eigenvalues. To the best of the authors' knowledge there is no sufficiently general condition that guarantees continuity of λ k (M, g, μ) which can be verified in an efficient manner in our current setting. As an example, we note that all examples of convergence covered in the present paper fail the integral distance convergence criterion given in [Kok14, Section 4.2]. Many stronger convergence criteria exists, see e.g. [AP21, BLLdC08, BL07], however they generally require explicit knowledge of some transition operators between Hilbert spaces, which usually means having explicit information about the eigenfunctions. Our goal is to obtain synthetic criteria for eigenvalue and eigenfunction convergence which depends only on the measures μ n , and potentially on domains Ω n on which it is supported.
Let Ω n ⊂ M be a sequence of domains viewed as Riemannian manifolds with the metric induced on M , and {μ n : n ∈ N}, μ be Radon measures so that supp(μ n ) ⊂ Ω n . We use the same notation g, μ n for their restrictions to Ω n . Suppose that (M1) μ n * − μ and Vol g (M \ Ω n ) → 0; (M2) the measures μ, μ n are admissible for all n; (M3) there is an equibounded family of extension maps J n : W 1,2 (Ω n , μ n ) → W 1,2 (M, μ n ).

CONTINUITY OF EIGENVALUES 543
The condition (M2) guarantees the existence of the μ n -orthonormal collection of eigenfunctions f n j ∈ W 1,2 (Ω n , μ n ) associated with λ j (Ω n , g, μ n ). In any situation where Ω n = M for all n, the third condition and the volume part of the first condition are automatically satisfied. The map J n is often built using harmonic extensions, and the collection J n f n j remains μ n orthonormal. We now describe two conditions for the eigenfunctions.
(EF2) For every j, k ∈ N, the functions f n j , f n k satisfy where δ jk is the Kronecker delta.
Condition (EF2) implies that f n j : n ∈ N is bounded in W 1,2 (M, μ), so that up to a subsequence, f n j f j weakly in W 1,2 (M, μ) and λ j (Ω n , g, μ n ) → λ j for some λ j 0.
Condition (EF1) implies that the functions f j are eigenfunctions associated with (M, g, μ) with the corresponding eigenvalues λ j . At this point it is unclear whether λ j is indeed the j-th eigenvalue λ j (M, g, μ). This will follow from condition (EF2), which says essentially that the eigenfunctions do not lose mass in the limit. We formalize this procedure in the following proposition. Proof. From the definition via Rayleigh quotient, we see that for all j and n, λ j (Ω n , g, μ n ) λ j (M, g, μ n ). By Proposition 4.7 along with Condition (M1), we have that up to a subsequence λ j (Ω n , g, μ n ) → λ j λ j (M, g, μ). For each fixed j, we have In view of condition (M3), the first term on the right-hand side converges to 1. By condition (EF2) there exists C > 0 such that for all n, J n 2 f n j 2 W 1,2 (Ωn,μn) C(λ j (Ω n , g, μ n ) + 1).
This means that the sequence f n j : n ∈ N is bounded in W 1,2 (M, μ) so that up to a subsequence, there exists f j such that J n f n

First Examples of Spectrum Convergence
In this section we collect several applications of the setup presented in the previous section. Most of the results in this section are generalisations of known results either to a manifold context or to higher dimensions. 5.1 Convergence for L p densities. The case d = 2, β n ∈ L p , p > 1 has previously appeared in [KNPP20, Lemma 6.2].
Proposition 5.1. Let β n be a sequence of non-negative densities converging in L d 2 (log L) a (M ) to a non-negative density β, where a = 0 for d 3 and a = 1 for d = 2. Then λ k (M, g, β n dv g ) → λ k (M, g, β dv g ) as n → ∞.
We deduce that β n dv g → βdv g in W 1, d d−1 (M ) * , so that by Proposition 4.11 the eigenvalues converge. For d = 2, proceed the same way but with the pairing of the spaces exp L 1 (M ) and L 1 (log L) 1 (M ), along with the optimal Sobolev embedding W 1,2,−1/2 (M ) → exp L 1 (M ), see [Cia96, Example 1].

Approximation of eigenvalues of measures supported on a hypersurface.
Let (M, g) be a compact Riemannian manifold. Let Σ ⊂ M be a compact, not necessarily connected, codimension 1 smooth submanifold without boundary and ρ ∈ C(Σ) be a non-negative density on Σ. Assume Σ = Σ i Σ b , where Σ i ∩ ∂M = ∅ and Σ b is either empty or coincides with ∂M . Let N ε,i be an ε-tubular neighbourhood of Σ i . For sufficiently small ε the exponential map exp Σi can be used to identify N ε,i with N ε Σ i , the ε-ball in the normal bundle of Σ i . Similarly, if Σ b = ∂M is not empty, its ε-tubular neighbourhood N ε,b can be identified with Σ b × [0, ε] using exp Σb . If n is an outward unit normal then we define otherwise.
The next theorem says that we can approximate the eigenvalues of weighted Steklov or transmission problems as in Example 4.4 using weighted Laplace eigenvalues. When Σ = Σ b = ∂M , our construction is similar to the one found for domains in R d in [LP15].
Proof. We give the proof in the case Σ = Σ i , the other case is analogous. The conditions (M1)-(M3) are obviously satisfied. We claim that for any u ∈ W 1,1 (M ) one has M uρ ε dv g − Σ uρ dA Σ g C u W 1,1 (Nε) (5.1) for ε small enough. In particular, for any p > 1 and u ∈ W 1,p (M ) one has by Hölder's inequality ρ ε dv g − ρdA Σ g , u Vol(N ε ) 1/p u W 1,p (M ) , With the notation introduced above, we may now state the main theorem of this section.
where μ is defined in (6.2) and the associated eigenfunctions extended to M converge strongly in W 1,2 (M ). The Neumann eigenvalues satisfy The proof of Theorem 6.2 is split into two parts: Proposition 6.3 where the convergence of the Neumann eigenvalues is shown and Proposition 6.5 where we prove convergence of the Steklov eigenvalues. In both cases, we prove that the associated measures converge in W 1,p (M ) * , for all p > 1. In the construction of perforated sets, we already have that conditions (M1)-(M3) are satisfied, so that by Proposition 4.11, this is enough to obtain eigenvalue and eigenfunction convergence. For the Steklov problem, we observe that Condition (4.2) follows directly from [GHL21, Lemma 12] so that strong convergence of the eigenfunctions also follows if we prove the appropriate convergence of the measures. We note that (6.3) could be deduced by an appropriate modification of the proofs in [RT75] or [AP21], however this would require introducing new concepts whereas the results from Sects. 3 and 4 can prove both convergence of the Neumann and Steklov eigenpairs at the same time. This also puts an emphasis on the fact that it is achieved for the same domains. Proposition 6.3. As ε → 0, the measures dx Ω ε converge to dx M in W 1,p (M ) * for every p ∈ (1, ∞). In particular, the Neumann eigenpairs for Ω ε converges to those of Ω.

Convergence of the Steklov eigenpairs.
Before proving convergence of the Steklov eigenpairs, we require the following useful lemma.
Proof. By density of smooth functions in W 1,p (B(0, R)) it is sufficient to prove the inequality for smooth f . Let f ∈ W 1,p (B(0, R)) be the radially constant function given by f (ρ, θ) = f (r, θ), since p < d we can assign any value of f at 0. Set F := f − f ∈ W 1,p (B(0, R)) and observe that F vanishes on ∂B(0, r), and that ∂ ρ F = ∂ ρ f . We directly compute that (6.4) To conclude, we will bound the norm of F with a radial Friedrichs' inequality. By simple integration and Hölder's inequality we have that for every ρ ∈ (0, R) and θ ∈ S Inserting this estimate into (6.5) and then (6.4) yields our claim.
The main purpose of this section is to prove the following proposition Proposition 6.5. As ε → 0, the measures μ ε α → μ α in W 1,p (M ) * for all p > 1.