Perturbed eigenvalues of polyharmonic operators in domains with small holes

Abstract

We study singular perturbations of eigenvalues of the polyharmonic operator on bounded domains under removal of small interior compact sets. We consider both homogeneous Dirichlet and Navier conditions on the external boundary, while we impose homogeneous Dirichlet conditions on the boundary of the removed set. To this aim, we develop a notion of capacity which is suitable for our higher-order context, and which permits to obtain a description of the asymptotic behaviour of perturbed simple eigenvalues in terms of a capacity of the removed set, in dependence of the respective normalized eigenfunction. Then, in the particular case of a subset which is scaling to a point, we apply a blow-up analysis to detect the precise convergence rate, which turns out to depend on the order of vanishing of the eigenfunction. In this respect, an important role is played by Hardy–Rellich inequalities in order to identify the appropriate functional space containing the limiting profile. Remarkably, for the biharmonic operator this turns out to be the same, regardless of the boundary conditions prescribed on the exterior boundary.

1 Introduction

The aim of the present work is to study perturbations of the eigenvalues of the polyharmonic operator $(-\Delta )^m$, $m\ge 2$, when from a given bounded domain $\Omega \subset {\mathbb {R}}^N$ an interior compact set K is removed, thus introducing a singular perturbation. We focus on the case in which K is small, in the sense that its capacity is asymptotically near 0, with respect to a notion of capacity suitably developed for our higher-order setting. More specifically, for $m\ge 2$ we consider the eigenvalue problems

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^mu=\lambda u&{}\text{ in }\;\Omega ,\\ u=\partial _nu=\cdots =\partial _n^{m-1}u=0&{}\text{ on }\;{\partial \Omega }, \end{array}\right. }\;{\text{ resp. }}\;{\left\{ \begin{array}{ll} (-\Delta )^mu=\lambda u&{}\text{ in }\;\Omega ,\\ u=\Delta u=\dots =\Delta ^{m-1} u=0&{}\text{ on }\;{\partial \Omega }, \end{array}\right. }\nonumber \\ \end{aligned}$$

(1.1)

with Dirichlet and Navier boundary conditions (BCs) respectively, and, given a compact set $K\subset \subset ~\!\Omega$, we are interested in the corresponding eigenvalue problems in case K is removed from $\Omega$, that is

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^mu=\lambda u&{}\quad \text{ in }\;\Omega \setminus K,\\ u=\partial _nu=\cdots =\partial _n^{m-1}u=0&{}\quad \text{ in }\;\partial (\Omega \!\setminus \!K), \end{array}\right. } \end{aligned}$$

(1.2)

in the Dirichlet case, and

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^mu=\lambda u&{}\quad \text{ in }\;\Omega \setminus K,\\ u=\Delta u=\dots =\Delta ^{m-1} u=0&{}\quad \text{ on }\;{\partial \Omega },\\ u=\partial _nu=\cdots =\partial _n^{m-1}u=0&{}\quad \text{ on }\;\partial K, \end{array}\right. } \end{aligned}$$

(1.3)

where, instead, Navier BCs on ${\partial \Omega }$ are considered. Note that in both cases we always deal with Dirichlet BCs on ${\partial K}$. The goal is to investigate spectral stability and sharp asymptotic estimates for the eigenvalues of problems (1.2) and (1.3) when K vanishes in a capacitary sense.

Qualitative properties of solutions to higher-order problem are deeply related to the boundary conditions that one prescribes. The most common ones in the literature are Dirichlet BCs

$$\begin{aligned} u=\partial _nu=\cdots =\partial _n^{m-1}u=0\qquad \text{ on }\;\,{\partial \Omega }, \end{aligned}$$

(1.4)

and Navier BCs

$$\begin{aligned} u=\Delta u=\cdots =\Delta ^{m-1}u=0\qquad \text{ on }\;\,{\partial \Omega }. \end{aligned}$$

(1.5)

Indeed, from the point of view of the applications, they correspond to the simplest Kirchhoff-Love models of a thin plate, either clamped or hinged at the boundary, respectively in the Dirichlet and the Navier case.

While the existence and regularity theory for linear problems is essentially the same in both cases (see e.g. [17]), however solutions have relevant differences, even when $\Omega$ is a smooth domain. The most striking and famous one is regarding positivity. In the Navier case the solution inherits its sign from the data, since one can decouple the problem into a system of second-order equations, for which a maximum principle holds. Instead, positivity preserving is in general lost in the Dirichlet case, even for smooth and convex domains, except for peculiar situations in which one can rely on a global analysis of the Green function, such as for the case of the ball and its smooth deformations, see [17, 19]. On the other hand, functions which undergo Dirichlet BCs can be trivially extended by 0 outside the domain, so that the extension continues to belong to the same higher-order Sobolev space, while this is not true anymore for solutions of Navier problems because of possible jumps on ${\partial \Omega }$ of the normal derivative. Motivated by these arguments, we investigate perturbations of the eigenvalues of $(-\Delta )^m$ in both context of Dirichlet and Navier BCs on ${\partial \Omega }$. Since we rely more on extension properties rather than positivity issues, our analysis will be harder in the Navier case.

In the second-order case (i.e. $m=1$), spectral stability under removal of small (condenser) capacity sets is proved in [10] in a very general context, see also [7] and [14]. More specifically, in [10] it is shown there that the function $\lambda (\Omega \setminus K)-\lambda (\Omega )$ is differentiable with respect to the capacity of the removed set K relative to $\Omega$. A sharp quantification of the vanishing order of the variation of simple eigenvalues is given in [2], when concentrating families of compact sets are considered: the precise rate of convergence is asymptotic to the u-capacity associated to the limit eigenfunction u (see [7, Definition 2.1] and [10, (14)] for the notion of u-capacity) and sharp asymptotic estimates are given in terms of the diameter of the removed set if the limit set is a point in ${\mathbb {R}}^2$, and either if the eigenfunction does not vanish there, or in case of specific concentrating sets such as disks or segments. Asymptotic estimates of u-capacities and eigenvalues of the Dirichlet Laplacian, on bounded planar domains with small holes of the more general form $\varepsilon \overline{\omega }$ with $\omega$ a bounded domain and $\varepsilon \rightarrow 0$, are given in [1]. In both [2] and [1], a tool that helps to provide precise asymptotic estimates in dimension two is given by elliptic coordinates, which allow rewriting the equations satisfied by the capacitary potentials in a rather explicit way and which however do not have a simple analogue in higher dimensions. In the complementary case $N\ge 3$, an approach based on a blow-up argument is used in [13] to derive sharp asymptotic estimates of the u-capacity, and consequently of the eigenvalue variation, for general families of sets which may also concentrate at the boundary. This method has been applied also to fractional problems in [3].

For the higher order setting $m\ge {2}$, asymptotic expansions of eigenvalues of biharmonic operators under removal of a family of sets which are uniformly vanishing to a point $\{x_0\}$ have been obtainsed in [8, 21, 22]. All these papers deal with the two-dimensional case and only Dirichlet boundary conditions, both on ${\partial \Omega }$ and on ${\partial K}$, are considered. The main difference with the corresponding two-dimensional second-order problem, is that the limiting problem involves the punctured domain $\Omega \setminus \{x_0\}$. In [8] formal recursive asymptotic expansions are found in the nondegenerate case, namely when the gradient of the corresponding eigenfunction does not vanish at $x_0$, as well as in the degenerate case. In the former case, these expansions are justified in a suitable functional setting which makes use of weighted Sobolev spaces, named after Kondrat’ev, in order to deal with the point constraint. On the other hand, motivated by the study of MEMS devices, in [21], the asymptotic behaviour of eigenpairs is formally obtained, using the method of matching asymptotic expansions. A more delicate situation is taken into account in [22], when both the removed subdomain is vanishing, as well as the biharmonic part of the operator, provided a second-order term is introduced in the equation. In all these works, the asymptotic expansions of the perturbed eigenvalues are of logarithmic kind, fact that recalls the expansion in the two-dimensional case for the Laplace operator given in [2, Theorem 1.7]. We note however that, unlike what happens for the second order problem, capacities cannot play there the role of perturbation parameters, since in dimension 2 the higher order capacity of a point (defined as in (1.11)) is different from zero; this is also the reason why the limiting problem is formulated in the punctured domain. We mention that the spectral behavior of higher-order elliptic operators upon domain perturbation is investigated also in [5] for Dirichlet, Neumann and intermediate boundary conditions.

The first aim of the present paper is a rigorous description of the asymptotic behaviour of the perturbed eigenvalues for polyharmonic operators $(-\Delta )^m$ for any $m\ge 2$ and for a large class of removed sets, in the spirit of [2, 3, 13]. Since we deal with sets of vanishing capacities, we are focused on the high dimensional case $N\ge 2m$. Furthermore, as second important objective, we investigate whether and how different boundary conditions on ${\partial \Omega }$ affect the analysis. As already remarked, in the present work we consider Dirichlet boundary conditions on ${\partial K}$. In order to have a complete picture of the influence of the boundary conditions, the complementary situation of Navier BCs on ${\partial K}$ should be addressed. However, the techniques developed in the present work strongly rely on extension properties which are characteristic of the Dirichlet case, so that a different approach should be devised to treat the Navier case on ${\partial K}$. We plan to address this in a future work.

In order to give the precise statements of the main results, we first describe the functional setting and the notation we are going to use throughout the paper.

Notation We denote the normal derivative of the function u by $\partial _nu$. For a set $D\subset {\mathbb {R}}^N$, ${\mathcal {U}}(D)$ denotes some open neighbourhood of D, $C^\infty _0(D)$ is the space of the infinitely differentiable functions which are compactly supported in D, and $L^p(D)$ with $p\in [1,+\infty ]$ is the space of p-integrable functions. The norm of $L^p(D)$ is denoted simply by $\Vert \cdot \Vert _p$ whenever the domain is clear from the context. For every $m\in {\mathbb {N}}$ and $u:D\rightarrow {\mathbb {R}}$ with $D\subset {\mathbb {R}}^N$, we denote as $D^mu$ the tensor of m-th order derivatives of u and define $|D^mu|^2=\sum _{|\alpha |=m}|D^\alpha u|^2$, where $|\alpha |$ is the length of the multi-index $\alpha$.

The symbol $\lesssim$ is used when an inequality is true up to an omitted structural constant, and we write $f={\mathcal {O}}(g)$ (resp. $f={\scriptstyle {\mathcal {O}}}(g))$ as $x\rightarrow x_0$ when there exists a constant $C>0$ such that $|f(x)|\le C|g(x)|$ in a neighbourhood of $x_0$ (resp. $\frac{f(x)}{g(x)}\rightarrow 0$ as $x\rightarrow x_0$).

1.1 The functional setting

Let $\Omega$ be a bounded smooth domain in ${\mathbb {R}}^N$. In order to treat at once different boundary conditions on ${\partial \Omega }$, i.e. the settings of problems (1.2) and (1.3), we introduce the following notation. For $m\ge 2$ the set $V^m(\Omega )\subset H^m(\Omega )$ is defined either as

$$\begin{aligned} V^m(\Omega ):=H^m_0(\Omega ) \end{aligned}$$

in case Dirichlet boundary conditions (1.4) are prescribed on ${\partial \Omega }$, where $H^m_0(\Omega )$ is the closure in $H^m(\Omega )$ of $C^\infty _0(\Omega )$, or by

$$\begin{aligned} V^m(\Omega ):=H^m_\vartheta (\Omega ) \end{aligned}$$

if Navier boundary conditions (1.5) are assumed on ${\partial \Omega }$. Here $H^m_\vartheta (\Omega )$ is the closure in $H^m(\Omega )$ of the space

$$\begin{aligned} C^m_\vartheta ({\overline{\Omega }}):=\left\{ u\in C^m({\overline{\Omega }})\ \big |\ \Delta ^ju|_{{\partial \Omega }} =0\,\text{ for } \text{ all }\,0\le j<\tfrac{m}{2}\right\} \end{aligned}$$

and it can be characterized as

$$\begin{aligned} H^m_\vartheta (\Omega )=\left\{ u\in H^m(\Omega )\ \big |\ \Delta ^ju|_{{\partial \Omega }}=0\ \text{ in } \text{ the } \text{ sense } \text{ of } \text{ traces }\ \text{ for } \text{ all }\ 0\le j<\tfrac{m}{2}\right\} . \end{aligned}$$

Note that for $m=2$ we have $H^2_\vartheta (\Omega )=H^2(\Omega )\cap H^1_0(\Omega )$. In both cases $V^m(\Omega )$ is a closed subspace of $H^m(\Omega )$. Moreover, for a bounded domain $\Omega \subset {\mathbb {R}}^N$, the norms

$$\begin{aligned} \Vert \cdot \Vert _{H^m(\Omega )}:=\sum _{|\alpha |\le m}\Vert D^\alpha \cdot \Vert _{L^2(\Omega )} \end{aligned}$$

(with the multi-index notation) and

$$\begin{aligned} \Vert \nabla ^m\cdot \Vert _{L^2(\Omega )},\qquad \text{ where }\quad \nabla ^mf:={\left\{ \begin{array}{ll} \Delta ^{\frac{m}{2}}f&{}\quad \text{ for }\;m\;\text{ even },\\ \nabla \Delta ^{\frac{m-1}{2}}f&{}\quad \text{ for }\;m\;\text{ odd }, \end{array}\right. } \end{aligned}$$

are equivalent on both $H^m_0(\Omega )$ and $H^m_\vartheta (\Omega )$, see e.g. [17, Theorem 2.2] for the Dirichlet case and [18] for the Navier case. In particular, there exists a constant $C=C(N,m,\Omega )>0$, depending only on N, m, and $\Omega$, such that

$$\begin{aligned} \Vert u\Vert _{H^m(\Omega )}\le C\Vert \nabla ^m u\Vert _{L^2(\Omega )}\quad \text {for all }u\in V^m(\Omega ). \end{aligned}$$

(1.6)

Note also that in the Dirichlet case all boundary conditions are stable, and therefore they are all included in the definition of the space $H^m_0(\Omega )$; on the other hand, only the first half of the Navier conditions are stable, while the boundary conditions $\Delta ^ju|_{{\partial \Omega }}=0$ for $\tfrac{m}{2}\le j\le m-1$ are natural and thus do not appear in the definition of $H^m_\vartheta (\Omega )$. For a comprehensive discussion, see [17, Sec.2.4].

The next spaces are relevant when a “hole” is produced in the domain. For a compact set $K\subset \Omega$, we define

$$\begin{aligned} V^m_0(\Omega \setminus K):={\left\{ \begin{array}{ll} H^m_0(\Omega \setminus K)&{}\quad \text{ in } \text{ the } \text{ Dirichlet } \text{ case },\\ H^m_{\vartheta ,0}(\Omega \setminus K)&{}\quad \text{ in } \text{ the } \text{ Navier } \text{ case }. \end{array}\right. } \end{aligned}$$

Here $H^m_{\vartheta ,0}(\Omega \setminus K)$ denotes the space suitable for Navier BCs on ${\partial \Omega }$ and Dirichlet BCs on ${\partial K}$. More precisely, $H^m_{\vartheta ,0}(\Omega \setminus K)$ is the closure in $H^m_{\vartheta }(\Omega )$ of

$$\begin{aligned} C^m_{\vartheta ,0}({\overline{\Omega }}\setminus K):=\left\{ u\in C^m_\vartheta ({\overline{\Omega }})\ \big |\ \text {supp}\, u\cap {\mathcal {U}}(K)=\emptyset \ \text{ for } \text{ some }\ {\mathcal {U}}(K)\right\} . \end{aligned}$$

In case $\partial K$ is smooth, $u\in H^m_{\vartheta ,0}(\Omega {\setminus } K)$ if and only if $u\in H^m(\Omega {\setminus } K)$ and

$$\begin{aligned} \Delta ^ju|_{{\partial \Omega }}=0\ \,\text{ for } \text{ all }\ \,0\le j<\tfrac{m}{2}\quad \,\text{ and }\quad \,\partial _n^hu|_{{\partial K}}=0\ \,\text{ for } \text{ all }\ \,0\le h\le m-1 \end{aligned}$$

in the sense of $L^2$-traces. Note that we have the following chain of inclusions

$$\begin{aligned} H^m_0(\Omega \setminus K)\subsetneq H^m_{\vartheta ,0}(\Omega \setminus K)\subsetneq H^m_\vartheta (\Omega )\subsetneq H^m(\Omega ), \end{aligned}$$

(1.7)

where the second inclusion holds by extending to 0 in K functions defined in $\Omega \setminus K$, thanks to the Dirichlet conditions imposed on ${\partial K}$. For the same reason, note also that, for any compact sets $K_1,K_2$ such that $K_1\subset K_2\subset \Omega$, one has

$$\begin{aligned} V^m(\Omega \setminus K_2)\subset V^m(\Omega \setminus K_1). \end{aligned}$$

All such spaces are Hilbert spaces with scalar product¹$q_m(u,v):=\int _{\Omega }\nabla ^mu\,\nabla ^mv$. Note that, unlike the general case, $q_m(\cdot ,\cdot )$ does not involve boundary integrals, see [17, Sec.2.4]. By standard arguments [17, Theorem 2.15], the linear problem $(-\Delta )^mu=f$ in $\Omega \setminus K$, with $f\in L^2(\Omega \setminus K)$ and boundary conditions either (1.4) or (1.5), admits a unique weak solution $u\in V^m_0(\Omega \setminus K)$, in the sense that

$$\begin{aligned} \int _\Omega \nabla ^mu\,\nabla ^m\varphi =\int _{\Omega }f\varphi \qquad \text{ for } \text{ all }\ \ \varphi \in V^m_0(\Omega \setminus K). \end{aligned}$$

Analogously, we define the eigenvalues of problems (1.2) and (1.3) in the weak sense. We say that $(\lambda ,u)$ is an eigenpair of (1.2) (resp. (1.3)) if $(\lambda ,u)\in {\mathbb {R}}\times V^m_0(\Omega \setminus K)$ satisfies

$$\begin{aligned} u\not \equiv 0\quad \text {and}\quad \int _\Omega \nabla ^mu\,\nabla ^m\varphi = \lambda \int _{\Omega }u\varphi \qquad \text{ for } \text{ all }\;\varphi \in V^m_0(\Omega \setminus K). \end{aligned}$$

(1.8)

By classical spectral theory, problems (1.2) and (1.3) admit a diverging sequence of positive eigenvalues

$$\begin{aligned} 0<\lambda _1(\Omega \setminus K)\le \cdots \le \lambda _j(\Omega \setminus K)\le \cdots \rightarrow +\infty , \end{aligned}$$

where each one is repeated as many times as its multiplicity. Of course the same holds for the unperturbed problems (1.1), whose eigenvalues are denoted as $\left( \lambda _j(\Omega )\right) _{j\in {\mathbb {N}}}$. We recall that the eigenvalues may be variationally characterized as

$$\begin{aligned} \lambda _j(\Omega \setminus K)=\min _{\begin{array}{c} {\mathcal {X}}_j\subset V^m_0(\Omega \setminus K) \\ {\textrm{dim}}\,{\mathcal {X}}_j=j \end{array}}\,\max _{v\in {\mathcal {X}}_j} \frac{\int _{\Omega \setminus K}|\nabla ^mv|^2}{\int _{\Omega \setminus K}|v|^2}. \end{aligned}$$

(1.9)

Finally, for $\Omega$ and K as before, we define

$$\begin{aligned} X^m(\Omega ):={\left\{ \begin{array}{ll} C^\infty _0(\Omega )&{}\quad \text{ in } \text{ the } \text{ Dirichlet } \text{ case },\\ C^m_\vartheta ({\overline{\Omega }})&{}\quad \text{ in } \text{ the } \text{ Navier } \text{ case }, \end{array}\right. } \end{aligned}$$

(1.10)

and

$$\begin{aligned} X^m_0(\Omega \setminus K):={\left\{ \begin{array}{ll} C^\infty _0(\Omega \setminus K)&{}\quad \text{ in } \text{ the } \text{ Dirichlet } \text{ case },\\ C^m_{\vartheta ,0}({\overline{\Omega }}\setminus K)&{}\quad \text{ in } \text{ the } \text{ Navier } \text{ case }, \end{array}\right. } \end{aligned}$$

for the sake of a compact notation in some of the proofs.

1.2 Main results

In the spirit of the previously cited works [2, 3, 13], asymptotic expansions of eigenvalues under removal of small sets can be established treating as a perturbation parameter a suitable notion of capacity. Extending to the higher-order Sobolev framework the classical definition in the second-order case, for every compact set $K\subset \Omega$ we define the (condenser) $V^m$-capacity of K in $\Omega$ as

$$\begin{aligned} {\textrm{cap}}_{V^{m}\!,\,\Omega }(K):=\inf \left\{ \int _\Omega |\nabla ^mf|^2\,\Big |\,f\in V^m(\Omega ),\, f-\eta _K\in V^m_0(\Omega \setminus K)\right\} , \end{aligned}$$

(1.11)

where $\eta _K$ is a fixed smooth function such that $\text {supp}\,\eta _K\subset \Omega$ and $\eta _K\equiv 1$ in a neighbourhood of K. The $V^m$-capacity of a set K gives an indication about its relevance for the higher-order Sobolev space $V^m$, in the sense that zero $V^m$-capacity sets do not affect the space $V^m(\Omega )$ when they are removed from $\Omega$, and hence nor the spectrum of the polyharmonic operator (Proposition 2.1).

In our analysis, a notion of “weighted” capacity, which represents the higher order analogue of the u-capacity introduced in [7, Definition 2.1] and [10, (14)] for second order problems, will be significant too. Given a function $u\in V^m(\Omega )$, we define the $(u,V^m)$-capacity of K in $\Omega$ as

$$\begin{aligned} {\textrm{cap}}_{V^{m}\!,\,\Omega }(K,u):=\inf \left\{ \int _\Omega |\nabla ^mf|^2\,\Big |\,f\in V^m(\Omega ),\, f-u\in V^m_0(\Omega \setminus K)\right\} . \end{aligned}$$

(1.12)

Note that u is relevant only in a neighbourhood of K. Hence,

$$\begin{aligned}{\textrm{cap}}_{V^{m}\!,\,\Omega }(K,u)={\textrm{cap}}_{V^{m}\!,\,\Omega }(K,\eta _Ku)\end{aligned}$$

for any cut-off function $\eta _K$ as before. This permits to extend the notion of $(u,V^m)$-capacity to functions $u\in H^m_{loc}({\mathbb {R}}^N)$.

For those cases in which we need to distinguish the capacities according to the boundary conditions on ${\partial \Omega }$, we use the following notation:

$$\begin{aligned} {\textrm{cap}}_{m,\,\Omega }(K):={\textrm{cap}}_{H^m_0,\Omega }(K)\qquad \text{ and }\qquad {\textrm{cap}}_{m,\vartheta ,\,\Omega }(K):={\textrm{cap}}_{H^m_\vartheta ,\Omega }(K), \end{aligned}$$

for the Dirichlet and Navier BCs on ${\partial \Omega }$, respectively. Similarly we denote

$$\begin{aligned} {\textrm{cap}}_{m,\,\Omega }(K,u):={\textrm{cap}}_{H^m_0,\Omega }(K,u)\qquad \text{ and }\qquad {\textrm{cap}}_{m,\vartheta ,\,\Omega }(K,u):={\textrm{cap}}_{H^m_\vartheta ,\Omega }(K,u). \nonumber \\ \end{aligned}$$

(1.13)

We point out that the $V^m$-capacity as well as the $(u,V^m)$-capacity of a compact set K are attained by a unique minimizer, which is called capacitary potential, and which we denote by $W_K$ and $W_{K,u}$ respectively. The proof of the attainment of both capacities, together with some basic properties which will be used throughout the paper, is presented in Sect. 2.

Our first result is about the stability of the spectrum of $(-\Delta )^m$, once a set of small $V^m$-capacity is removed.

Theorem 1.1

Let $N\ge 2m$ and $\Omega \subset {\mathbb {R}}^N$ be a smooth bounded domain. Suppose one of the following:

• (D) $V^m(\Omega )=H^m_0(\Omega )$ and $K\subset \Omega$ is compact;

• (N) $V^m(\Omega )=H^m_\vartheta (\Omega )$ and the exists $K_0\subset \Omega$ compact such that K is compact and $K\subset K_0$.

Denote by $\lambda _j(\Omega )$ and $\lambda _j(\Omega \setminus K)$, $j\in {\mathbb {N}}\setminus \{0\}$, the eigenvalues respectively for (1.1) and (1.8). For all $j\in {\mathbb {N}}\setminus \{0\}$, there exist $\delta >0$ and $C>0$ (which depends on $K_0$ in the Navier case (N)) such that, if ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)<\delta$, one has

$$\begin{aligned} |\lambda _j(\Omega \setminus K)-\lambda _j(\Omega )|\le C\left( {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)\right) ^{1/2}. \end{aligned}$$

In particular $\lambda _j(\Omega \setminus K)\rightarrow \lambda _j(\Omega )$ as ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)\rightarrow 0$.

The proof of Theorem 1.1 is based on the variational characterization of the eigenvalues (1.9) and it is detailed in Sect. 3.1. We remark that spectral stability in a more general higher-order context was also established in [5] with a different approach. Here we propose a self-contained and simple proof for our Dirichlet and Navier-Dirichlet settings.

Aiming now at a more precise estimate of the convergence rate, we introduce the following notion of convergence of sets.

Definition 1.1

Let $\{K_\varepsilon \}_{\varepsilon >0}$ be a family of compact sets contained in $\Omega$. We say that $K_\varepsilon$ is concentrating to a compact set $K\subset \Omega$ as $\varepsilon \rightarrow 0$ if, for every open set $U\subseteq \Omega$ such that $U\supset K$, there exists $\varepsilon _U>0$ such that $U\supset K_\varepsilon$ for every $\varepsilon \in (0,\varepsilon _U)$.

An example is given by a decreasing family of compact sets, see e.g. [13, Example 3.7]. Note that this property alone is not sufficient to have the standard (i.e. metric) convergence of sets. For instance, the uniqueness of the limit set is not assured (e.g. if $K_\varepsilon$ is concentrating to K then $K_\varepsilon$ is concentrating also to ${\widetilde{K}}$ for any compact set ${\widetilde{K}}$ which contains K). However, as for second-order problems, in the case of a 0-capacity limit set, this concept of convergence of sets is enough to prove the continuity of the capacity (Proposition 3.1) and the Mosco convergence [11, 25] of the respective $V^m$-spaces (Proposition 3.2). These will be the tools needed for a sharp asymptotic expansion of a perturbed simple eigenvalue $\lambda _J(\Omega \setminus K_\varepsilon )$ in terms of the $(u_J,V^m)$-capacity of the vanishing compact sets $K_\varepsilon$, where $u_J$ is a normalized eigenfunction relative to $\lambda _J(\Omega )$.

Theorem 1.2

Let $N\ge 2m$ and $\Omega \subset {\mathbb {R}}^N$ be a smooth bounded domain. Let $\lambda _J(\Omega )$ be a simple eigenvalue of (1.1) and $u_J\in V^m(\Omega )$ be a corresponding eigenfunction normalized in $L^2(\Omega )$. Let $\{K_\varepsilon \}_{\varepsilon >0}$ be a family of compact sets concentrating, as $\varepsilon \rightarrow 0$, to a compact set K with ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)=0$. Then, as $\varepsilon \rightarrow 0$,

$$\begin{aligned} \lambda _J(\Omega \setminus K_\varepsilon )=\lambda _J(\Omega )+{\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)+{\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)). \end{aligned}$$

(1.14)

Theorem 1.2 is the higher-order counterpart of [2, Theorem 1.4] and its proof is presented in Sect. 3.2. In the expansion (1.14), the asymptotic parameter is the $(u_J,V^m)$-capacity of the vanishing set. The next aim is to quantify ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)$ as a function of the diameter of $K_\varepsilon$. In this respect, we focus on the particular case in which the limit set K is a point $x_0\in \Omega$ (which in dimension $N\ge 2m$ has zero $V^m$-capacity, see Proposition 2.3); without loss of generality, we consider $x_0=0$. We deal with a uniformly shrinking family of compact sets $K_\varepsilon$, the model case being $K_\varepsilon =\varepsilon {\mathcal {K}}\ni 0$ for some fixed compact set ${\mathcal {K}}\subset {\mathbb {R}}^N$. In this case, assuming 0 to be an interior point of $\Omega$, and having the operator $(-\Delta )^m-\lambda$ constant coefficients, the eigenfunction $u_J$ is analytic at 0, see [20], and hence it does not have infinite order of vanishing there. Therefore, there exist $\gamma \in {\mathbb {N}}$ and a $\gamma$-homogeneous polyharmonic polynomial $U_0\in H^m_{loc}({\mathbb {R}}^N)$ such that

$$\begin{aligned} U_\varepsilon :=\frac{u_J(\varepsilon \,\cdot )}{\varepsilon ^\gamma }\rightarrow U_0\qquad \text{ in }\;H^m(B_R(0)) \end{aligned}$$

(1.15)

for all $R>0$ as $\varepsilon \rightarrow 0$. This fact follows from a general result about elliptic equations by Bers [6, Sec.4 Theorem 1], see also [9, Theorem 2.1], provided—as in our case—one discards the possibility of an infinite order of vanishing.

In light of (1.15), our strategy to find an asymptotic expansion of ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)$ is based on a blow-up argument: we rescale the boundary value problem defining the capacitary potential $W_{K_\varepsilon ,u_J}$, find a limit equation on ${\mathbb {R}}^N\setminus {\mathcal {K}}$, and prove the convergence of the family of rescaled capacitary potentials to the one for the limiting problem. To this aim, a suitable notion of capacity in ${\mathbb {R}}^N$, involving homogeneous higher-order Sobolev spaces $D^{m,2}_0({\mathbb {R}}^N)$ and denoted by ${\textrm{cap}}_{m,{\mathbb {R}}^N}$, will be needed, see Sect. 2.2. The asymptotic expansion of ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)$ obtained by these arguments turns out to depend on the order of vanishing of $u_J$ at the point 0. More precisely, we have the following results, which we state below for the model case $K_\varepsilon =\varepsilon {\mathcal {K}}$ and prove in more generality in Sect. 4.1. For the Dirichlet case we have the following:

Theorem 1.3

(Dirichlet case) Let $N>2m$ and $\Omega \subset {\mathbb {R}}^N$ be a bounded smooth domain with $0\in \Omega$. Let ${\mathcal {K}}\subset {\mathbb {R}}^N$ be a fixed compact set and, for all $\varepsilon >0$, $K_\varepsilon =\varepsilon {\mathcal {K}}$. Let $\lambda _J$ be an eigenvalue of (1.1) with Dirichlet boundary conditions and $u_J\in H^m_0(\Omega )$ be a corresponding eigenfunction normalized in $L^2(\Omega )$. Then

$$\begin{aligned} \textrm{cap}_{m\!,\,\Omega }(K_\varepsilon ,u_J) =\varepsilon ^{N-2m+2\gamma }\left( {\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)+{\scriptstyle {\mathcal {O}}}(1)\right) \end{aligned}$$

(1.16)

as $\varepsilon \rightarrow 0$, with $\gamma$ and $U_0$ as in (1.15).

The dimensional restriction $N>2m$ is mainly due to the possibility of characterizing higher-order homogeneous Sobolev spaces as concrete functional spaces satisfying Sobolev and Hardy-type inequalities (see Sects. 2.2.1 and 2.2.2 ). In the conformal case $N=2m$ such spaces are instead made of classes of functions defined up to additive polynomials, see [15, II.6-7]. In the Navier setting, we need to restrict to the biharmonic case $m=2$.

Theorem 1.4

(Navier case) Let $N>4$ and $\Omega \subset {\mathbb {R}}^N$ be a bounded smooth domain with $0\in \Omega$. Let ${\mathcal {K}}\subset {\mathbb {R}}^N$ be a fixed compact set and, for all $\varepsilon >0$, $K_\varepsilon =\varepsilon {\mathcal {K}}$. Let $\lambda _J$ be an eigenvalue of (1.1) with $m = 2$ and Navier boundary conditions and $u_J\in H^2_\vartheta (\Omega )$ be a corresponding eigenfunction normalized in $L^2(\Omega )$. Then

$$\begin{aligned} \textrm{cap}_{2,\vartheta \!,\,\Omega }(K_\varepsilon ,u_J)=\varepsilon ^{N-4+2\gamma }\left( {\textrm{cap}}_{2,{\mathbb {R}}^N}({\mathcal {K}},U_0)+{\scriptstyle {\mathcal {O}}}(1)\right) \end{aligned}$$

as $\varepsilon \rightarrow 0$, with $\gamma$ and $U_0$ as in (1.15) with $m=2$.

It is remarkable that the same asymptotic expansion (1.16) for $m=2$ holds true for both Dirichlet and Navier BCs on ${\partial \Omega }$. As a consequence, imposing different conditions on the external boundary does not affect the first term of the asymptotic expansion of the perturbed eigenvalues. In the proof of Theorems 1.3 and 1.4 we will need to distinguish between the two settings. If in the case of Dirichlet BCs on ${\partial \Omega }$ the natural candidate as functional space for the limiting problem is $D^{m,2}_0({\mathbb {R}}^N\setminus {\mathcal {K}})$, on the other hand, in the Navier case, because of the impracticability of the trivial extension of a function outside $\Omega$, this is not evident and follows after a more involved analysis which makes use of suitable Hardy–Rellich inequalities. In Sect. 2.2.2 we give the precise statement and proofs of such inequalities. This is the main reason for the restriction to the case $m=2$, see Sect. 4.1.

Braiding together Theorem 1.2 and Theorems 1.3–1.4, we obtain the following sharp asymptotic expansions of $\lambda _J(\Omega \setminus K_\varepsilon )$, stated here for the model case $K_\varepsilon =\varepsilon {\mathcal {K}}$.

Theorem 1.5

(Dirichlet case) Let $N>2m$ and $\Omega \subset {\mathbb {R}}^N$ be a bounded smooth domain containing 0. Let ${\mathcal {K}}\subset {\mathbb {R}}^N$ be a fixed compact set and, for all $\varepsilon >0$, $K_\varepsilon =\varepsilon {\mathcal {K}}$. Let $\lambda _J$ be a simple eigenvalue of (1.1) with Dirichlet boundary conditions and let $u_J\in H^m_0(\Omega )$ be a corresponding eigenfunction normalized in $L^2(\Omega )$. Then

$$\begin{aligned} \lambda _J(\Omega \setminus K_\varepsilon )=\lambda _J(\Omega ) +\varepsilon ^{N-2m+2\gamma }\left( {\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)+{\scriptstyle {\mathcal {O}}}(1)\right) \end{aligned}$$

as $\varepsilon \rightarrow 0$, with $\gamma$ and $U_0$ as in (1.15).

Theorem 1.6

(Navier case) Let $N>4$ and $\Omega \subset {\mathbb {R}}^N$ be a bounded smooth domain containing 0. Let ${\mathcal {K}}\subset {\mathbb {R}}^N$ be a fixed compact set and, for all $\varepsilon >0$, $K_\varepsilon =\varepsilon {\mathcal {K}}$. Let $\lambda _J$ be a simple eigenvalue of (1.1) with $m = 2$ and Navier boundary conditions and let $u_J\in H^2_\vartheta (\Omega )$ be a corresponding eigenfunction normalized in $L^2(\Omega )$. Then

$$\begin{aligned} \lambda _J(\Omega \setminus K_\varepsilon ) =\lambda _J(\Omega )+\varepsilon ^{N-4+2\gamma }\left( {\textrm{cap}}_{2,{\mathbb {R}}^N}({\mathcal {K}},U_0)+{\scriptstyle {\mathcal {O}}}(1)\right) \end{aligned}$$

as $\varepsilon \rightarrow 0$, with $\gamma$ and $U_0$ as in (1.15) with $m=2$.

Theorems 1.3–1.6 deal with the model case $K_\varepsilon =\varepsilon {\mathcal {K}}$. Section 4.1 will be devoted to the proof of their analogues for a more comprehensive setting of general families of concentrating compact sets $\{K_\varepsilon \}_{\varepsilon >0}$ which uniformly shrink to a point, see Theorems 4.4–4.7.

The asymptotic expansion provided by Theorems 1.5–1.6 detects the sharp vanishing rate of the eigenvalue variation whenever ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)\ne 0$. In Sect. 4.2 we establish sufficient conditions for this to hold. In particular, this will always be the case when the Lebesgue measure of ${\mathcal {K}}$ is positive (Proposition 4.8), or when either the eigenfunction $u_J$ does not vanish at the point $x_0$ (Proposition 4.9), or it does vanish but the compactum ${\mathcal {K}}$ and the null-set of the limiting polynomial $U_0$ in (1.15) are “transversal enough” (Proposition 4.10).

The paper is then concluded by the short Sect. 5 which contains a discussion about questions which are left open by our analysis and possible directions in which our results may be extended.

2 Definition of higher-order capacity with Dirichlet and Navier BCs and first properties

The aim of this section is to introduce a notion of capacity which agrees with the higher-order framework of the problem and which turns out to be an important tool in order to study the asymptotics of the eigenvalues of the perturbed problems (1.2)–(1.3). The concept of (condenser) capacity, well-known for the second-order case, was first considered in the higher-order setting by Maz’ya for bounded domains on which Dirichlet boundary conditions are imposed, or for the whole space, see² e.g. [23, 24]. In Sect. 2.1 we propose an unified treatment for both Dirichlet and Navier settings and establish the main properties of the capacities defined by (1.11)–(1.12). In Sect. 2.2 we recall the main properties of the homogeneous Sobolev spaces and establish a Hardy–Rellich inequality with intermediate derivatives. Moreover we introduce the notion of capacity of a compact set in the whole space ${\mathbb {R}}^N$ for large dimensions $N>2m$.

2.1 Higher-order capacities in $\varvec{V^m_0}$

Let $m\in {\mathbb {N}}\setminus \{0\}$, $\Omega$ be a bounded smooth domain in ${\mathbb {R}}^N$ and K be a compact subset of $\Omega$. First, we observe that both capacities (1.11)–(1.12) are attained. Indeed, for any $u\in V^m(\Omega )$, we have that $S_u:=\left\{ g\in V^m(\Omega )\,|\,g-u\in V^m_0(\Omega {\setminus } K)\right\}$ is an affine hyperplane in $V^m(\Omega )$, so in particular a convex set. This implies that there exists a unique element in $V^m(\Omega )$ which minimizes the distance from the origin, i.e. the norm $\Vert \nabla ^m \cdot \Vert _2$ in $S_u$, which is called capacitary potential and is denoted by $W_{K,u}$ (in case u is replaced by $\eta _K$, we simply denote it by $W_K$). This means that $W_{K,u}$ is such that

$$\begin{aligned} {\textrm{cap}}_{V^{m}\!,\,\Omega }(K,u)=\int _\Omega |\nabla ^mW_{K,u}|^2 \end{aligned}$$

and it is the unique (weak) solution of the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^mW_{K,u}=0\quad \ \text{ in }\;\Omega \setminus K,\\ W_{K,u}\in V^m(\Omega ),\\ W_{K,u}-u\in V^m_0(\Omega \setminus K), \end{array}\right. } \end{aligned}$$

(2.1)

in the sense that $W_{K,u}\in V^m(\Omega )$, $W_K-u\in V^m_0(\Omega {\setminus } K)$ and

$$\begin{aligned} \int _{\Omega \setminus K}\nabla ^mW_{K,u}\nabla ^m\varphi =0\qquad \text{ for } \text{ all }\; \varphi \in V^m_0(\Omega \setminus K). \end{aligned}$$

In (2.1) we are in fact prescribing homogeneous Dirichlet or Navier boundary conditions on ${\partial \Omega }$ and, in case $\partial K$ is smooth, an “m-Dirichlet-matching” between $W_{K,u}$ and u on ${\partial K}$, i.e. the m conditions $W_{K,u}=u$, $\partial _nW_{K,u}=\partial _nu$, $\dots$, $\partial _n^{m-1}W_{K,u}=\partial _n^{m-1}u$ on ${\partial K}$.

In particular, the minimizer $W_K$ of the $V^m$-capacity is such that

$$\begin{aligned} {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)=\int _\Omega |\nabla ^mW_K|^2 \end{aligned}$$

and it is the unique (weak) solution of the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^mW_K=0\quad \text{ in }\;\Omega \setminus K,\\ W_K\in V^m(\Omega ),\\ W_K-\eta _K\in V^m_0(\Omega \setminus K), \end{array}\right. } \end{aligned}$$

in the sense that $W_K\in V^m(\Omega )$, $W_K-\eta _K\in V^m_0(\Omega {\setminus } K)$ and

$$\begin{aligned} \int _{\Omega \setminus K}\nabla ^mW_K\nabla ^m\varphi =0 \qquad \text{ for } \text{ all }\;\varphi \in V^m_0(\Omega \setminus K). \end{aligned}$$

(2.2)

We observe that ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)=0$ implies that $0\in S_{\eta _K}$, i.e. $\eta _K\in V^m_0(\Omega \setminus K)$. Since $\eta _K\equiv 1$ on K, this can only hold true when the Sobolev space “does not see” K, i.e. when $V^m_0(\Omega \setminus K)=V^m(\Omega )$. As a consequence, the eigenvalues of problems (1.2) and (1.3) coincide with those of (1.1). More precisely, in the spirit of [10, Propositions 2.1 and 2.2] (see also [13, Proposition 3.3]), we prove the following.

Proposition 2.1

The following statements are equivalent:

• i) ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)=0$;

• ii) $V^m_0(\Omega \setminus K)=V^m(\Omega )$;

• iii) $\lambda _n(\Omega \setminus K)=\lambda _n(\Omega )\,$ for all $n\in {\mathbb {N}}$.

Proof

To show $(i)\Rightarrow (ii)$, by density of $X^m(\Omega )$ in $V^m(\Omega )$, see (1.10), it is enough to prove that each $u\in X^m(\Omega )$ may be approximated by functions in $V^m_0(\Omega \setminus K)$ in the $V^m$-norm. Since ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)=0$, there exists $(w_i)_i\subset V^m(\Omega )$ with $w_i-\eta _K\in V^m_0(\Omega {\setminus } K)$ so that $\Vert \nabla ^mw_i\Vert _2^2\rightarrow 0$ as $i\rightarrow +\infty$. Hence, defining $v_i:=u(1-\eta _Kw_i)$, one has that $v_i\in V^m_0(\Omega \setminus K)$ and, in view of (1.6),

$$\begin{aligned} \begin{aligned} \Vert \nabla ^m(u-v_i)\Vert _2^2&=\Vert \nabla ^m(u\eta _Kw_i)\Vert _2^2 \lesssim \sum _{j=0}^m\int _{\Omega }|D^{m-j}(\eta _Ku)|^2|D^{j}w_i|^2\\&\le \Vert \eta _Ku\Vert _{W^{m,\infty }(\Omega )}^2\sum _{j=0}^m\int _{\Omega }|D^j w_i|^2=\Vert \eta _Ku\Vert _{W^{m,\infty }(\Omega )}^2\Vert w_i\Vert ^2_{H^m(\Omega )}\\&\le C^2\Vert \eta _Ku\Vert _{W^{m,\infty }(\Omega )}^2 \Vert \nabla ^m w_i\Vert _2^2\rightarrow 0 \end{aligned} \end{aligned}$$

as $i\rightarrow +\infty$.

The reversed implication $(ii)\Rightarrow (i)$ is due to the fact that $\varphi =W_K$ may be used as a test function in (2.2) to obtain that $\Vert W_K\Vert _{V^m_0(\Omega {\setminus } K)}=\Vert W_K\Vert _{V^m(\Omega )}=0$, which is equivalent to (i).

$(ii)\Rightarrow (iii)$ easily follows from the minimax characterization of the eigenvalues (1.9). The converse is implied by the spectral theorem, because by (iii) one is able to find an orthonormal basis of $V^m(\Omega )$ made of $V^m_0(\Omega \setminus K)$-functions. $\square$

Remark 1

From Proposition 2.1, in particular from the implication $(i)\Rightarrow (ii)$, one derives the following equivalence:

$$\begin{aligned} {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)=0\quad \;\Leftrightarrow \quad \;{\textrm{cap}}_{V^{m}\!,\,\Omega }(K,u)=0\;\,\text{ for } \text{ all }\;\, u\in V^m(\Omega ). \end{aligned}$$

Next, we investigate some properties of the above defined capacities, in particular the monotonicity properties with respect to $\Omega$ and K, and the relation between the Dirichlet and the Navier capacities.

Proposition 2.2

(Monotonicity properties of the capacity) The following properties hold.

• i) If $K_1\subset K_2\subset \Omega$, $K_1,K_2$ are compact, and $h\in V^m(\Omega )$, then

  $$\begin{aligned} {\textrm{cap}}_{V^{m}\!,\,\Omega }(K_1,h)\le {\textrm{cap}}_{V^{m}\!,\,\Omega }(K_2,h). \end{aligned}$$

• ii) If $K\subset \Omega _1\subset \Omega _2$, K is compact, and $h\in H^m(\Omega _2)$, then

  $$\begin{aligned}{\textrm{cap}}_{m,\Omega _2}(K,h)\le {\textrm{cap}}_{m,\Omega _1}(K,h).\end{aligned}$$

• iii) For every $K\subset \Omega$ compact and $h\in H^m(\Omega )$, there holds

  $$\begin{aligned}{\textrm{cap}}_{m,\vartheta ,\,\Omega }(K,h)\le {\textrm{cap}}_{m,\,\Omega }(K,h).\end{aligned}$$

Proof

i) It is enough to notice that, for $u\in V^m(\Omega )$, the condition $u-h\in V^m_0(\Omega \setminus K_2)$ is more restrictive than $u-h\in V^m_0(\Omega \setminus K_1)$.

ii) Any $u\in H^m_0(\Omega _1)$ can be extended by 0 to a function in $H^m_0(\Omega _2)$, so the minimization for ${\textrm{cap}}_{m,\Omega _2}(K,h)$ takes into consideration a larger set of test functions than the one for ${\textrm{cap}}_{m,\Omega _1}(K,h)$, and consequently the $\inf$ decreases.

iii) It follows directly from the inclusions in (1.7). $\square$

Remark 2

Note that the argument used in the proof of (ii) for Dirichlet BCs is no more available in the case of Navier BCs on ${\partial \Omega }$.

As an example, which is also relevant for our purposes, we compute the capacity of a point in ${\mathbb {R}}^N$.

Proposition 2.3

(Capacity of a point) Let $x_0\in \Omega$. Then ${\textrm{cap}}_{V^{m}\!,\,\Omega }(\{x_0\})=0$ if $N\ge 2m$, while ${\textrm{cap}}_{V^{m}\!,\,\Omega }(\{x_0\})>0$ when $N\le 2m-1$.

Proof

It is not restrictive to assume that $x_0=0\in \Omega$. If $N\le 2m-1$, then the embedding $V^m(\Omega )\hookrightarrow C^0({\overline{\Omega }})$ is continuous i.e. $\Vert \nabla ^m u\Vert _2\ge C(m,N,\Omega )\Vert u\Vert _\infty$ for all $u\in V^m(\Omega )$, with a constant $C(m,N,\Omega )>0$ which does not depend on u. In particular for those functions in $V^m(\Omega )$ for which $u(0)=1$, one has $\Vert \nabla ^m u\Vert _2\ge C(m,N,\Omega )$. Hence the infimum in the definition of ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)$ is strictly positive.

In view of Proposition 2.2(iii), it is sufficient to prove the result for the Dirichlet case. Let $N\ge 2m+1$ and take a sequence of shrinking cut-off in the following way: let $\zeta \in C^\infty _0(B_2(0))$ such that $\zeta \equiv 1$ on $B_1(0)$ and consider $\zeta _k(x):=\zeta (kx)$. One has that $\zeta _k\in C^\infty _0(B_{\frac{2}{k}}(0))$ and $\zeta _k\equiv 1$ on $B_{\frac{1}{k}}(0)$, hence $\text {supp}\,\zeta _k\subset \Omega$ for $k\ge k_0=k_0(\text {dist}(0,{\partial \Omega }))$. We compute

$$\begin{aligned} \int _\Omega |\nabla ^m\zeta _k|^2=\int _{B_{\frac{2}{k}}(0)}|k^m(\nabla ^m\zeta )(kx)|^2dx=k^{2m-N}\int _{B_2(0)}|\nabla ^m\zeta |^2\rightarrow 0 \end{aligned}$$

as $k\rightarrow \infty$ since $2m-N<0$. Being such functions admissible for the minimization of ${\textrm{cap}}_{m,\,\Omega }$, we deduce ${\textrm{cap}}_{m,\,\Omega }(\{0\})=0$. The argument is similar for the case $N=2m$, provided we choose accurately the sequence of cut-off functions, see [24, Proposition 7.6.1/2 and Proposition 13.1.2/2]. For the sake of completeness, we retrace here the proof. Let $\alpha$ denote a function in $C^\infty \left( [0,1]\right)$ equal to zero near $t=0$, to 1 near $t=1$, and such that $0\le \alpha (t)\le 1$. Define then $\zeta _\varepsilon :=\alpha (v_\varepsilon )$, where

$$\begin{aligned} v_\varepsilon (x):={\left\{ \begin{array}{ll} 1&{}\text {if }|x|\le \varepsilon ,\\ \frac{\log |x|-\log \sqrt{\varepsilon }}{\log \varepsilon -\log \sqrt{\varepsilon }}&{}\text {if }\varepsilon \le |x|\le \sqrt{\varepsilon },\\ 0&{}\text {if }|x|\ge \sqrt{\varepsilon }. \end{array}\right. } \end{aligned}$$

Notice that $v_\varepsilon$ is continuous but not $C^1$; on the other hand $\zeta _\varepsilon \in C^\infty _0(B_{\sqrt{\varepsilon }}(0))$, since $\alpha$ is constant in a neighbourhood of 0 and in a neighbourhood of 1 by construction. Therefore $\zeta _\varepsilon \in H^m_0(B_1(0))$ for any $\varepsilon \in (0,1)$. Moreover $\zeta _\varepsilon \equiv 1$ in $B_{\varepsilon }(0)$ so that $\zeta _\varepsilon$ is an admissible test function in the minimization of ${\textrm{cap}}_{m,\,\Omega }(\{0\})$. By direct calculations, there exists a constant $C=C(m)>0$ (independent of $\varepsilon$) such that

$$\begin{aligned} |\nabla ^m\zeta _\varepsilon (x)|\le \frac{C}{|\log \varepsilon |}\frac{1}{|x|^m}\quad \text {for all }\varepsilon<|x|<\sqrt{\varepsilon }, \end{aligned}$$

whereas

$$\begin{aligned} \nabla ^m\zeta _\varepsilon (x)=0\quad \text {if either }|x|\le \varepsilon \text { or }|x|\ge \sqrt{\varepsilon }. \end{aligned}$$

Therefore

$$\begin{aligned} \int _\Omega |\nabla ^m\zeta _\varepsilon |^2\lesssim \frac{1}{\log ^2\varepsilon }\int _\varepsilon ^{\sqrt{\varepsilon }}\frac{1}{r}\,dr=\frac{1}{2|\log \varepsilon |}\rightarrow 0 \end{aligned}$$

as $\varepsilon \rightarrow 0$. The argument is concluded as above. $\square$

2.2 Homogeneous Sobolev spaces and capacities in $\varvec{{\mathbb {R}}^N}$

2.2.1 The homogeneous Sobolev spaces $\varvec{D^{m,2}_0({\mathbb {R}}^N)}$

So far, we defined the notion of $V^m$-capacity for compact sets contained in an open bounded smooth domain $\Omega \subset {\mathbb {R}}^N$. An analogous definition can be given when $\Omega ={\mathbb {R}}^N$, provided the underlined space is of homogeneous kind. We introduce the homogeneous Sobolev spaces (sometimes referred to as Beppo Levi spaces) $D^{m,2}_0({\mathbb {R}}^N)$ as the completion of $C^\infty _0({\mathbb {R}}^N)$ with respect to the norm

$$\begin{aligned} \Vert u\Vert _{D^{m,2}_0({\mathbb {R}}^N)}:=\left( \int _{{\mathbb {R}}^N}|\nabla ^mu|^2\right) ^\frac{1}{2}. \end{aligned}$$

Actually, the spaces $D^{m,2}_0({\mathbb {R}}^N)$ are more commonly defined as the completion with respect to the norm $\Vert D^m\cdot \Vert _2$, i.e. with respect to the full tensor of all highest derivatives. However, the two definitions are equivalent since, by integration by parts, $\Vert D^m\cdot \Vert _2$ and $\Vert \nabla ^m\cdot \Vert _2$ are equivalent norms on $C^\infty _0({\mathbb {R}}^N)$, see e.g. [17, Sec.2.2.1].

For large dimensions $N>2m$, the following Sobolev inequalities are well-known: for every $0\le j\le m$ there exists a constant $S(N,m,j)>0$ (depending only on N, m and j) such that

$$\begin{aligned} S(N,m,j)\left( \int _{{\mathbb {R}}^N}|D^ju|^{2^*_{m,j}}\right) ^{\frac{2}{2^*_{m,j}}}\le \Vert D^mu\Vert ^2_{L^2({\mathbb {R}}^N)}\quad \text {for all }u\in C^\infty _0({\mathbb {R}}^N), \end{aligned}$$

(2.3)

where $2^*_{m,j}:=\frac{2N}{N-2(m-j)}$. In particular, for $j=0$, there exists a constant $S(N,m)>0$ such that

$$\begin{aligned} S(N,m)\left( \int _{{\mathbb {R}}^N}|u|^{2^*_m}\right) ^{\frac{2}{2^*_m}}\le \Vert u\Vert _{D^{m,2}_0({\mathbb {R}}^N)}\quad \text {for all }u\in C^\infty _0({\mathbb {R}}^N), \end{aligned}$$

where $2^*_m:=2^*_{m,0}=\frac{2N}{N-2m}$, see [17, Theorem 2.3]. In view of (2.3), if $N>2m$, one may also characterize $D^{m,2}_0({\mathbb {R}}^N)$ as

$$\begin{aligned} D^{m,2}_0({\mathbb {R}}^N)=\big \{u\in L^{2^*_m}({\mathbb {R}}^N)\,\big |\, D^j u\in L^{2^*_{m,j}}({\mathbb {R}}^N)\text { for all }0<j\le m\big \}. \end{aligned}$$

Analogously, for $K\subset {\mathbb {R}}^N$ compact, one may consider the exterior domain $\Omega ={\mathbb {R}}^N\setminus K$ and define $D^{m,2}_0({\mathbb {R}}^N\setminus K)$ as the completion of $C^\infty _0({\mathbb {R}}^N\setminus K)$ with respect to the norm $\Vert \nabla ^m\cdot \Vert _2$, which is characterized, for $N>2m$, as

$$\begin{aligned} D^{m,2}_0({\mathbb {R}}^N\setminus K)=\bigg \{ \begin{array}{ll} u\in L^{2^*_m}({\mathbb {R}}^N\setminus K)\,\big |\,&{} D^j u\in L^{2^*_{m,j}}({\mathbb {R}}^N\setminus K)\text { for all }0<j\le m \\ {} &{}\text{ and }\ \psi u\in H^m_0({\mathbb {R}}^N\setminus K)\ \text{ for } \text{ all }\ \psi \in C^\infty _0({\mathbb {R}}^N) \end{array} \bigg \}, \end{aligned}$$

see [15, Theorem II.7.6].

2.2.2 A Hardy–Rellich-type inequality with intermediate derivatives

Besides Sobolev inequalities, an important tool in the theory of Sobolev spaces in large dimensions $N>2m$ is represented by Hardy–Rellich inequalities, which state that the Sobolev norm of the highest order derivatives controls a singularly weighted Sobolev norm of the function. We refer to [12] for such inequalities in $H^m_0(\Omega )$ and to [16, 18] for their extensions to $H^m_\vartheta (\Omega )$. In this section, inspired by [26], we prove a Hardy–Rellich-type inequality for the space $H^2_\vartheta (\Omega )$ including also the gradient term, which provides a further characterization of the space $D^{2,2}_0({\mathbb {R}}^N)$ for $N>4$. It will be needed in Sect. 4.1 to identify the functional space containing the limiting profile in the blow-up argument, when Navier BCs are imposed on ${\partial \Omega }$.

Theorem 2.4

Let $N>4$ and $\Omega \subset {\mathbb {R}}^N$ be a smooth bounded domain. Then, for every function $u\in H^2(\Omega )\cap H^1_0(\Omega )$, one has that $\frac{u}{|x|^2},\,\frac{\nabla u}{|x|}\in L^2(\Omega )$ and

$$\begin{aligned} (N-4)^2\int _{\Omega }\frac{|u|^2}{|x|^4}\,dx+2(N-4)\int _{\Omega }\frac{|\nabla u|^2}{|x|^2}\,dx\le \int _{\Omega }|\Delta u|^2\,dx. \end{aligned}$$

(2.4)

Proof

Let $u\in C^\infty ({\overline{\Omega }})$ be such that $u|_{{\partial \Omega }}=0$. Let us assume that $0\in \Omega$. Let us introduce a parameter $\lambda$ to be fixed later and, for $\varepsilon >0$ small, let us denote $\Omega _\varepsilon :=\Omega \setminus B_\varepsilon (0)$. We have that

$$\begin{aligned} \begin{aligned} 0&\le \int _{\Omega _\varepsilon }\left( \frac{x}{|x|}\Delta u+\lambda u\frac{x}{|x|^3}\right) ^2dx=\int _{\Omega _\varepsilon }(\Delta u)^2+\lambda ^2\int _{\Omega _\varepsilon }\frac{u^2}{|x|^4}\,dx+2\lambda \int _{\Omega _\varepsilon }\frac{u}{|x|^2}\Delta u\,dx. \end{aligned}\nonumber \\ \end{aligned}$$

(2.5)

We can rewrite the third term as

$$\begin{aligned} \begin{aligned} \int _{\Omega _\varepsilon }\frac{u}{|x|^2}\Delta u\,dx&=-\int _{\Omega _\varepsilon }\nabla u\left( \frac{\nabla u}{|x|^2}-2u\frac{x}{|x|^4}\right) dx+\int _{\partial \Omega }\frac{u}{|x|^2}\partial _\nu u\,d\sigma -\frac{1}{\varepsilon ^2}\int _{\partial B_\varepsilon }u\nabla u\cdot \frac{x}{\varepsilon }\,d\sigma \\&=-\int _{\Omega _\varepsilon }\frac{|\nabla u|^2}{|x|^2}\,dx+\int _{\Omega _\varepsilon }\nabla (u^2)\frac{x}{|x|^4}\,dx+{\mathcal {O}}(\varepsilon ^{N-3})\\&=-\int _{\Omega _\varepsilon }\frac{|\nabla u|^2}{|x|^2}\,dx-(N-4)\int _{\Omega _\varepsilon }\frac{u^2}{|x|^4}\,dx+\int _{\partial \Omega }u^2\frac{x\cdot \nu }{|x|^4}\\&\quad -\int _{\partial B_\varepsilon }\frac{u^2}{\varepsilon ^3}\,d\sigma +{\mathcal {O}}(\varepsilon ^{N-3}) \end{aligned} \end{aligned}$$

as $\varepsilon \rightarrow 0$. Since the third term vanishes and the second to last term is ${\mathcal {O}}(\varepsilon ^{N-4})$ as $\varepsilon \rightarrow 0$, from (2.5) we get

$$\begin{aligned} 0\le \int _{\Omega _\varepsilon }(\Delta u)^2+\lambda ^2\int _{\Omega _\varepsilon }\frac{u^2}{|x|^4}\,dx-2\lambda \int _{\Omega _\varepsilon }\frac{|\nabla u|^2}{|x|^2}\,dx-2\lambda (N-4)\int _{\Omega _\varepsilon }\frac{u^2}{|x|^4}\,dx+{\mathcal {O}}(\varepsilon ^{N-4}). \end{aligned}$$

Choosing now $\lambda =N-4$, we obtain

$$\begin{aligned} (N-4)^2\int _{\Omega _\varepsilon }\frac{u^2}{|x|^4}\,dx+2(N-4)\int _{\Omega _\varepsilon }\frac{|\nabla u|^2}{|x|^2}\,dx\le \int _{\Omega _\varepsilon }(\Delta u)^2+{\mathcal {O}}(\varepsilon ^{N-4}) \quad \text {as }\varepsilon \rightarrow 0. \end{aligned}$$

Inequality (2.4) follows by letting $\varepsilon \rightarrow 0$ and by density of the set $\{u\in C^\infty ({\overline{\Omega }})\,|\,u|_{{\partial \Omega }}=0\}$ in $H^2(\Omega )\cap H^1_0(\Omega )$. If $0\not \in \Omega$ the above argument can be repeated by considering directly in (2.5) the integral on the whole $\Omega$. $\square$

We observe that (2.4) holds also for all functions in $C^\infty _0({\mathbb {R}}^N)$ (since any $u\in C^\infty _0({\mathbb {R}}^N)$ is contained in some $H^2_\vartheta (\Omega )$). Therefore, by density of $C^\infty _0({\mathbb {R}}^N)$ in $D^{2,2}_0({\mathbb {R}}^N)$ and Fatou’s Lemma we easily deduce that, if $N>4$, then

$$\begin{aligned} (N-4)^2\int _{{\mathbb {R}}^N}\frac{|u|^2}{|x|^4}\,dx+2(N-4)\int _{{\mathbb {R}}^N}\frac{|\nabla u|^2}{|x|^2}\,dx\le \int _{{\mathbb {R}}^N}|\Delta u|^2\,dx \end{aligned}$$

for all $u\in D^{2,2}_0({\mathbb {R}}^N)$. In particular we have that $D^{2,2}_0({\mathbb {R}}^N)$ is contained in the space

$$\begin{aligned} {\mathcal {S}}^2({\mathbb {R}}^N):=\left\{ u\in H^2_{loc}({\mathbb {R}}^N)\,\Big |\,\frac{\nabla ^{2-k}u}{|x|^k}\in L^2({\mathbb {R}}^N)\text { for }k\in \{0,1,2\}\right\} . \end{aligned}$$

We prove now that the two functional spaces coincide.

Proposition 2.5

${\mathcal {S}}^2({\mathbb {R}}^N)=D^{2,2}_0({\mathbb {R}}^N)$ for all $N>4$.

Proof

We have already observed above that ${\mathcal {S}}^2({\mathbb {R}}^N)\supseteq D^{2,2}_0({\mathbb {R}}^N)$. Let now $u\in {\mathcal {S}}^2({\mathbb {R}}^N)$, $\eta$ be a cutoff function with support in $B_2(0)$ and which takes the value 1 in $B_1(0)$, and define $\eta _R:=\eta \left( \tfrac{\cdot }{R}\right)$ for all $R>0$. Hence $\eta _Ru\in H^2_0(B_{2R}(0))$ and we claim that $\Vert \Delta (\eta _Ru- u)\Vert _{2}\rightarrow 0$ as $R\rightarrow +\infty$. Indeed,

$$\begin{aligned} \Vert \Delta \big (\left( \eta _R-1\right) u\big )\Vert _2^2\lesssim \Vert (\Delta \eta _R)u\Vert _2^2+\Vert \nabla \eta _R\nabla u\Vert _2^2+\Vert \left( \eta _R-1\right) \Delta u\Vert _2^2, \end{aligned}$$

where

$$\begin{aligned} \Vert \left( \eta _R-1\right) \Delta u\Vert _2^2\le \int _{{\mathbb {R}}^N\setminus B_R}|\Delta u|^2\rightarrow 0 \end{aligned}$$

as $R\rightarrow +\infty$, and for $k\in \{1,2\}$,

$$\begin{aligned} \begin{aligned} \Vert \nabla ^k\eta _R\nabla ^{2-k}u\Vert _2^2&=\int _{R<|x|<2R}\frac{1}{R^{2k}}\left| \big (\nabla ^k\eta \big )\left( \frac{x}{R}\right) \right| ^2|\nabla ^{2-k}u|^2\,dx\\&\lesssim 2^{2k}\int _{{\mathbb {R}}^N\setminus B_R(0)}\frac{|\nabla ^{2-k}u|^2}{|x|^{2k}}\,dx\rightarrow 0 \end{aligned} \end{aligned}$$

as $R\rightarrow +\infty$. By density of $C^\infty _0(B_{2R}(0))$ in $H^2_0(B_{2R}(0))$, this implies that $C^\infty _0({\mathbb {R}}^N)$ is dense in ${\mathcal {S}}^2({\mathbb {R}}^N)$ in the $D^{2,2}_0$-norm, thus concluding the proof. $\square$

2.2.3 Capacities in $\varvec{{\mathbb {R}}^N}$

Similarly to the case of a bounded set $\Omega$ described in Sect. 2.1, for any compact set $K\subset {\mathbb {R}}^N$ and any $u\in D^{m,2}_0({\mathbb {R}}^N)$ with $N>2m$, we define

$$\begin{aligned} {\textrm{cap}}_{m,{\mathbb {R}}^N}(K,u):=\inf \left\{ \int _{{\mathbb {R}}^N}|\nabla ^mf|^2\,\Big |\,f\in D^{m,2}_0({\mathbb {R}}^N),\, f-u\in D^{m,2}_0({\mathbb {R}}^N\setminus K)\right\} , \end{aligned}$$

(2.6)

which we simply denote by ${\textrm{cap}}_{m,{\mathbb {R}}^N}(K)$ when $u=\eta _K$. The argument for the attainability of the capacity is easily adapted from the one for ${\textrm{cap}}_{V^m,\Omega }(K,u)$. Analogous properties hold also in this setting, in particular it is true that

$$\begin{aligned} D^{m,2}_0({\mathbb {R}}^N)=D^{m,2}_0({\mathbb {R}}^N\setminus K)\quad \text {if and only if}\quad {\textrm{cap}}_{m,{\mathbb {R}}^N}(K)=0 \end{aligned}$$

(2.7)

which directly implies that

$$\begin{aligned} {\textrm{cap}}_{m,{\mathbb {R}}^N}(K)=0\quad \;\Leftrightarrow \quad \;{\textrm{cap}}_{m,{\mathbb {R}}^N}(K,u)=0\;\,\text{ for } \text{ all }\;\, u\in D^{m,2}_0({\mathbb {R}}^N). \end{aligned}$$

The analogue of (2.7) in the case of a bounded domain $\Omega$ is contained in Proposition 2.1 and its proof relies on (1.6), which in turn is based on a Poincaré inequality, the latter being no longer valid in ${\mathbb {R}}^N$. However, if $N>2m$, the role played by Poincaré inequalities can be replaced by the critical Sobolev embedding. Although known, here we retrace the proof of (2.7) for the sake of completeness. Let $u\in C^\infty _0({\mathbb {R}}^N)$, set $\Sigma :=\text {supp}(u)$, and consider $(w_i)_i\subset D^{m,2}_0({\mathbb {R}}^N)$ with $w_i-\eta _K\in D^{m,2}_0({\mathbb {R}}^N{\setminus } K)$ such that $\Vert \nabla ^mw_i\Vert _2^2\rightarrow 0$ as $i\rightarrow +\infty$. Then $v_i:=u(1-w_i)\in D^{m,2}_0({\mathbb {R}}^N\setminus K)$ and, defining $q_j:=2^*_{m,j}=\frac{2N}{N-2(m-j)}\ge 2$ for $j\in \{0,\dots ,m\}$, one has that

$$\begin{aligned} \begin{aligned} \Vert \nabla ^m(u-v_i)\Vert _2^2&=\Vert \nabla ^m(uw_i)\Vert _{L^2(\Sigma )}^2\lesssim \Vert u\Vert _{W^{m,\infty }({\mathbb {R}}^N)}^2\sum _{j=0}^m\int _{\Sigma }|D^jw_i|^2\\&\lesssim \sum _{j=0}^m\left( \int _{\Sigma }|D^jw_i|^{q_j}\right) ^{\frac{2}{q_j}}\le \sum _{j=0}^m\Vert D^jw_i\Vert _{L^{q_j}({\mathbb {R}}^N)}^2\\&\lesssim \Vert D^mw_i\Vert _{L^2({\mathbb {R}}^N)}^2\lesssim \Vert \nabla ^mw_i\Vert _{L^2({\mathbb {R}}^N)}^2\rightarrow 0, \end{aligned} \end{aligned}$$

where in the last steps we used Hölder inequality, the Sobolev inequality (2.3), and the equivalence of the norms $\Vert D^m\cdot \Vert _2$ and $\Vert \nabla ^m\cdot \Vert _2$.

For later use, we also recall the right continuity of the capacity, see [24, Sec. 13.1.1].

Lemma 2.6

Let K be a compact subset of $\Omega \subset {\mathbb {R}}^N$. For any $\varepsilon >0$ there exists a neighbourhood ${\mathcal {U}}(K)\subset \Omega$ such that for any compact set ${\widetilde{K}}$ with $K\subset {\widetilde{K}}\subset {\mathcal {U}}(K)$, there holds

$$\begin{aligned} {\textrm{cap}}_{m,\Omega }({\widetilde{K}})\le {\textrm{cap}}_{m,\Omega }(K)+\varepsilon . \end{aligned}$$

Although the notion of capacity needed for the blow-up analysis in Sect. 4.1 is the one given in (2.6), sometimes it is useful to consider a second one defined as

$$\begin{aligned} {\textrm{Cap}}_{m,\,{\mathbb {R}}^{N}}^{\ge }(K):=\inf \left\{ \int _{{\mathbb {R}}^N}|\nabla ^mf|^2\,\Big |\,f\in D^{m,2}_0({\mathbb {R}}^N),\, f\ge 1\;\text{ a.e. } \text{ on }\;K\right\} , \end{aligned}$$

(2.8)

which is well-defined for $N>2m$, and similarly $\textrm{Cap}_{m,\Omega }^\ge$ for $\Omega \subset \subset {\mathbb {R}}^N$, see [23, 24]. One of the advantages in this approach is that the capacitary potential associated to ${\textrm{Cap}}_{m,\,{\mathbb {R}}^{N}}^{\ge }$ is positive, see [17, Sec. 3.1.2]. Note that for all $\Omega \subset {\mathbb {R}}^N$ one has $\textrm{Cap}_{m,\Omega }^\ge (K)\le {\textrm{cap}}_{m,\Omega }(K)$ because the class of test functions considered in (2.8) includes the one considered for the minimization in (2.6). Actually it turns out that the two definitions are equivalent, in the sense that the two capacities estimate each other, as stated below. We report here the result for $\Omega ={\mathbb {R}}^N$, referring to [23] for the general case $\Omega \subsetneq {\mathbb {R}}^N$.

Lemma 2.7

([24], Theorem 13.3.1) Let $m\in {\mathbb {N}}\setminus \{0\}$ and $N>2m$. There exists a constant $c>0$ such that

$$\begin{aligned} c\,{\textrm{cap}}_{m,{\mathbb {R}}^N}(K)\le {\textrm{Cap}}_{m,\,{\mathbb {R}}^{N}}^{\ge }(K)\le {\textrm{cap}}_{m,{\mathbb {R}}^N}(K) \end{aligned}$$

for any compact set $K\subset {\mathbb {R}}^N$.

Remark 3

The constant c appearing in Lemma 2.7 can be taken 1 in the second-order case $m=1$, so the two definitions coincide, see e.g. [24, Sec. 13.3]. Whether this is the case also for the higher-order case it is still an open question.

Remark 4

As an extension of Proposition 2.3, it is known that a regular manifold of dimension d has zero capacity in the sense of (2.8) if and only if $d\le N-2m$, see [4, Corollary 5.1.15]. By Lemma 2.7, this result holds also for the notion (2.6) of capacity.

3 Convergence and asymptotic expansion of the perturbed eigenvalues

In this section we study stability and asymptotic expansion of the perturbed eigenvalues of (1.2) and (1.3), when from a bounded domain $\Omega \subset {\mathbb {R}}^N$ one removes a compact set K of small $V^m$-capacity. The main goal is to extend the results obtained in the second-order framework (in particular [2, Theorem 1.4]) to the higher-order settings described in the introduction. The first part is devoted to the proof of the stability result of Theorem 1.1, which applies for rather general domains, while in the second part we focus on the asymptotic expansion of simple eigenvalues contained in Theorem 1.2, for which we require the notion of concentrating family of compact sets.

3.1 Spectral stability: Proof of Theorem 1.1

We present here a simple and self-contained proof of the stability of the point spectrum of the polyharmonic operator with respect to the capacity of the removed compactum, in both Dirichlet and Navier settings described in Sect. 1.1. It is essentially based on the variational characterization of the eigenvalues (1.9) and on the properties of the capacitary potentials, and it follows some ideas exploited for the same question in the second-order case in [3, Theorem 1.2].

Proof of Theorem 1.1

Denote by $(u_i)_{i=1}^\infty$ an orthonormal basis of $L^2(\Omega )$ such that each $u_i$ is an eigenfunction of problem (1.1) associated to the eigenvalue $\lambda _i(\Omega )$. By classical elliptic regularity theory (see e.g. [17, Section 2.5]), the smoothness of ${\partial \Omega }$ yields $u_i\in C^m({\overline{\Omega }})$ for all $i\in {\mathbb {N}}$. In order to deal at once with both cases (D) and (N), we introduce the function H defined by $H\equiv 1$ in the Dirichlet case, and by $H=\eta _{K_0}$ in the Navier case. Here $\eta _{K_0}$ is a cutoff function which is equal to 1 in a neighbourhood on $K_0$ and with support contained in some compact set $\widetilde{K_0}$ such that $K_0\subset \widetilde{K_0}\subset \Omega$. The cutoff $\eta _{K_0}$ is introduced in order to enforce the boundary conditions on ${\partial \Omega }$ in the Navier case.

Fix $j\in {\mathbb {N}}\setminus \{0\}$. For any $\ell \in \{1,\dots , j\}$, we define $\Phi _\ell :=u_\ell (1-HW_K)$ and introduce the subspace $X_j:={{\,\textrm{span}\,}}\{\Phi _\ell \}_{\ell =1}^j$. Note that $\Phi _\ell \in V^m_0(\Omega {\setminus } K)$ by definition of the capacitary potential $W_K$, so $X_j\subset V^m_0(\Omega \setminus K)$. The aim is to prove that $X_j$ is a j-dimensional subspace of $V^m_0(\Omega \setminus K)$ so that the right hand side of (1.9) is smaller than the maximum of the Rayleigh quotient over $X_j$. Note that, by trivially extending the functions $\left\{ \Phi _\ell \right\} _{\ell =1}^j$ in K, the integrals may be evaluated on $\Omega$. First,

$$\begin{aligned} \int _\Omega \Phi _h\Phi _\ell =\int _\Omega u_h u_\ell -2\int _\Omega u_h u_\ell \,H W_K+\int _{\Omega }u_h u_\ell \,H^2 W_K^2, \end{aligned}$$

therefore, by orthonormality of $\{u_\ell \}_{\ell =1}^j$ in $L^2(\Omega )$ and (1.6),

$$\begin{aligned} \begin{aligned} \left| \int _\Omega \Phi _h\Phi _\ell -\delta _{h,\ell }\right|&\le \max _{1\le h\le j}\Vert u_h\Vert _{L^\infty (\Omega )}^2\left( 2|\Omega |^{1/2}\Vert W_K\Vert _2+\Vert W_K\Vert _2^2\right) \\&\lesssim \left( {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)\right) ^{1/2}+{\textrm{cap}}_{V^{m}\!,\,\Omega }(K), \end{aligned} \end{aligned}$$

(3.1)

where $\delta _{h,\ell }$ stands for the Kroenecker delta. Let now $(W_n)_n\subset X^m(\Omega )$, see (1.10), be a sequence of smooth functions which approximates in the $V^m$-norm the capacitary potential $W_K$ and satisfying $W_n=1$ in ${\mathcal {U}}(K)$. The existence of such a sequence is guaranteed by the definition of $W_K$. Define moreover $\Phi ^\ell _n:=u_\ell (1-HW_n)$ for $n\in {\mathbb {N}}$. Note that $\Phi ^\ell _n\rightarrow \Phi _\ell$ in $V^m(\Omega )$ as $n\rightarrow +\infty$. We get

$$\begin{aligned} \begin{aligned} \int _{\Omega }\nabla ^m\Phi ^h_n\nabla ^m\Phi ^\ell _n&=\int _{\Omega }\nabla ^m\left( u_h(1-HW_n)\right) \nabla ^m\left( u_\ell (1-HW_n)\right) \\&=\int _{\Omega }\nabla ^mu_h\nabla ^mu_\ell (1-HW_n)^2+ T_m(u_h,u_\ell ,W_n), \end{aligned} \end{aligned}$$

(3.2)

where the term $T_m$ contains all remaining products between the derivatives of $u_h$, $u_\ell$, and $1-HW_n$. To deal with the first term on the right in (3.2), consider $u_\ell (1-HW_n)^2\in V^m(\Omega )$ by regularity of the factors, as a test function for the eigenvalue problem (1.8) for $\lambda _h(\Omega )$. One obtains

$$\begin{aligned} \begin{aligned} \lambda _h(\Omega )\int _{\Omega }\Phi ^h_n\Phi ^\ell _n&=\lambda _h(\Omega )\int _{\Omega }u_hu_\ell (1-HW_n)^2=\int _{\Omega }\nabla ^mu_h\nabla ^m\left( u_\ell (1-HW_n)^2\right) \\&=\int _{\Omega }\nabla ^mu_h\nabla ^mu_\ell (1-HW_n)^2+ \int _{\Omega }\nabla ^mu_h S_m(u_\ell ,W_n), \end{aligned} \end{aligned}$$

where again all remaining products involving intermediate derivatives of $u_\ell$ and $1-HW_n$ are collected in the term $S_m$ (which is a vector if m is odd). Isolating the first term on the right hand-side, and substituting it into (3.2), we get

$$\begin{aligned} \int _{\Omega }\nabla ^m\Phi ^h_n\nabla ^m\Phi ^\ell _n-\lambda _h(\Omega )\int _{\Omega }\Phi ^h_n\Phi ^\ell _n=-\int _{\Omega }\nabla ^mu_h S_m(u_\ell ,W_n)+T_m(u_h,u_\ell ,W_n). \nonumber \\ \end{aligned}$$

(3.3)

Moreover,

$$\begin{aligned} \begin{aligned} \Big |\int _{\Omega }&\nabla ^mu_hS_m(u_\ell ,W_n)\Big |\\&\le \sum _{i=1}^m\sum _{\tau =0}^i\int _{\Omega }|\nabla ^mu_h||D^{m-i}u_\ell ||D^{i-\tau }(1-HW_n)||D^\tau (1-HW_n)\Big |\\&\le \Vert \nabla ^mu_h\Vert _\infty \Vert u_\ell \Vert _{W^{m,\infty }(\Omega )}\sum _{i=1}^m\sum _{\tau =0}^i\Vert D^{i-\tau }(1-HW_n)\Vert _2\Vert D^\tau (1-HW_n)\Vert _2\\&\le \max _{1\le k\le j}\Vert u_k\Vert _{W^{m,\infty }(\Omega )}^2\sum _{i=1}^m\!\bigg (2\Vert D^i(HW_n)\Vert _2\Vert 1-HW_n\Vert _2\\&\quad +\sum _{\tau =1}^{i-1}\Vert D^{i-\tau }(HW_n)\Vert _2\Vert D^\tau (HW_n)\Vert _2\bigg )\\&\le C(\Omega ,j,m)\Big (\Vert H\Vert _{W^{m,\infty }(\Omega )}\Vert W_n\Vert _{H^m(\Omega )}(|\Omega |^{1/2}+\Vert W_n\Vert _2)\\&\quad +\Vert H\Vert _{W^{m,\infty }(\Omega )}^2\Vert W_n\Vert _{H^m(\Omega )}^2\Big )\\&\lesssim C(\Omega ,j,m)\Big (\Vert H\Vert _{W^{m,\infty }(\Omega )}\left( {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)+o_n(1)\right) ^{1/2}\\&\quad +\Vert H\Vert _{W^{m,\infty }(\Omega )}^2({\textrm{cap}}_{V^{m}\!,\,\Omega }(K)+o_n(1))\Big )\quad \text {as } n\rightarrow \infty , \end{aligned} \end{aligned}$$

(3.4)

having used the equivalence of the norms $\Vert \cdot \Vert _{H^m(\Omega )}$ and $\Vert \nabla ^m\cdot \Vert _2$ in $V^m(\Omega )$. Here $o_n(1)$ denotes a real sequence converging to 0 as $n\rightarrow +\infty$. Analogously one may estimate the last term in (3.3):

$$\begin{aligned} \begin{aligned} |T_m&(u_h,u_\ell ,W_n)|\le \sum _{\begin{array}{c} i,\tau \in \{0,\dots ,m\}\\ (i,\tau )\ne (0,0) \end{array}}\int _{\Omega }|D^{m-i}u_h||D^i(1-HW_n)||D^{m-\tau }u_\ell ||D^\tau (1-HW_n)|\\&\le \max _{1\le h\le j}\Vert u_h\Vert _{W^{m,\infty }(\Omega )}^2\bigg (2\sum _{\tau =1}^m\Vert D^\tau (HW_n)\Vert _2\Vert 1-HW_n\Vert _2\\&\quad +\sum _{i,\tau \in \{1,\dots ,m\}}\Vert D^i(HW_n)\Vert _2\Vert D^\tau (HW_n)\Vert _2\bigg )\\&\le C(\Omega ,j,m)\Big (\Vert H\Vert _{W^{m,\infty }(\Omega )}\Vert W_n\Vert _{H^m(\Omega )}\left( |\Omega |^{1/2}+\Vert W_n\Vert _2\right) \\&\quad +\Vert H\Vert _{W^{m,\infty }(\Omega )}^2\Vert W_n\Vert _{H^m(\Omega )}^2\Big )\\&\lesssim C(\Omega ,j,m)\Big (\Vert H\Vert _{W^{m,\infty }(\Omega )}\left( {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)+o_n(1)\right) ^{1/2}\\&\quad +\Vert H\Vert _{W^{m,\infty }(\Omega )}^2({\textrm{cap}}_{V^{m}\!,\,\Omega }(K)+o_n(1))\Big )\quad \text {as } n\rightarrow \infty . \end{aligned} \end{aligned}$$

(3.5)

All in all, from (3.3)–(3.5), one concludes

$$\begin{aligned}{} & {} \left| \int _{\Omega }\nabla ^m\Phi ^h_n\nabla ^m\Phi ^\ell _n-\lambda _h(\Omega )\int _{\Omega }\Phi ^h_n\Phi ^\ell _n\right| \\{} & {} \quad \le {\widetilde{C}}\left( \left( {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)+o_n(1)\right) ^{1/2}+{\textrm{cap}}_{V^{m}\!,\,\Omega }(K)+o_n(1)\right) , \end{aligned}$$

where ${\widetilde{C}}$ depends on $K_0$ in the Navier case. Letting now $n\rightarrow +\infty$ in both sides of the inequality, and taking into account (3.1), one infers

$$\begin{aligned} \left| \int _{\Omega }\nabla ^m\Phi _h\nabla ^m\Phi _\ell -\lambda _h(\Omega )\delta _{h,\ell }\right| \le {\widetilde{C}}\left( \left( {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)\right) ^{1/2}+{\textrm{cap}}_{V^{m}\!,\,\Omega }(K)\right) . \end{aligned}$$

(3.6)

Hence, from (3.1) and (3.6) one sees that, when ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)$ is small enough, the functions $\{\Phi _\ell \}_{\ell =1}^j$ are linearly independent in $V^m_0(\Omega \setminus K)$, and so the subspace $X_j$ has dimension j. Therefore, recalling that $\lambda _h(\Omega )\le \lambda _j(\Omega )$ for all $h\in \{1,\dots ,j\}$, again from (3.1) and (3.6) one finally infers that

$$\begin{aligned} \begin{aligned} \lambda _j(\Omega \setminus K)&\le \max _{\begin{array}{c} \left( \alpha _1,\dots ,\alpha _j\right) \in {\mathbb {R}}^j\\ \sum _{i=1}^j\alpha _i=1 \end{array}}\frac{\sum \limits _{h,\ell =1}^j\alpha _h\alpha _\ell \int _{\Omega }\nabla ^m\Phi _h\nabla ^m\Phi _\ell }{\sum \limits _{h,\ell =1}^j\alpha _h\alpha _\ell \int _{\Omega }\Phi _h\Phi _\ell }\\&\le \max _{\begin{array}{c} \left( \alpha _1,\dots ,\alpha _j\right) \in {\mathbb {R}}^j\\ \sum _{i=1}^j\alpha _i=1 \end{array}}\frac{\sum \limits _{h=1}^j\alpha _h^2\lambda _h(\Omega )+{\mathcal {O}}\left( \left( {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)\right) ^{1/2}\right) }{\sum \limits _{h=1}^j\alpha _h^2+{\mathcal {O}}\left( \left( {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)\right) ^{1/2}\right) }\\&\le \frac{\lambda _j(\Omega )+{\mathcal {O}}\left( \left( {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)\right) ^{1/2}\right) }{1+{\mathcal {O}}\left( \left( {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)\right) ^{1/2}\right) }=\lambda _j(\Omega )+{\mathcal {O}}\left( \left( {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)\right) ^{1/2}\right) \end{aligned} \end{aligned}$$

as ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)\rightarrow 0$. $\square$

3.2 Asymptotic expansion of eigenvalues: Proof of Theorem 1.2

Let $\{K_\varepsilon \}_{\varepsilon >0}$ be a family of compact subsets of $\Omega$ and denote by $\lambda _J(\Omega {\setminus } K_\varepsilon )$ the J-th eigenvalue of $(-\Delta )^m$ in $V^m_0(\Omega \setminus K_\varepsilon )$, i.e. of problem (1.8) with $K=K_\varepsilon$. If there exists a limiting set K for which ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon )\rightarrow {\textrm{cap}}_{V^{m}\!,\,\Omega }(K)=0$, Theorem 1.1 and Proposition 2.1 guarantee that $\lambda _J(\Omega {\setminus } K_\varepsilon )\rightarrow \lambda _J(\Omega {\setminus } K)=\lambda _J(\Omega )$, if we denote by $\lambda _J(\Omega \setminus K)$ the corresponding eigenvalue of the limiting problem in $V^m_0(\Omega {\setminus } K)=V^m_0(\Omega )$. Moreover, Theorem 1.1 gives us a first estimate on the eigenvalue convergence rate in terms of the $V^m$-capacity of the removed set $K_\varepsilon$. Inspired by [2], we are now going to sharpen this result, by detecting the first term of the asymptotic expansion of $\lambda _J(\Omega \setminus K_\varepsilon )$, provided the family of compact sets $\{K_\varepsilon \}_{\varepsilon >0}$ converges to K as specified in Definition 1.1. Indeed, as the next two propositions show, this definition of convergence, although very general, is enough to prove the stability of the $(u,V^m)$-capacity in case ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)=0$, as well as the Mosco convergence of the functional spaces.

Proposition 3.1

Let $\{K_\varepsilon \}_{\varepsilon >0}$ be a family of compact sets contained in $\Omega \subset {\mathbb {R}}^N$ concentrating to a compact set $K\subset \Omega$ with ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)=0$ as $\varepsilon \rightarrow 0$. Then, for every function $u\in V^m(\Omega )$, one has that $W_{K_\varepsilon ,u}\rightarrow W_{K,u}=0$ strongly in $V^m(\Omega )$ and

$$\begin{aligned}{\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u)\rightarrow {\textrm{cap}}_{V^{m}\!,\,\Omega }(K,u)=0\quad \hbox {as} \quad \varepsilon \rightarrow 0.\end{aligned}$$

Proof

It is analogous to the one for the case $m=1$ given in [2, Proposition B.1]. It is in fact essentially based on the fact that $V^m_0(\Omega \setminus K)=V^m(\Omega )$ for sets of null $V^m$-capacity, as shown in Proposition 2.1, and on the consequent Remark 1. $\square$

Definition 3.1

Let $\{K_\varepsilon \}_{\varepsilon >0}$ be a family of compact sets compactly contained in a bounded domain $\Omega$. We say that $\Omega \setminus K_\varepsilon$ converges to $\Omega \setminus K$ in the sense of Mosco in $V^m_0$ if the following two conditions are satisfied:

(i):

the weak limit points in $V^m(\Omega )$ of every family of functions $u_\varepsilon \in V^m_0(\Omega \setminus K_\varepsilon )$ belong to $V^m_0(\Omega \setminus K)$;

(ii):

for every $u\in V^m_0(\Omega \setminus K)$, there exists a family of functions $\{u_\varepsilon \}_{\varepsilon >0}$ such that, for every $\varepsilon >0$, $u_\varepsilon \in V^m_0(\Omega {\setminus } K_\varepsilon )$ and $u_\varepsilon \rightarrow u$ in $V^m(\Omega )$.

In order to stress the underlined functional space, we also say that $V^m_0(\Omega \setminus K_\varepsilon )$ converges to $V^m_0(\Omega \setminus K)$ in the sense of Mosco.

Lemma 3.2

Let $\{K_\varepsilon \}_{\varepsilon >0}$ be a family of compact sets concentrating to a compact set $K\subset \Omega$ with ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)=0$ as $\varepsilon \rightarrow 0$. Then $V^m_0(\Omega \setminus K_\varepsilon )$ converges to $V^m_0(\Omega \setminus K)$ as $\varepsilon \rightarrow 0$ in the sense of Mosco.

Proof

Verification of (i). Let $\{u_\varepsilon \}_\varepsilon \subset V^m(\Omega )$ be such that $u_\varepsilon \in V^m_0(\Omega \setminus K_\varepsilon )$ and $u_\varepsilon \rightharpoonup u$ in $V^m(\Omega )$. Since ${\textrm{cap}}_{m,\Omega }(K)=0$, we have that $V^m(\Omega )=V^m_0(\Omega \setminus K)$ by Proposition 2.1, hence u belongs to $V^m_0(\Omega {\setminus } K)$.

Verification of (ii). Let $u\in V^m_0(\Omega {\setminus } K)=V^m(\Omega )$. For every $k\in {\mathbb {N}}{\setminus }\{0\}$, by density there exists $\chi _k\in X^m_0(\Omega \setminus K)$ such that $\Vert \nabla ^m(\chi _k-u)\Vert _2<\frac{1}{k}$. Note that, if $K_\varepsilon$ is concentrating to K in the sense of Definition 1.1, for a chosen cutoff function $\eta _K\in C^\infty _0(\Omega )$ such that $\eta _K\equiv 1$ in a neighbourhood of K, one has that $\eta _K\equiv 1$ in a neighbourhood of $K_\varepsilon$ for $\varepsilon$ small enough. By definition of $W_K$, one may find $(W_n)_n\subset V^m(\Omega )$ and a sequence $(\varepsilon _n)_n\searrow 0$ such that $\Vert \nabla ^mW_n\Vert _2<\frac{1}{n}$ and $W_n\equiv 1$ in a neighbourhood of $K_\varepsilon$ for all $\varepsilon \in (0,\varepsilon _n]$. Defining, for all $n,k\in {\mathbb {N}}{\setminus }\{0\}$, $Z_n^k:=\chi _k\left( 1-\eta _KW_n\right)$, one has that $Z_n^k\in V^m_0(\Omega \setminus K_\varepsilon )$ for all $\varepsilon \in (0,\varepsilon _n]$ and

$$\begin{aligned} \Vert \nabla ^m\left( Z_n^k-\chi _k\right) \Vert _2 \lesssim \Vert \eta _K\Vert _{W^{m,\infty }(\Omega )}\Vert W_n\Vert _{V^m(\Omega )}\Vert \chi _k\Vert _{W^{m,\infty }(\Omega )}\le \frac{C_k}{n} \end{aligned}$$

for some $C_k>0$ depending on k. Hence, for each $k\in {\mathbb {N}}{\setminus }\{0\}$, there exists $n_k\in {\mathbb {N}}$ such that $n_k\nearrow \infty$ as $k\rightarrow \infty$ and $\Vert \nabla ^m\left( Z_{n_k}^k-\chi _k\right) \Vert _2<\frac{1}{k}$. In order to construct the family required for the Mosco convergence, for any $\varepsilon \in (0,\varepsilon _{n_1})$ it is sufficient to define $u_\varepsilon :=Z_{n_k}^k$, choosing k such that $\varepsilon \in (\varepsilon _{n_{k+1}},\varepsilon _{n_k}]$. Indeed, for any $\delta >0$, letting $k\in {\mathbb {N}}\setminus \{0\}$ be such that $\frac{2}{k}<\delta$, we have that, for all $\varepsilon \in (0,\varepsilon _{n_k}]$, $u_\varepsilon =Z_{n_j}^j$ for some $j\ge k$, so that

$$\begin{aligned} \Vert \nabla ^m\left( u_\varepsilon -u\right) \Vert _2\le \Vert \nabla ^m\big (Z_{n_j}^j -\chi _j\big )\Vert _2+\Vert \nabla ^m\left( \chi _j-u\right) \Vert _2<\frac{2}{j}\le \frac{2}{k}<\delta , \end{aligned}$$

thus proving that $u_\varepsilon \rightarrow u$ in $V^m(\Omega )$ as $\varepsilon \rightarrow 0$. $\square$

Remark 5

Note that the Mosco convergence of sets implies the convergence of the spectra of the polyharmonic operators, see [5]. For the Dirichlet case, in particular this can be seen combining [5, Proposition 2.9, footnote 2 p.8, and Theorem 4.3].

Lemma 3.3

Let $K\subset \Omega$ be a compact set and $\{K_\varepsilon \}_{\varepsilon >0}$ be a family of compact subsets of $\Omega$ concentrating to K as $\varepsilon \rightarrow 0$. If ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)=0$, then, for every $f\in V^m(\Omega )$, we have that $\Vert W_{K_\varepsilon ,f}\Vert _{H^{m-1}(\Omega )}^2={\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,f))$ as $\varepsilon \rightarrow 0$.

Proof

The proof is inspired by [2, Lemma A.1]. Suppose by contradiction that there exist $C>0$ and a sequence $\varepsilon _n\rightarrow 0$ such that

$$\begin{aligned} \Vert W_{K_{\varepsilon _n},f}\Vert _{H^{m-1}(\Omega )}^2\ge C\,{\textrm{cap}}_{V^{m}\!,\,\Omega }(K_{\varepsilon _n},f)\quad \text {for all } n. \end{aligned}$$

(3.7)

Let us consider

$$\begin{aligned} Z_n:=\frac{W_{K_{\varepsilon _n},f}}{\Vert W_{K_{\varepsilon _n},f}\Vert _{H^{m-1}(\Omega )}}. \end{aligned}$$

We have

$$\begin{aligned} \Vert Z_n\Vert _{H^{m-1}(\Omega )}=1\qquad \text{ and }\qquad \Vert \nabla ^mZ_n\Vert _2^2=\frac{\Vert \nabla ^mW_{K_{\varepsilon _n},f}\Vert _2^2}{\Vert W_{K_{\varepsilon _n},f}\Vert ^2_{H^{m-1}(\Omega )}}\le \frac{1}{C} \end{aligned}$$

with $C>0$ as in (3.7).

Then one may find a subsequence (still denoted by $Z_n$) and $Z\in V^m(\Omega )$, so that $Z_n\rightharpoonup Z$ in $V^m(\Omega )$. By the compact embedding $H^m(\Omega )\hookrightarrow \hookrightarrow H^{m-1}(\Omega )$, Z is also the strong limit in the $H^{m-1}(\Omega )$ topology. This implies that $\Vert Z\Vert _{H^{m-1}(\Omega )}=1$. However, by the Mosco convergence of Lemma 3.2 one may show that

$$\begin{aligned} \int _{\Omega \setminus K}\nabla ^mZ\,\nabla ^m\varphi =0\qquad \text{ for } \text{ all }\;\varphi \in V^m_0(\Omega \setminus K), \end{aligned}$$

(3.8)

and hence for all $\varphi \in V^m(\Omega )$ by Proposition 2.1, since we assumed ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K)=0$. Indeed, given $\varphi \in V^m_0(\Omega \setminus K)$ there exists a sequence $\{\varphi _{\varepsilon _n}\}_n$ so that $\varphi _{\varepsilon _n}\in V^m_0(\Omega {\setminus } K_{\varepsilon _n})$ for each $n\in {\mathbb {N}}$ and $\varphi _{\varepsilon _n}\rightarrow \varphi$ in $V^m(\Omega )$, for which then

$$\begin{aligned} \int _{\Omega \setminus K_{\varepsilon _n}}\nabla ^mZ_n\,\nabla ^m\varphi _{\varepsilon _n}=0 \end{aligned}$$

for all $n\in {\mathbb {N}}$ by definition of $Z_n$ as a multiple of the capacitary potential $W_{K_{\varepsilon _n},f}$. Then, (3.8) follows by weak-strong convergence in $V^m(\Omega )$, yielding $Z=0$, a contradiction. $\square$

We are now in the position to prove the asymptotic expansion of the perturbed eigenvalues. The suitable asymptotic parameter turns out to be the $(u_J,V^m)$-capacity of the removed set, where $u_J$ is an eigenfunction normalized in $L^2(\Omega )$ associated to the eigenvalue $\lambda _J$.

In the following, $(-\Delta )^m_\varepsilon$ stands for the polyharmonic operator acting on $V^m_0(\Omega {\setminus } K_\varepsilon )$. Similarly, to shorten notation, we write $\lambda _\varepsilon :=\lambda _J(\Omega \setminus K_\varepsilon )$ and the corresponding $(u_J,V^m)$-capacitary potential is denoted by $W_\varepsilon :=W_{K_\varepsilon ,u_J}\in V^m(\Omega )$; we also write $\lambda _J:=\lambda _J(\Omega )$.

Proof of Theorem 1.2

First note that the simplicity of $\lambda _J$, i.e. of $\lambda _J(\Omega \setminus K)$ by Proposition 2.1, together with the convergence of the perturbed eigenvalues given by Theorem 1.1, implies the simplicity of $\lambda _\varepsilon$ for $\varepsilon$ sufficiently small.

Let $\psi _\varepsilon :=u_J-W_\varepsilon \in V^m_0(\Omega {\setminus } K_\varepsilon )$ and $\varphi \in V^m_0(\Omega {\setminus } K_\varepsilon )$. Then

$$\begin{aligned} \int _{\Omega }\nabla ^m\psi _\varepsilon \nabla ^m\varphi -\lambda _J\int _{\Omega }\psi _\varepsilon \varphi =\int _{\Omega \setminus K_\varepsilon }\!\nabla ^m u_J\nabla ^m\varphi -\lambda _J\int _{\Omega }\psi _\varepsilon \varphi =\lambda _J\int _{\Omega }W_\varepsilon \varphi . \end{aligned}$$

This means that $\psi _\varepsilon$ satisfies weakly in $V^m_0(\Omega \setminus K_\varepsilon )$ the equation

$$\begin{aligned} \left( (-\Delta )^m-\lambda _J\right) \psi _\varepsilon =\lambda _JW_\varepsilon . \end{aligned}$$

(3.9)

Since by Lemma 3.3 with $f=u_J$ one has $\Vert W_\varepsilon \Vert _2={\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)^{1/2})$ as $\varepsilon \rightarrow 0$, we infer

$$\begin{aligned} \text {dist}(\lambda _J,\sigma ((-\Delta )^m_\varepsilon ))\le \frac{\Vert ((-\Delta )^m-\lambda _J)\psi _\varepsilon \Vert _2}{\Vert \psi _\varepsilon \Vert _2}={\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)^{1/2}) \end{aligned}$$

as $\varepsilon \rightarrow 0$. Since we know by Theorem 1.1 and Proposition 3.1 that the spectrum of $(-\Delta )^m$ in $V^m_0(\Omega {\setminus } K_\varepsilon )$ varies continuously with respect to $\varepsilon$ and that the eigenvalue $\lambda _\varepsilon$ is simple for $\varepsilon$ small enough, one first deduces

$$\begin{aligned} |\lambda _\varepsilon -\lambda _J|={\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)^{1/2})\quad \text {as }\varepsilon \rightarrow 0. \end{aligned}$$

Denote now by $\Pi _\varepsilon$ the projector (with respect to the scalar product in $L^2$) onto the eigenspace related to $\lambda _\varepsilon$ and take $u_\varepsilon :=\frac{\Pi _\varepsilon \psi _\varepsilon }{\Vert \Pi _\varepsilon \psi _\varepsilon \Vert _2}$ as normalized eigenfunction. The first goal is to estimate the difference of the two eigenfunctions $u_J$ and $u_\varepsilon$:

$$\begin{aligned} \begin{aligned} \Vert u_J-u_\varepsilon \Vert _2&\le \Vert u_J-\psi _\varepsilon \Vert _2+\Vert \psi _\varepsilon -\Pi _\varepsilon \psi _\varepsilon \Vert _2+\left\| \Pi _\varepsilon \psi _\varepsilon -\frac{\Pi _\varepsilon \psi _\varepsilon }{\Vert \Pi _\varepsilon \psi _\varepsilon \Vert _2}\right\| _2\\&=\Vert W_\varepsilon \Vert _2+\Vert \psi _\varepsilon -\Pi _\varepsilon \psi _\varepsilon \Vert _2+\left| 1-\Vert \Pi _\varepsilon \psi _\varepsilon \Vert _2^{-1}\right| \Vert \Pi _\varepsilon \psi _\varepsilon \Vert _2. \end{aligned} \end{aligned}$$

Note that Lemma 3.3 yields $\Vert W_\varepsilon \Vert _2={\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)^{1/2})$ and, moreover, we have

$$\begin{aligned} \Vert \Pi _\varepsilon \psi _\varepsilon \Vert _2\le \Vert \psi _\varepsilon \Vert _2\le \Vert u_J\Vert _2+\Vert W_\varepsilon \Vert _2={\mathcal {O}}(1)\quad \text {as }\varepsilon \rightarrow 0. \end{aligned}$$

Hence, we need to estimate $\Vert \psi _\varepsilon -\Pi _\varepsilon \psi _\varepsilon \Vert _2$ and $\left| 1-\Vert \Pi _\varepsilon \psi _\varepsilon \Vert _2^{-1}\right|$. We claim that both quantities are ${\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)^{1/2})$, obtaining thus

$$\begin{aligned} \Vert u_J-u_\varepsilon \Vert _2={\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)^{1/2})\quad \text {as }\varepsilon \rightarrow 0, \end{aligned}$$

(3.10)

and postpone the proof of such claim to the end of the proof. Then we have

$$\begin{aligned} \begin{aligned} {\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)&=\int _{\Omega }|\nabla ^m W_\varepsilon |^2=\int _{\Omega }\nabla ^m(u_J-\psi _\varepsilon )\nabla ^m W_\varepsilon =\int _{\Omega }\nabla ^m u_J\nabla ^m W_\varepsilon \\&=\lambda _J\int _{\Omega }u_JW_\varepsilon =\lambda _J\int _{\Omega }u_\varepsilon W_\varepsilon +\lambda _J\int _{\Omega }(u_J-u_\varepsilon )W_\varepsilon \\&{\mathop {=}\limits ^{(3.9)}}\int _{\Omega }\nabla ^m\psi _\varepsilon \nabla ^m u_\varepsilon -\lambda _J\int _{\Omega }u_\varepsilon \psi _\varepsilon +\lambda _J\int _{\Omega }(u_J-u_\varepsilon )W_\varepsilon \\&=\left( \lambda _\varepsilon -\lambda _J\right) \int _{\Omega }u_\varepsilon \psi _\varepsilon +\lambda _J\int _{\Omega }(u_J-u_\varepsilon )W_\varepsilon , \end{aligned} \end{aligned}$$

and therefore

$$\begin{aligned} \left( \lambda _\varepsilon -\lambda _J\right) \int _{\Omega }u_\varepsilon \psi _\varepsilon ={\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)-\lambda _J\int _{\Omega }\left( u_J-u_\varepsilon \right) W_\varepsilon . \end{aligned}$$

(3.11)

Since now

$$\begin{aligned} \int _{\Omega }u_\varepsilon \psi _\varepsilon =\Vert u_\varepsilon \Vert _2^2+\int _{\Omega }u_\varepsilon \left( \psi _\varepsilon -u_\varepsilon \right) =1+\int _{\Omega }u_\varepsilon \left( \psi _\varepsilon -u_\varepsilon \right) \end{aligned}$$

and

$$\begin{aligned} \left| \int _{\Omega }u_\varepsilon \left( \psi _\varepsilon -u_\varepsilon \right) \right| \le \Vert u_\varepsilon \Vert _2\Vert \psi _\varepsilon -u_\varepsilon \Vert _2={\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)^{1/2}), \end{aligned}$$

where the last equality is again due to the claims above, from (3.11) and (3.10), we infer

$$\begin{aligned} \lambda _\varepsilon -\lambda _J=\frac{{\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)+{\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J))}{1+{\scriptstyle {\mathcal {O}}}(1)}={\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)\left( 1+{\scriptstyle {\mathcal {O}}}(1)\right) \end{aligned}$$

as $\varepsilon \rightarrow 0$, as desired. To conclude, we prove the claims above. Since $\lambda _\varepsilon$ is a simple eigenvalue, denoting by $T_\varepsilon$ the restriction of $(-\Delta )^m_\varepsilon$ on $\ker \Pi _\varepsilon$, we have that $\sigma (T_\varepsilon )=\sigma ((-\Delta )^m_\varepsilon )\setminus \{\lambda _\varepsilon \}$ and, by simplicity, $\text {dist}(\lambda _\varepsilon ,\sigma (T_\varepsilon ))\ge \delta$ for some $\delta >0$, uniformly with respect to $\varepsilon$. Hence,

$$\begin{aligned} \begin{aligned}&\Vert \psi _\varepsilon -\Pi _\varepsilon \psi _\varepsilon \Vert _2\le \frac{1}{\delta }\left\| \left( T_\varepsilon -\lambda _\varepsilon \right) \left( \psi _\varepsilon -\Pi _\varepsilon \psi _\varepsilon \right) \right\| _2\lesssim \Vert \left( (-\Delta )^m-\lambda _\varepsilon \right) \psi _\varepsilon \Vert _2\\&~~~~~~~\le \Vert \left( (-\Delta )^m-\lambda _J\right) \psi _\varepsilon \Vert _2+|\lambda _J-\lambda _\varepsilon |\Vert \psi _\varepsilon \Vert _2=|\lambda _J|\Vert W_\varepsilon \Vert _2+|\lambda _J-\lambda _\varepsilon |\Vert \psi _\varepsilon \Vert _2\\&~~~~~~~={\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)^{1/2}). \end{aligned} \end{aligned}$$

Since, by definition of $\psi _\varepsilon$ and Lemma 3.3, $\Vert \psi _\varepsilon \Vert _2=1+{\scriptstyle {\mathcal {O}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)^{1/2})}$ as $\varepsilon \rightarrow 0$, one thus finds that $\Vert \Pi _\varepsilon \psi _\varepsilon \Vert _2=1+{\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)^{1/2})$, which in particular yields the desired estimate $1-\Vert \Pi _\varepsilon \psi _\varepsilon \Vert _2^{-1}={\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)^{1/2})$. This concludes the proof. $\square$

Remark 6

We observe that, in the proof of Theorem 1.2, the following estimate for the normalized eigenfunction $u_\varepsilon \in V^m_0(\Omega \setminus K_\varepsilon )$ of $(-\Delta )^m$ relative to $\lambda _J(\Omega \setminus K_\varepsilon )$ was established:

$$\begin{aligned} \Vert u_\varepsilon -u_J\Vert _2={\scriptstyle {\mathcal {O}}}({\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)^{1/2})\quad \text {as }\varepsilon \rightarrow 0. \end{aligned}$$

4 Sharp asymptotic expansions of perturbed eigenvalues: the case of uniformly shrinking holes.

4.1 A blow-up analysis

In Theorem 1.2 we obtained an asymptotic expansion of a perturbed simple eigenvalue in terms of ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)$, in case the limiting removed set has zero $V^m$-capacity. However, in view of possible applications, the dependence on the removed set $K_\varepsilon$ is quite implicit using such an asymptotic parameter. Therefore, we aim to understand how this quantity behaves with respect to the diameter of the hole, in the case of a uniformly shrinking family of compact sets which concentrate to a point, a set with zero $V^m$-capacity in large dimensions by Proposition 2.3.

First, we only suppose that $\{K_\varepsilon \}_{\varepsilon >0}$ uniformly shrinks to a point, which is assumed to be 0 in the following, in the sense that

$$\begin{aligned} K_\varepsilon \subset \overline{B_{C\varepsilon }(0)} \end{aligned}$$

(4.1)

for some constant $C>0$ and $\varepsilon$ small enough. The following is a generalization of [2, Lemma 2.2] to the higher-order setting.

Proposition 4.1

Let $\Omega \subset {\mathbb {R}}^N$ be a smooth bounded domain such that $0\in \Omega$ and let $\{K_\varepsilon \}_{\varepsilon >0}$ be a family of compact sets satisfying (4.1). Let $h\in H^m(\Omega )$ be such that

$$\begin{aligned}|D^kh(x)|={\mathcal {O}}(|x|^{\gamma -k})\quad \hbox {as}\quad |x|\rightarrow 0\end{aligned}$$

for some $\gamma \in {\mathbb {N}}$ and all $k\in \{0,\dots ,m\}$. Then

$$\begin{aligned} {\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,h)={\mathcal {O}}(\varepsilon ^{N-2m+2\gamma })\quad \text {as }\varepsilon \rightarrow 0. \end{aligned}$$

(4.2)

Proof

By Proposition 2.2(iii), it is sufficient to prove (4.2) for the Dirichlet case $V^m(\Omega )=H^m_0(\Omega )$. Let $\varphi \in C^\infty _0({\mathbb {R}}^N)$ with $\text {supp}\,\varphi \subset B_2(0)$ and $\varphi \equiv 1$ in a neighbourhood of $\overline{B_1(0)}$, and define $\varphi _\varepsilon (x):=\varphi ((C\varepsilon )^{-1}x)$ for all $\varepsilon >0$ small. Then $h_\varepsilon :=\varphi _\varepsilon h$ coincides with h in a neighbourhood of $\overline{B_{C\varepsilon }(0)}$. By monotonicity

$$\begin{aligned} \begin{aligned} {\textrm{cap}}_{m,\,\Omega }(K_\varepsilon ,h)&\le {\textrm{cap}}_{m,\,\Omega }(\overline{B_{C\varepsilon }(0)},h)\le \int _{\Omega }|\nabla ^m h_\varepsilon |^2\\&\lesssim \sum _{k=0}^m\int _{B_{2C\varepsilon }(0)}|D^{m-k}\varphi _\varepsilon (x)|^2|D^kh(x)|^2\,dx\\&\lesssim \sum _{k=0}^m(C\varepsilon )^{2k-2m}\int _{B_{2C\varepsilon }(0)}\left| D^{m-k}\varphi \left( \frac{x}{C\varepsilon }\right) \right| ^2|D^kh(x)|^2\,dx\\&\lesssim \sum _{k=0}^m(C\varepsilon )^{2k-2m+N}\int _{B_2(0)}|D^{m-k}\varphi (y)|^2|D^kh(C\varepsilon y)|^2\,dy\\&\lesssim \varepsilon ^{N-2m+2\gamma }\sum _{k=0}^m\int _{B_2(0)}|D^{m-k}\varphi (y)|^2\,dy\lesssim \varepsilon ^{N-2m+2\gamma }, \end{aligned} \end{aligned}$$

having used the assumption that $\Vert D^kh\Vert _\infty \lesssim \varepsilon ^{\gamma -k}$ in $B_{2C\varepsilon }(0)$. $\square$

Next, having in mind the model case $K_\varepsilon :=\varepsilon {\mathcal {K}}$ for a fixed compactum ${\mathcal {K}}$, we consider families of compact sets which uniformly shrink to $\{0\}$ as in (4.1) but enjoying a more specific structure. To this aim we assume

• (M1) there exists $M\subset {\mathbb {R}}^N$ compact such that $\varepsilon ^{-1}K_\varepsilon \subseteq M$ for all $\varepsilon \in (0,1)\,$;

• (M2) there exists ${\mathcal {K}}\subset {\mathbb {R}}^N$ compact such that ${\mathbb {R}}^N\setminus \varepsilon ^{-1}K_\varepsilon \rightarrow {\mathbb {R}}^N\setminus {\mathcal {K}}$ in the sense of Mosco as $\varepsilon \rightarrow 0$.

In our context (M2) means the following:

(i):

if $u_\varepsilon \in D^{m,2}_0({\mathbb {R}}^N\setminus \varepsilon ^{-1}K_\varepsilon )$ is so that $u_\varepsilon \rightharpoonup u$ in $D^{m,2}_0({\mathbb {R}}^N)$ as $\varepsilon \rightarrow 0$, then we have that $u\in D^{m,2}_0({\mathbb {R}}^N\setminus {\mathcal {K}})$;

(ii):

if $u\in D^{m,2}_0({\mathbb {R}}^N\setminus {\mathcal {K}})$, then there exists a family of functions $\{u_\varepsilon \}_{\varepsilon >0}$ such that $u_\varepsilon \in D^{m,2}_0({\mathbb {R}}^N{\setminus }\varepsilon ^{-1}K_\varepsilon )$ for all $\varepsilon >0$ and $u_\varepsilon \rightarrow u$ in $D^{m,2}_0({\mathbb {R}}^N)$ as $\varepsilon \rightarrow 0$.

In this case we also say that $D^{m,2}_0({\mathbb {R}}^N\setminus \varepsilon ^{-1}K_\varepsilon )$ converges to $D^{m,2}_0({\mathbb {R}}^N\setminus {\mathcal {K}})$ in the sense of Mosco.

Remark 7

Assumption (M1) is actually equivalent to the condition (4.1), since $M\subset B_C(0)$ for some $C>0$.

Lemma 4.2

Let $N>2m$. Under the assumption (M1) the following are equivalent:

• 1. $D^{m,2}_0({\mathbb {R}}^N\setminus \varepsilon ^{-1}K_\varepsilon )$ converges to $D^{m,2}_0({\mathbb {R}}^N\setminus {\mathcal {K}})$ in the sense of Mosco;

• 2. $H^m_0(B_R(0)\setminus \varepsilon ^{-1}K_\varepsilon )$ converges to $H^m_0(B_R(0)\setminus {\mathcal {K}})$ in the sense of Mosco for all $R>r(M)$, where $r(M):=\inf \{\rho >0\,|\,B_\rho (0)\supset M\}$.

We denote by (M$_R$2.i) and (M$_R$2.ii) the correspondent conditions (M2.i) and (M2.ii) which enter in the definition of the Mosco convergence relative to the space $H^m_0(B_R(0))$. In the following we use the shorter notation $B_R:=B_R(0)$.

Proof

$\varvec{1)\Rightarrow 2)}$. Verification of (M$_R$2.i). Let $\{u_\varepsilon \}_{\varepsilon >0}\subset H^m_0(B_R)$ be a family of functions such that $u_\varepsilon \in H^m_0(B_R{\setminus }\varepsilon ^{-1}K_{\varepsilon })$ and $u_\varepsilon \rightharpoonup u$ in $H^m_0(B_R)$. We show that $u\in H^m_0(B_R\setminus ~\!{\mathcal {K}})$. Denoting by $u_\varepsilon ^E$ and $u^E$ the trivial extension of $u_\varepsilon$ and u outside $B_R$ respectively, then $u_\varepsilon ^E\in D^{m,2}_0({\mathbb {R}}^N\setminus \varepsilon ^{-1}K_{\varepsilon })$ and $u_\varepsilon ^E\rightharpoonup u^E$ in $D^{m,2}_0({\mathbb {R}}^N)$. Hence condition (M2.i) guarantees that $u^E\in D^{m,2}_0({\mathbb {R}}^N{\setminus }{\mathcal {K}})$, which by construction implies that $u\in H^m_0(B_R\setminus {\mathcal {K}})$.

Verification of (M$_R$2.ii). Let $v\in C^\infty _0(B_R{\setminus }{\mathcal {K}})$ and $\Lambda _1,\Lambda _2\subset \Omega$ be two open sets such that $\text {supp}\,v\subset \subset \Lambda _1\subset \subset \Lambda _2\subset \subset B_R$. Take $\eta \in C^\infty _0(\Lambda _2)$ with $\eta \equiv 1$ on $\Lambda _1$. Since $v^E\in D^{m,2}_0({\mathbb {R}}^N\setminus {\mathcal {K}})$, then by (M2.ii) there exists a family $\{v_\varepsilon \}_{\varepsilon >0}\subset D^{m,2}_0({\mathbb {R}}^N)$ with $v_\varepsilon \in D^{m,2}_0({\mathbb {R}}^N\setminus \varepsilon ^{-1}K_{\varepsilon })$ for all $\varepsilon >0$ such that $v_\varepsilon \rightarrow v^E$ in $D^{m,2}_0({\mathbb {R}}^N)$, i.e.

$$\begin{aligned}\Vert \nabla ^m(v_\varepsilon -v^E)\Vert _{L^2({\mathbb {R}}^N)}\rightarrow 0\quad \hbox {as}\quad \varepsilon \rightarrow 0.\end{aligned}$$

By construction, $\eta v_\varepsilon \in H^m_0(B_R{\setminus }\varepsilon ^{-1}K_{\varepsilon })$. We claim that $\eta v_\varepsilon \rightarrow v$ in $H^m_0(B_R)$. Indeed, denoting by $q_j:=2^*_{m,j}=\frac{2N}{N-2(m-j)}\ge 2$ and $p_j:=2\left( \tfrac{q_j}{2}\right) '=\frac{N}{m-j}$ for $j\in \{0,\dots ,m\}$, one has

$$\begin{aligned} \begin{aligned} \Vert \nabla ^m(\eta v_\varepsilon -v)\Vert _{L^2(B_R)}&=\Vert \nabla ^m(\eta (v_\varepsilon -v))\Vert _{L^2(B_R)}\le \sum _{j=0}^m\Vert D^{m-j}\eta \,D^j(v_\varepsilon -v)\Vert _{L^2(B_R)}\\&\le \sum _{j=0}^m\Vert D^{m-j}\eta \Vert _{L^{p_j}(\text {supp}\,\eta )}\Vert D^j(v_\varepsilon -v^{E})\Vert _{L^{q_j}({\mathbb {R}}^N)}\\&\le \sum _{j=0}^m|\text {supp}\,\eta |^{\frac{1}{p_j}}\Vert D^{m-j}\eta \Vert _\infty \Vert D^m(v_\varepsilon -v^{E})\Vert _{L^2({\mathbb {R}}^N)}\\&\le C(m,N,R)\Vert \eta \Vert _{W^{m,\infty }({\mathbb {R}}^N)}\Vert \nabla ^m(v_\varepsilon -v^{E})\Vert _{L^2({\mathbb {R}}^N)}\rightarrow 0. \end{aligned} \end{aligned}$$

(4.3)

The last steps are due to the critical Sobolev embedding on ${\mathbb {R}}^N$ (for which it is fundamental that $N>2m$), see (2.3), and to the equivalence of the norms $\Vert D^m\cdot \Vert _2$ and $\Vert \nabla ^m\cdot \Vert _2$, see e.g. [17, Chp. 2.2].

The above argument and the density of $C^\infty _0(B_R\setminus {\mathcal {K}})$ in $H^m_0(B_R\setminus {\mathcal {K}})$ imply that, fixing any $v\in H^m_0(B_R{\setminus }{\mathcal {K}})$, for every $\delta >0$ there exists a family $\{v_{\delta ,\varepsilon }\}_{\varepsilon >0}$ such that $v_{\delta ,\varepsilon }\in H^m_0(B_R\setminus K_\varepsilon )$ and $\Vert v_{\delta ,\varepsilon }-v\Vert _{H^m(B_R)}<\delta$ for all $\varepsilon \in (0,{\bar{\varepsilon }}_\delta ]$ for some ${\bar{\varepsilon }}_\delta >0$. Therefore there exists a vanishing sequence $(\varepsilon _n)_n\searrow 0$ such that $\big \Vert v_{\frac{1}{k},\varepsilon }-v\big \Vert _{H^m(B_R)}<\frac{1}{k}$ for all $\varepsilon \in (0,\varepsilon _k]$. Defining $v_\varepsilon =v_{\frac{1}{n},\varepsilon }$ for $\varepsilon \in (\varepsilon _{n+1}, \varepsilon _{n}]$, we have that, for all $\varepsilon \in (0, \varepsilon _{1}]$, $v_\varepsilon \in H^m_0(B_R{\setminus } K_\varepsilon )$ and $v_\varepsilon \rightarrow v$ in $H^m_0(B_R)$ as $\varepsilon \rightarrow 0$.

$\varvec{2)\Rightarrow 1)}$. Verification of (M2.i). Let $\{u_\varepsilon \}_{\varepsilon >0}\subset D^{m,2}_0({\mathbb {R}}^N)$ be a family of functions such that $u_\varepsilon \in D^{m,2}_0({\mathbb {R}}^N{\setminus }\varepsilon ^{-1}K_{\varepsilon })$ and $u_\varepsilon \rightharpoonup u$ in $D^{m,2}_0({\mathbb {R}}^N)$. Taking $\eta \in C^\infty ({\mathbb {R}}^N)$ and $R>0$ such that $\text {supp}\,\eta \subset B_R(0)$, due to the continuity of the map $D^{m,2}_0({\mathbb {R}}^N)\rightarrow H^m_0(B_R)$, $u\mapsto \eta u$, which can be easily proved arguing as in (4.3), one has that $\eta u_\varepsilon \rightharpoonup \eta u$ in $H^m_0(B_R)$. Hence, (M$_R$2.i) implies $\eta u\in H^m_0(B_R\setminus {\mathcal {K}})$. Hence $\eta u\in D^{m,2}_0({\mathbb {R}}^N{\setminus }{\mathcal {K}})$ for every $\eta \in C^\infty _0({\mathbb {R}}^N)$. Let us now take $\eta _1\in C^\infty _0({\mathbb {R}}^N)$ with $0\le \eta _1\le 1$, $\eta _1\equiv 1$ on $B_{\frac{1}{2}}$ and $\text {supp}\,\eta _1\subset B_1$, and define $\eta _k:=\eta _1\left( \frac{\cdot }{k}\right)$, so that $\text {supp}\,\eta _k\subset B_k$. We are going to prove that $\eta _ku\rightarrow u$ in $D^{m,2}_0({\mathbb {R}}^N)$ as $k\rightarrow +\infty$, in order to conclude that $u\in D^{m,2}_0({\mathbb {R}}^N\setminus {\mathcal {K}})$. We estimate as follows:

$$\begin{aligned} \Vert \nabla ^m(\eta _ku-u)\Vert _{L^2({\mathbb {R}}^N)}^2&\le \int _{{\mathbb {R}}^N}|\eta _k-1|^2|\nabla ^mu|^2+\sum _{j=0}^{m-1}\int _{B_k\setminus B_{\frac{k}{2}}}|D^{m-j}\eta _k|^2|D^ju|^2\nonumber \\&\le \int _{{\mathbb {R}}^N\setminus B_{\frac{k}{2}}}|\nabla ^mu|^2+\sum _{j=0}^{m-1}\Vert D^{m-j}\eta _k\Vert _{L^{p_j}({\mathbb {R}}^N)}^2\Vert D^ju\Vert _{L^{q_j}({\mathbb {R}}^N\setminus B_{\frac{k}{2}})}^{2}\nonumber \\&=\int _{{\mathbb {R}}^N\setminus B_{\frac{k}{2}}}|\nabla ^mu|^2+\sum _{j=0}^{m-1}\Vert D^{m-j}\eta _1\Vert _{L^{p_j}({\mathbb {R}}^N)}^2\Vert D^ju\Vert _{L^{q_j}({\mathbb {R}}^N\setminus B_{\frac{k}{2}})}^{2}, \end{aligned}$$

(4.4)

where we have used the fact that $\text {supp}\, (D^{m-j}\eta _k)\subset B_k{\setminus } B_{\frac{k}{2}}$ if $j\le m-1$ and

$$\begin{aligned} \Vert D^{m-j}\eta _k\Vert _{L^{p_j}({\mathbb {R}}^N)}^2&=k^{-2(m-j)}\left( \int _{{\mathbb {R}}^N}|D^{m-j}\eta _1(x/k)|^{\frac{N}{m-j}}\,dx\right) ^{\!\!\frac{2(m-j)}{N}}\\ {}&=\left( \int _{{\mathbb {R}}^N}|D^{m-j}\eta _1(y)|^{\frac{N}{m-j}}\,dy\right) ^{\!\!\frac{2(m-j)}{N}}. \end{aligned}$$

Since $u\in D^{m,2}_0({\mathbb {R}}^N)$, the first term at the right-hand side of (4.4) tends to 0 as $k\rightarrow +\infty$; moreover, the critical Sobolev embedding (2.3) implies that $D^j u\in L^{q_j}({\mathbb {R}}^N)$ for all $0\le j\le m$, so that also the second term goes to 0. We conclude that $\eta _ku\rightarrow u$ in $D^{m,2}_0({\mathbb {R}}^N)$ as $k\rightarrow +\infty$, which yields $u\in D^{m,2}_0({\mathbb {R}}^N{\setminus }{\mathcal {K}})$.

Verification of (M2.ii). Let $u\in D^{m,2}_0({\mathbb {R}}^N{\setminus }{\mathcal {K}})$. Let $\delta >0$. By density, there exists a function $v\in C^\infty _0({\mathbb {R}}^N\setminus {\mathcal {K}})$ such that $\Vert \nabla ^m(u-v)\Vert _{L^2({\mathbb {R}}^N)}<\frac{\delta }{2}$. Take $R>0$ so that $\text {supp}\,v\subset B_R$. Then $v\in H^m_0(B_R\setminus {\mathcal {K}})$ and by (M$_R$2.ii) there exist ${\bar{\varepsilon }}_\delta >0$ and a family of functions $\{\varphi ^\delta _\varepsilon \}_{\varepsilon \in (0, {\bar{\varepsilon }}_\delta )}$ such that $\varphi ^\delta _\varepsilon \in H^m_0(B_R\setminus \varepsilon ^{-1}K_{\varepsilon })$ and $\Vert \nabla ^m(v-\varphi ^\delta _\varepsilon )\Vert _{L^2(B_R)}<\frac{\delta }{2}$ for all $\varepsilon \in (0, {\bar{\varepsilon }}_\delta )$. Hence, for all $\varepsilon \in (0, {\bar{\varepsilon }}_\delta )$, $(\varphi _\varepsilon ^\delta )^E\in D^{m,2}_0({\mathbb {R}}^N\setminus \varepsilon ^{-1}K_{\varepsilon })$ and $\Vert \nabla ^m(u-(\varphi _\varepsilon ^\delta )^E)\Vert _{L^2({\mathbb {R}}^N)}<\delta$.

As a consequence, there exists a strictly decreasing and vanishing sequence $\{\varepsilon _n\}_n$ such that, for every $n\in {\mathbb N}{\setminus }\{0\}$, there exists a family of functions $\{u^n_\varepsilon \}_{\varepsilon \in (0,\varepsilon _n)}$ such that

$$\begin{aligned}u^n_\varepsilon \in D^{m,2}_0({\mathbb {R}}^N\setminus \varepsilon ^{-1}K_{\varepsilon })\quad \hbox {and}\quad \Vert \nabla ^m(u-u_\varepsilon ^n)\Vert _{L^2({\mathbb {R}}^N)}<\frac{1}{n}\end{aligned}$$

for all $\varepsilon \in (0, \varepsilon _n)$. For every $\varepsilon \in (0,\varepsilon _1)$, we define $u_\varepsilon :=u_\varepsilon ^n$ if $\varepsilon _{n+1}\le \varepsilon <\varepsilon _n$. It is easy to verify that, by construction, $u_\varepsilon \in D^{m,2}_0({\mathbb {R}}^N{\setminus }\varepsilon ^{-1}K_{\varepsilon })$ for all $\varepsilon \in (0,\varepsilon _1)$ and $\Vert \nabla ^m(u-u_\varepsilon )\Vert _{L^2({\mathbb {R}}^N)}\rightarrow 0$ as $\varepsilon \rightarrow 0$. (M2.ii) is thereby verified. $\square$

Before stating the main results of the section, we prepose a lemma about the stability of the $(h,V^m)$-capacitary potential with respect to the function h.

Lemma 4.3

Let $K\subset \Omega \subset {\mathbb {R}}^N$, K compact, $\{h_n\}_{n\in {{\mathbb {N}}}}\subset H^m_{loc}(\Omega )$ and $h\in H^m_{loc}(\Omega )$. Let us suppose that, for some $\mathcal U(K)\subset \Omega$ open neighbourhood of K, $h_n\rightarrow h$ in $H^m(\mathcal U(K))$ as $n\rightarrow \infty$ and denote by $W_{K,h_n}$ (resp. $W_{K,h}$) the capacitary potential for ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K,h_n)$ (resp. ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K,h)$). Then $W_{K,h_n}\rightarrow W_{K,h}$ in $V^m(\Omega )$ and ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K,h_n)\rightarrow {\textrm{cap}}_{V^{m}\!,\,\Omega }(K,h)$.

Proof

Being capacitary potentials, the functions $W_{K,h_n}$ and $W_{K,h}$ satisfy

$$\begin{aligned} \int _{\Omega }\nabla ^m W_{K,h}\nabla ^m\varphi =0\quad \text {and}\quad \int _{\Omega }\nabla ^m W_{K,h_n}\nabla ^m\varphi =0\quad \text {for all }n\in {{\mathbb {N}}}\text { and } \varphi \in V^m_0(\Omega \setminus K). \end{aligned}$$

Let $\eta _K\in C^\infty ({\mathbb {R}}^N)$ be a cutoff function such that $0\le \eta _K\le 1$, $\text {supp}\,\eta _K\subset {\mathcal {U}}(K)$ and $\eta _K\equiv 1$ in a neighbourhood of K. Hence, by construction, one has that

$$\begin{aligned} W_{K,h_n}-\eta _Kh_n\in V^m_0(\Omega \setminus K)\qquad \text{ and }\qquad W_{K,h}-\eta _Kh\in V^m_0(\Omega \setminus K). \end{aligned}$$

Therefore

$$\begin{aligned}&\Vert \nabla ^m(W_{K,h_n}-W_{K,h})\Vert _2^2\\&\quad =\int _{\Omega }\left( \nabla ^mW_{K,h_n}-\nabla ^mW_{K,h}\right) \left( \nabla ^mW_{K,h_n}-\nabla ^mW_{K,h}\right) \\&\quad =\int _{\Omega }\nabla ^mW_{K,h_n}\nabla ^m\left( W_{K,h_n}-\eta _Kh_n\right) +\int _{\Omega }\nabla ^mW_{K,h_n}\nabla ^m\left( \eta _Kh_n-\eta _Kh\right) \\&\qquad +\int _{\Omega }\nabla ^mW_{K,h_n}\nabla ^m\left( \eta _Kh-W_{K,h}\right) -\int _{\Omega }\nabla ^mW_{K,h}\nabla ^m\left( W_{K,h_n}-\eta _Kh_n\right) \\&\qquad -\int _{\Omega }\nabla ^mW_{K,h}\nabla ^m\left( \eta _Kh_n-\eta _Kh\right) -\int _{\Omega }\nabla ^mW_{K,h}\nabla ^m\left( \eta _Kh-W_{K,h}\right) \\&\quad =\int _{\Omega }\nabla ^m\left( W_{K,h_n}-W_{K,h}\right) \nabla ^m\left( \eta _Kh_n-\eta _Kh\right) \\&\quad \le \Vert \nabla ^mW_{K,h_n}-\nabla ^mW_{K,h}\Vert _2\Vert \nabla ^m\left( \eta _K\left( h_n-h\right) \right) \Vert _2. \end{aligned}$$

This yields

$$\begin{aligned} \begin{aligned} \Vert \nabla ^mW_{K,h_n}-\nabla ^mW_{K,h}\Vert _2&\le \Vert \nabla ^m\left( \eta _K\left( h_n-h\right) \right) \Vert _2 \lesssim \Vert h_n-h\Vert _{H^m({\mathcal {U}}(K))}\rightarrow 0, \end{aligned} \end{aligned}$$

i.e. $W_{K,h_n}\rightarrow W_{K,h}$ in $V^m(\Omega )$, directly implying that ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K,h_n)\rightarrow {\textrm{cap}}_{V^{m}\!,\,\Omega }(K,h)$ as $n\rightarrow \infty$. $\square$

Remark 8

In case $\Omega ={\mathbb {R}}^N$ the same result holds with $W_{K,h_n}\rightarrow W_{K,h}$ in $D^{m,2}_0({\mathbb {R}}^N)$.

We are now in a position to prove the main results of this section, namely a generalized version of Theorems 1.3–1.6, which take into account families of domains which satisfy (M1)–(M2), rather than just the model case $K_\varepsilon =\varepsilon {\mathcal {K}}$.

Motivated by the asymptotic scaling properties of the eigenfunctions (1.15), we apply a blow-up argument to a rescaled problem, in order to find a limit equation on ${\mathbb {R}}^N\setminus {\mathcal {K}}$ and to prove the convergence of the family of scaled capacitary potentials to the one for the limiting problem. The capacity ${\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)$ will behave then as the limit capacity on ${\mathbb {R}}^N\setminus {\mathcal {K}}$ multiplied by a suitable power of $\varepsilon$ given by the scaling. In this argument, we work with the homogeneous Sobolev spaces and, in particular, for the Navier case the characterization via Hardy–Rellich inequalities of Sect. 2.2.2 will be needed. This is the main reason for the restriction to the fourth-order case in the Navier setting, since, up to our knowledge, the extension of Proposition 2.5 to the full generality $m\ge 2$ is an open problem.

Theorem 4.4

(Asymptotic expansion of the capacity, Dirichlet case) Let $N>2m$ and $\Omega \subset {\mathbb {R}}^N$ be a bounded smooth domain with $0\in \Omega$. Let $\{K_\varepsilon \}_{\varepsilon >0}$ be a family of compact sets uniformly concentrating to $\{0\}$ satisfying (M1)–(M2) for some compact set ${\mathcal {K}}$. Let $\lambda _J$ be an eigenvalue of (1.1) with Dirichlet boundary conditions and $u_J\in H^m_0(\Omega )$ be a corresponding eigenfunction normalized in $L^2(\Omega )$. Then

$$\begin{aligned} \textrm{cap}_{m\!,\,\Omega }(K_\varepsilon ,u_J)=\varepsilon ^{N-2m+2\gamma }\left( {\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)+{\scriptstyle {\mathcal {O}}}(1)\right) \end{aligned}$$

(4.5)

as $\varepsilon \rightarrow 0$, with $\gamma$ and $U_0$ as in (1.15).

Theorem 4.5

(Asymptotic expansion of the capacity, Navier case) Let $N>4$ and $\Omega \subset {\mathbb {R}}^N$ be a bounded smooth domain with $0\in \Omega$. Let $\{K_\varepsilon \}_{\varepsilon >0}$ be a family of compact sets uniformly concentrating to $\{0\}$ satisfying (M1)–(M2) for some compact set ${\mathcal {K}}$. Let $\lambda _J$ be an eigenvalue of (1.1) with $m = 2$ and Navier boundary conditions and $u_J\in H^2_\vartheta (\Omega )$ be a corresponding eigenfunction normalized in $L^2(\Omega )$. Then

$$\begin{aligned} \textrm{cap}_{2, \vartheta \!,\,\Omega }(K_\varepsilon ,u_J)=\varepsilon ^{N-4+2\gamma }\left( {\textrm{cap}}_{2,{\mathbb {R}}^N}({\mathcal {K}},U_0)+{\scriptstyle {\mathcal {O}}}(1)\right) \end{aligned}$$

(4.6)

as $\varepsilon \rightarrow 0$, with $\gamma$ and $U_0$ as in (1.15) with $m=2$.

As a direct consequence, braiding together Theorem 1.2 and Theorems 4.4–4.5 respectively, and recalling that for $N\ge 2m$ the point has null $V^m$-capacity by Proposition 2.3, we obtain Theorems 4.6 and 4.7 below.

Theorem 4.6

(Asymptotic expansion of perturbed eigenvalues, Dirichlet case) Let $N>2m$ and $\Omega \subset {\mathbb {R}}^N$ be a bounded smooth domain containing 0. Let $\{K_\varepsilon \}_{\varepsilon >0}$ be a family of compact sets uniformly concentrating to $\{0\}$ satisfying (M1)–(M2) for some compact set ${\mathcal {K}}$. Let $\lambda _J$ be a simple eigenvalue of (1.1) with Dirichlet boundary conditions and let $u_J\in H^m_0(\Omega )$ be a corresponding eigenfunction normalized in $L^2(\Omega )$. Then

$$\begin{aligned} \lambda _J(\Omega \setminus K_\varepsilon )=\lambda _J(\Omega )+\varepsilon ^{N-2m+2\gamma }\left( {\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)+{\scriptstyle {\mathcal {O}}}(1)\right) \end{aligned}$$

(4.7)

as $\varepsilon \rightarrow 0$, with $\gamma$ and $U_0$ as in (1.15).

Theorem 4.7

(Asymptotic expansion of perturbed eigenvalues, Navier case) Let $N>4$ and $\Omega \subset ~\!\!{\mathbb {R}}^N$ be a bounded smooth domain containing 0. Let $\{K_\varepsilon \}_{\varepsilon >0}$ be a family of compact sets uniformly concentrating to $\{0\}$ satisfying (M1)–(M2) for some compact set ${\mathcal {K}}$. Let $\lambda _J$ be a simple eigenvalue of (1.1) with $m = 2$ and Navier boundary conditions and let $u_J\in H^2_\vartheta (\Omega )$ be a corresponding eigenfunction normalized in $L^2(\Omega )$. Then

$$\begin{aligned} \lambda _J(\Omega \setminus K_\varepsilon )=\lambda _J(\Omega )+\varepsilon ^{N-4+2\gamma }\left( {\textrm{cap}}_{2,{\mathbb {R}}^N}({\mathcal {K}},U_0)+{\scriptstyle {\mathcal {O}}}(1)\right) \end{aligned}$$

(4.8)

as $\varepsilon \rightarrow 0$, with $\gamma$ and $U_0$ as in (1.15) with $m=2$.

The proofs of Theorems 4.4 and 4.5 follow a similar structure. We proceed hence to prove them at once using the introduced unifying notation, detailing the differences when needed.

Proof of Theorems 4.4–4.5

Motivated by (1.15), we define the analogously scaled potentials

$$\begin{aligned} {\widetilde{W}}_\varepsilon :=\frac{W_\varepsilon (\varepsilon \cdot )}{\varepsilon ^\gamma },\qquad \text{ where }\quad W_\varepsilon :=W_{K_\varepsilon ,u_J}. \end{aligned}$$

It is easy to verify that ${\widetilde{W}}_\varepsilon$ is the capacitary potential for $U_\varepsilon$ in $\varepsilon ^{-1}\Omega \setminus \varepsilon ^{-1}K_{\varepsilon }$, i.e.

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^m{\widetilde{W}}_\varepsilon =0\quad \text{ in }\;\varepsilon ^{-1}\Omega \setminus \varepsilon ^{-1}K_{\varepsilon },\\ {\widetilde{W}}_\varepsilon \in V^m(\varepsilon ^{-1}\Omega ),\\ {\widetilde{W}}_\varepsilon -U_\varepsilon \in V^m_0(\varepsilon ^{-1}\Omega \setminus \varepsilon ^{-1}K_{\varepsilon }), \end{array}\right. } \end{aligned}$$

(4.9)

where $m=2$ in the Navier case. The first goal now is to prove that the so-rescaled capacitary potentials weakly converge to some function ${\widetilde{W}}$ and prove that ${\widetilde{W}}$ is a capacitary potential in ${\mathbb {R}}^N\setminus {\mathcal {K}}$. To this aim, we need to distinguish between Dirichlet and Navier conditions on ${\partial \Omega }$. Indeed, by extension by zero outside the rescaled domains, in the first case it is rather natural to prove that the limit functional space is $D^{m,2}_0({\mathbb {R}}^N\setminus {\mathcal {K}})$; that the same holds true in the Navier case is not evident and requires a finer analysis. A fundamental role in this second case is played by the Hardy–Rellich inequality discussed in Sect. 2.2.2, which is however available just for $m=2$. The two cases will converge then in the final step where the asymptotic expansions (4.5)–(4.6) are proved.

Step 1 (Dirichlet case ${V^m=H^m_0}$). By (M1) there exists $R>0$ such that, for $\varepsilon$ small enough, $\varepsilon ^{-1}K_{\varepsilon }\subset M\subset B_R(0)\subset \varepsilon ^{-1}\Omega$, and hence, in view of Proposition 2.2,

$$\begin{aligned} {\textrm{cap}}_{m,\varepsilon ^{-1}\Omega }(\varepsilon ^{-1}K_{\varepsilon },U_\varepsilon )\le {\textrm{cap}}_{m,B_R(0)}(M,U_\varepsilon ). \end{aligned}$$

Since $U_\varepsilon \rightarrow U_0$ in $H^m(B_R(0))$ by (1.15), applying Lemma 4.3 in $B_R(0)$, we infer that

$$\begin{aligned} {\textrm{cap}}_{m,B_R(0)}(M,U_\varepsilon )\rightarrow {\textrm{cap}}_{m,B_R(0)}(M,U_0)\quad \text {as }\varepsilon \rightarrow 0. \end{aligned}$$

This yields in particular that $\Vert \nabla ^m{\widetilde{W}}_\varepsilon \Vert _{L^2(\varepsilon ^{-1}\Omega )}^2={\textrm{cap}}_{m,\varepsilon ^{-1}\Omega }(\varepsilon ^{-1}K_{\varepsilon },U_\varepsilon )$ is bounded uniformly with respect to $\varepsilon$. Letting ${\widetilde{W}}^{E}_{\varepsilon }$ be the extension by 0 of ${\widetilde{W}}_\varepsilon$ outside $\varepsilon ^{-1}\Omega$, we have thus that $\Vert {\widetilde{W}}^{E}_{\varepsilon }\Vert _{D^{m,2}_0({\mathbb {R}}^N)}\le C$. Since $D^{m,2}_0({\mathbb {R}}^N)$ is a Hilbert space, and so reflexive, for every sequence $\varepsilon _n\rightarrow 0^+$ there exist a subsequence $\varepsilon _{n_k}$ and ${\widetilde{W}}\in D^{m,2}_0({\mathbb {R}}^N)$ such that

$$\begin{aligned} {\widetilde{W}}^E_{\varepsilon _{n_k}}\rightharpoonup {\widetilde{W}}\quad \text{ weakly } \text{ in } D^{m,2}_0({\mathbb {R}}^N)\text { as }k\rightarrow \infty . \end{aligned}$$

(4.10)

We claim now that $\Vert \nabla ^m{\widetilde{W}}\Vert _2^2={\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)$.

Let $\varphi \in C^\infty _0({\mathbb {R}}^N{\setminus }{\mathcal {K}})$ and $R>r(M)$ be such that $\text {supp}\,\varphi \subset B_R$, then by (M$_R$2-ii) of Lemma 4.2, one may find a family $\{\varphi _\varepsilon \}_{\varepsilon >0}\subset H^m_0(B_R)$ such that $\varphi _\varepsilon \in H^m_0(B_R\setminus \varepsilon ^{-1}K_{\varepsilon })$ and $\varphi _\varepsilon \rightarrow \varphi$ in $H^m_0(B _R)$ as $\varepsilon \rightarrow 0$. In particular, for $\varepsilon$ small enough, one has that $B_R\subset \varepsilon ^{-1}\Omega$, so $\varphi _\varepsilon$ may be taken as test function for the capacitary potential ${\widetilde{W}}_\varepsilon$. Hence,

$$\begin{aligned} 0&=\int _{\varepsilon _{n_k}^{-1}\Omega \setminus \varepsilon _{n_k}^{-1}K_{\varepsilon _{n_k}}}\nabla ^m\widetilde{W}_{ \varepsilon _{n_k}}\,\nabla ^m\varphi _{\varepsilon _{n_k}}\\&=\int _{{\mathbb {R}}^N\setminus \varepsilon _{n_k}^{-1}K_{\varepsilon _{n_k}}} \nabla ^m\widetilde{W}^E_{ \varepsilon _{n_k}}\,\nabla ^m\varphi _{\varepsilon _{n_k}} \rightarrow \int _{{\mathbb {R}}^N\setminus {\mathcal {K}}}\nabla ^m{\widetilde{W}}\,\nabla ^m\varphi \quad \text {as }k\rightarrow \infty \end{aligned}$$

by weak-strong convergence in $D^{m,2}_0({\mathbb {R}}^N)$. We are left to show that ${\widetilde{W}}-\eta U_0\in D^{m,2}_0({\mathbb {R}}^N\setminus {\mathcal {K}})$, for some cutoff function $\eta$ which is equal to 1 in a neighbourhood of ${\mathcal {K}}$. Let $\eta \in C^\infty _0({\mathbb {R}}^N)$ be equal to 1 on an open set ${\mathcal {U}}$ with ${\mathcal {K}}\cup M\subset {\mathcal {U}}$; hence $\eta$ is also equal to 1 on neighbourhoods of each $\varepsilon ^{-1}K_{\varepsilon }$ by (M1). Then ${\widetilde{W}}_\varepsilon ^E-\eta U_\varepsilon \in D^{m,2}_0({\mathbb {R}}^N{\setminus }\varepsilon ^{-1}K_{\varepsilon })$ and ${\widetilde{W}}^E_{\varepsilon _{n_k}}-\eta U_{\varepsilon _{n_k}}\rightharpoonup {\widetilde{W}}-\eta U_0$ in $D^{m,2}_0({\mathbb {R}}^N)$ as $k\rightarrow \infty$, and so by (M2-i) one infers that ${\widetilde{W}}-\eta U_0\in D^{m,2}_0({\mathbb {R}}^N\setminus {\mathcal {K}})$. All in all, we deduce that ${\widetilde{W}}$ is the capacitary potential relative to ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)$, i.e.

$$\begin{aligned} \Vert \nabla ^m{\widetilde{W}}\Vert ^2_{L^2({\mathbb {R}}^N)}={\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0). \end{aligned}$$

(4.11)

Since the limit ${\widetilde{W}}$ in (4.10) depends neither on the sequence $\{\varepsilon _n\}$ nor on the subsequence $\{\varepsilon _{n_k}\}$, we conclude that

$$\begin{aligned} {\widetilde{W}}^E_{\varepsilon }\rightharpoonup {\widetilde{W}}\quad \text{ weakly } \text{ in } D^{m,2}_0({\mathbb {R}}^N)\text { as }\varepsilon \rightarrow 0. \end{aligned}$$

(4.12)

Step 1 (Navier case ${V^2=H^2_\vartheta }$). We recall that here we are assuming $m=2$. The boundedness of $\Vert \Delta {\widetilde{W}}_\varepsilon \Vert _{L^2(\varepsilon ^{-1}\Omega )}$ with a constant independent of $\varepsilon$ follows from the Dirichlet case, by recalling Proposition 2.2(iii). However, unlike the former case, one cannot now extend ${\widetilde{W}}_\varepsilon$ to 0 outside $\varepsilon ^{-1}\Omega$ and still obtain a function in $D^{2,2}_0({\mathbb {R}}^N)$. To overcome this problem we rely on the Hardy–Rellich inequality proved in Theorem 2.4. In fact, we have

$$\begin{aligned} \int _{\varepsilon ^{-1}\Omega }\frac{|{\widetilde{W}}_\varepsilon |^2}{|x|^4}\,dx+\int _{\varepsilon ^{-1}\Omega }\frac{|\nabla {\widetilde{W}}_\varepsilon |^2}{|x|^2}\,dx\lesssim \int _{\varepsilon ^{-1}\Omega }|\Delta {\widetilde{W}}_\varepsilon |^2\le C \end{aligned}$$

and therefore, by a diagonal process of extracted subsequences, for every sequence $\varepsilon _n\rightarrow 0^+$ there exist a subsequence $\varepsilon _{n_j}$ and ${\widetilde{W}}\in H^2_{loc}({\mathbb {R}}^N)$ for which

$$\begin{aligned} \frac{\nabla ^{2-k}{\widetilde{W}}_{\varepsilon _{n_j}}}{|x|^k}\rightharpoonup \frac{\nabla ^{2-k}{\widetilde{W}}}{|x|^k}\qquad \text{ in }\ L^2(B_R) \end{aligned}$$

(4.13)

as $j\rightarrow \infty$ for any $R>0$ and $k\in \{0,1,2\}$. By weak lower semicontinuity of the norm, we infer that

$$\begin{aligned} \int _{B_R}\frac{|\nabla ^{2-k}{\widetilde{W}}|^2}{|x|^{2k}}\,dx\le \liminf _{j\rightarrow \infty }\int _{B_R}\frac{|\nabla ^{2-k}{\widetilde{W}}_{\varepsilon _{n_j}}|^2}{|x|^{2k}}\,dx\le C, \end{aligned}$$

so that, letting $R\rightarrow +\infty$,

$$\begin{aligned} \int _{{\mathbb {R}}^N}\frac{|\nabla ^{2-k}{\widetilde{W}}|^2}{|x|^{2k}}\,dx\le C\quad \text {for all }k\in \{0,1,2\}. \end{aligned}$$

By Proposition 2.5, this is equivalent to ${\widetilde{W}}\in D^{2,2}_0({\mathbb {R}}^N)$.

It remains to prove that ${\widetilde{W}}$ is the capacitary potential relative to ${\textrm{cap}}_{2,{\mathbb {R}}^N}({\mathcal {K}},U_0)$. Let $\eta$ be as in the former case. Let $\varphi \in C^\infty _0(B_{1})$ be such that $\varphi \equiv 1$ in $B_{1/2}(0)$ and consider the scaled functions $\varphi _R:=\varphi \big (\tfrac{\cdot }{R}\big )$ with $R>r(M)$. Then $\varphi _R\big ({\widetilde{W}}_{\varepsilon _{n_j}}-\eta U_{\varepsilon _{n_j}}\big )\rightharpoonup \varphi _R\big ({\widetilde{W}}-\eta U_0\big )$ weakly in $H^2_0(B_R)$ and $\varphi _R({\widetilde{W}}_\varepsilon -\eta U_\varepsilon \big )\in H^2_0(B_R{\setminus }\varepsilon ^{-1}K_{\varepsilon })$. By (M$_R$2-i) we know then that $\varphi _R\big ({\widetilde{W}}-\eta U_0\big )\in H^2_0(B_R{\setminus }{\mathcal {K}})$. Now we claim that $\varphi _R\big ({\widetilde{W}}-\eta U_0\big )\rightarrow {\widetilde{W}}-\eta U_0$ as $R\rightarrow +\infty$ in $D^{2,2}_0({\mathbb {R}}^N)$, thus concluding that ${\widetilde{W}}-\eta U_0\in D^{2,2}_0({\mathbb {R}}^N{\setminus }{\mathcal {K}})$. Indeed,

$$\begin{aligned} \begin{aligned} \Vert \Delta \big (\left( \varphi _R-1\right) \big ({\widetilde{W}}-\eta U_0\big )\big )\Vert _2^2&\lesssim \Vert \Delta \varphi _R\big ({\widetilde{W}}-\eta U_0\big )\Vert _2^2+\Vert \nabla \varphi _R\nabla \big ({\widetilde{W}}-\eta U_0\big )\Vert _2^2\\&\quad +\Vert \left( \varphi _R-1\right) \Delta \big ({\widetilde{W}}-\eta U_0\big )\Vert _2^2, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \Vert \left( \varphi _R-1\right) \Delta \big ({\widetilde{W}}-\eta U_0\big )\Vert _2^2 \le \int _{{\mathbb {R}}^N\setminus B_{R/2}}|\Delta \big ({\widetilde{W}}-\eta U_0\big )|^2\rightarrow 0 \end{aligned}$$

as $R\rightarrow +\infty$, and, for any $k\in \{1,2\}$,

$$\begin{aligned} \begin{aligned} \Vert \nabla ^k\varphi _R\nabla ^{2-k}\big ({\widetilde{W}}-\eta U_0\big )\Vert _2^2&=\int _{\frac{R}{2}<|x|<R}\frac{1}{R^{2k}}\left| \big (\nabla ^k\varphi \big )\left( \frac{x}{R}\right) \right| ^2|\nabla ^{2-k}\big ({\widetilde{W}}-\eta U_0\big )|^2\,dx\\&\lesssim \int _{{\mathbb {R}}^N\setminus B_R(0)}\frac{|\nabla ^{2-k}\big ({\widetilde{W}}-\eta U_0\big )|^2}{|x|^{2k}}\,dx\rightarrow 0 \end{aligned} \end{aligned}$$

as $R\rightarrow +\infty$ since ${\widetilde{W}}-\eta U_0\in D^{2,2}_0({\mathbb {R}}^N)$ together with Proposition 2.5.

Next, we verify that

$$\begin{aligned} \int _{{\mathbb {R}}^N\setminus {\mathcal {K}}}\Delta {\widetilde{W}}\,\Delta \varphi =0\quad \text {for all }\varphi \in D^{2,2}_0({\mathbb {R}}^N\setminus {\mathcal {K}}). \end{aligned}$$

(4.14)

By density, it is enough to prove (4.14) for all $\varphi \in C^\infty _0({\mathbb {R}}^N{\setminus }{\mathcal {K}})$. Letting $\varphi \in C^\infty _0({\mathbb {R}}^N{\setminus }{\mathcal {K}})$, there exist $R>r(M)$ and $\varepsilon _0>0$ such that $\text {supp}\,\varphi \subset B_R\subset \varepsilon ^{-1}\Omega$ for all $\varepsilon <\varepsilon _0$, so that $\varphi \in H^2_0(B_R{\setminus }{\mathcal {K}})$. By (M$_R$2-ii) there exists a family $\{\varphi _\varepsilon \}_\varepsilon \subset H^2_0(B_R)$ such that $\varphi _\varepsilon \in H^2_0(B_R\setminus \varepsilon ^{-1}K_{\varepsilon })$ and $\varphi _\varepsilon \rightarrow \varphi$ in $H^2_0(B_R)$. Hence,

$$\begin{aligned} 0=\int _{\varepsilon _{n_j}^{-1}\Omega }\Delta {\widetilde{W}}_{\varepsilon _{n_j}}\Delta \varphi _{\varepsilon _{n_j}}=\int _{B_R}\Delta {\widetilde{W}}_{\varepsilon _{n_j}}\Delta \varphi _{\varepsilon _{n_j}}\rightarrow \int _{B_R}\Delta {\widetilde{W}}\,\Delta \varphi =\int _{{\mathbb {R}}^N\setminus {\mathcal {K}}}\Delta {\widetilde{W}}\,\Delta \varphi \end{aligned}$$

as $j\rightarrow \infty$, by weak-strong convergence in $H^2_0(B_R)$. We have thereby proved the claim that ${\widetilde{W}}$ is the capacitary potential relative to ${\textrm{cap}}_{2,{\mathbb {R}}^N}({\mathcal {K}},U_0)$. Since the limit ${\widetilde{W}}$ in (4.13) depends neither on the sequence $\{\varepsilon _n\}$ nor on the subsequence $\{\varepsilon _{n_k}\}$, we conclude that the convergences in (4.13) actually hold as $\varepsilon \rightarrow 0$, i.e.

$$\begin{aligned} \frac{\nabla ^{2-k}{\widetilde{W}}_{\varepsilon }}{|x|^k}\rightharpoonup \frac{\nabla ^{2-k}{\widetilde{W}}}{|x|^k}\qquad \text{ in }\ L^2(B_R)\quad \text {as }\varepsilon \rightarrow 0\quad \text {for all }R>0\text { and }k\in \{0,1,2\}. \nonumber \\ \end{aligned}$$

(4.15)

Step 2. ($m=2$ in the Navier case, $m\ge 2$ in the Dirichlet case). We aim now to prove the asymptotic expansions (4.5)–(4.6). As above, let $\eta \in C^\infty _0({\mathbb {R}}^N)$ be equal to 1 on an open set ${\mathcal {U}}$ with ${\mathcal {K}}\cup M\subset {\mathcal {U}}$. Let $R>0$ be such that $\text {supp}\,\eta \subset B_R$. Since ${\widetilde{W}}_\varepsilon -\eta U_\varepsilon \in V^m_0(\varepsilon ^{-1}\Omega {\setminus }\varepsilon ^{-1}K_{\varepsilon })$, by (4.9) and (4.11), together with (4.12) or (4.15), we obtain that

$$\begin{aligned} \begin{aligned} \Vert \nabla ^m{\widetilde{W}}_\varepsilon \Vert ^2_{L^2(\varepsilon ^{-1}\Omega )}&=\int _{\varepsilon ^{-1}\Omega }\nabla ^m{\widetilde{W}}_\varepsilon \,\nabla ^m\left( \eta U_\varepsilon \right) =\int _{B_R}\nabla ^m{\widetilde{W}}_\varepsilon \,\nabla ^m\!\left( \eta U_\varepsilon \right) \\&\rightarrow \int _{{\mathbb {R}}^N}\nabla ^m{\widetilde{W}}\,\nabla ^m\!\left( \eta U_0\right) =\Vert \nabla ^m{\widetilde{W}}\Vert ^2_{L^2({\mathbb {R}}^N)} \end{aligned} \end{aligned}$$

(4.16)

as $\varepsilon \rightarrow 0$ by weak-strong convergence. On the other hand, by rescaling one has that

$$\begin{aligned} \begin{aligned} \Vert \nabla ^m{\widetilde{W}}_\varepsilon \Vert ^2_{L^2(\varepsilon ^{-1}\Omega )}&=\frac{1}{\varepsilon ^{2\gamma }}\int _{\varepsilon ^{-1}\Omega }\left| \nabla ^m\!\left( W_\varepsilon (\varepsilon x)\right) \right| ^2\,dx=\varepsilon ^{-N+2m-2\gamma }\int _{\Omega }|\nabla ^m W_\varepsilon (y)|^2\,dy\\&=\varepsilon ^{-N+2m-2\gamma }{\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J). \end{aligned} \nonumber \\ \end{aligned}$$

(4.17)

Hence, from (4.11) and (4.16)–(4.17) we finally infer that

$$\begin{aligned} {\textrm{cap}}_{V^{m}\!,\,\Omega }(K_\varepsilon ,u_J)= \varepsilon ^{N-2m+2\gamma } \Vert \nabla ^m{\widetilde{W}}_\varepsilon \Vert ^2_{L^2(\varepsilon ^{-1}\Omega )}=\varepsilon ^{N-2m+2\gamma }\left( {\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)+{\scriptstyle {\mathcal {O}}}(1)\right) \end{aligned}$$

as $\varepsilon \rightarrow 0$. $\square$

4.2 Sufficient conditions for a sharp asymptotic expansion

Looking at the asymptotic expansions we have found in Theorems 4.6–4.7, one may ask whether the results are sharp, in the sense that the vanishing rate of the eigenvalue variation $\lambda _J(\Omega \setminus K_\varepsilon )-~\!\lambda _J(\Omega )$ is equal to $N-2m+2\gamma$. The next results provide sufficient conditions on ${\mathcal {K}}$ and $U_0$ in order to ensure that ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)\ne 0$.

Proposition 4.8

Under the assumptions of Theorems 4.6 or 4.7, suppose that the Lebesgue measure of ${\mathcal {K}}$ is positive. Then

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{\lambda _J(\Omega \setminus K_\varepsilon )-\lambda _J(\Omega )}{\varepsilon ^{N-2m+2\gamma }}={\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)>0. \end{aligned}$$

Proof

Denote by $W^{(0)}_{{\mathcal {K}}}$ the capacitary potential for ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)$ and suppose by contradiction that ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)=0$. Then, by the Hardy inequality for the polyharmonic operator (see [12, Theorem 12]), there exists a constant $c=c(N,m)$ such that

$$\begin{aligned} 0=\Vert \nabla ^mW^{(0)}_{{\mathcal {K}}}\Vert _{L^2({\mathbb {R}}^N)}^2\ge c\int _{{\mathbb {R}}^N}\frac{|W^{(0)}_{{\mathcal {K}}}|^2}{|x|^{2m}}\,dx\ge c\int _{{\mathcal {K}}}\frac{|W^{(0)}_{{\mathcal {K}}}|^2}{|x|^{2m}}\,dx= c\int _{{\mathcal {K}}}\frac{|U_0|^2}{|x|^{2m}}\,dx \end{aligned}$$

since $W^{(0)}_{{\mathcal {K}}}\equiv U_0$ on ${\mathcal {K}}$. Since $|{\mathcal {K}}|>0$, this readily implies that $U_0$ vanishes a.e. on ${\mathcal {K}}$. However, by construction, $U_0$ is a polyharmonic polynomial on ${\mathbb {R}}^N$ which is not identically zero (see [6, Sec.4 Theorem 1]), so it cannot vanish on a set of positive measure (since nontrivial analytic functions cannot vanish on positive measure sets). This, together with Theorems 4.6 and 4.7, concludes the proof. $\square$

The next results apply to some specific situations in which, although K has vanishing Lebesgue measure, one may anyway have that ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)\ne 0$.

Proposition 4.9

Let $N>2m$ and ${\mathcal {K}}\subset {\mathbb {R}}^N$ be a compactum with ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}})>0$. Suppose moreover that $u_J(0)\ne 0$. Then, in the setting of Theorems 4.6 or 4.7, we have that

$$\begin{aligned} \lambda _J(\Omega \setminus K_\varepsilon )=\lambda _J(\Omega )+\varepsilon ^{N-2m}u_J^2(0){\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}})+{\scriptstyle {\mathcal {O}}}(\varepsilon ^{N-2m}) \end{aligned}$$

(4.18)

as $\varepsilon \rightarrow 0$.

We mention that an expansion of type (4.18) was obtained in [10, Theorem 1.4] and [2, Theorem 1.7] in the case $N=2m=2$, in which the vanishing rate of the eigenvalue variation is logarithmic.

Proof of Proposition 4.9

Since $\{K_\varepsilon \}_{\varepsilon >0}$ is concentrating at $\{0\}$ and $u_J(0)\ne 0$, then the degree $\gamma$ of the polynomial $U_0$ is 0, and $U_0=u_J(0)$. It is then easy to see that

$$\begin{aligned} {\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},u_J(0))=u_J^2(0){\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}})>0, \end{aligned}$$

so that (4.7) and (4.8) can be rewritten as in (4.18). $\square$

In the case $u_J(0)=0$, the next result, inspired by [13, Lemma 3.11], may be useful. It tells that, if the compactum ${\mathcal {K}}$ and the null-set of the polynomial $U_0$ are “transversal enough”, then again ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)>0$.

Proposition 4.10

Let $N>2m$ and ${\mathcal {K}}\subset {\mathbb {R}}^N$ be a compactum with ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}})>0$. Letting $f\in ~\!\!C^\infty ({\mathbb {R}}^N)$, let us consider the set

$$\begin{aligned}Z_f^{{\mathcal {K}}}:=\{x\in {\mathcal {K}}\,|\,f(x)=0\}.\end{aligned}$$

If ${\textrm{cap}}_{m,{\mathbb {R}}^N}(Z_f^{{\mathcal {K}}})<{\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}})$, then ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},f)>0$.

Proof

Let $\{{\mathcal {U}}_n\}$ be a sequence of nested open sets in ${\mathbb {R}}^N$ so that $Z_f^{{\mathcal {K}}}\subset {\mathcal {U}}_n$ for all $n\in {\mathbb {N}}$ and $Z_f^{{\mathcal {K}}}=\bigcap _{n\in {\mathbb {N}}}\overline{{\mathcal {U}}_n}$ and let ${\mathcal {K}}_n:={\mathcal {K}}\setminus {\mathcal {U}}_n$, which is a sequence of compact sets. By subadditivity and monotonicity of the capacity (see e.g. [24]) one has that

$$\begin{aligned} {\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}})\le {\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}}_n)+{\textrm{cap}}_{m,{\mathbb {R}}^N}(\overline{{\mathcal {U}}_n}). \end{aligned}$$

Moreover, fixing $0<\delta <{\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}})-{\textrm{cap}}_{m,{\mathbb {R}}^N}(Z_f^{{\mathcal {K}}})$, one may find a neighbourhood ${\mathcal {U}}(Z_f^{{\mathcal {K}}})$ such that one has $\overline{{\mathcal {U}}_n}\subset {\mathcal {U}}(Z_f^{{\mathcal {K}}})$ by construction and ${\textrm{cap}}_{m,{\mathbb {R}}^n}(\overline{{\mathcal {U}}_n})\le {\textrm{cap}}_{m,{\mathbb {R}}^n}(Z_f^{{\mathcal {K}}})+\delta$ by Lemma 2.6, provided n is large enough. This implies

$$\begin{aligned} {\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}}_n)\ge {\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}})-{\textrm{cap}}_{m,{\mathbb {R}}^N}(Z_f^{{\mathcal {K}}})-\delta >0 \end{aligned}$$

(4.19)

for n large enough. We define ${\mathcal {K}}_n^+:=\{ x\in {\mathcal {K}}_n:f(x)>0\}$ and ${\mathcal {K}}_n^-:=\{ x\in {\mathcal {K}}_n:f(x)<0\}$ for all $n\in {\mathbb {N}}$. Noticing that ${\mathcal {K}}_n$ is the union of ${\mathcal {K}}_n^+$ and ${\mathcal {K}}_n^-$, we necessarily have that either ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}}_n^+)>0$ or ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}}_n^-)>0$; let us e.g. consider the case ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}}_n^+)>0$. By regularity of f and since ${\mathcal {K}}_n^+$ is compact, then $c_n^+:=\inf _{{\mathcal {K}}_n^+}f$ is attained and strictly positive. Take now any $u_n\in D^{m,2}_0({\mathbb {R}}^N)$ so that $u_n-\eta _{{\mathcal {K}}_n^+}f\in D^{m,2}_0({\mathbb {R}}^N{\setminus }{\mathcal {K}}_n^+)$ and define $v_n:=\frac{u_n}{c_n^+}$. Then it is clear that $v_n\in D^{m,2}_0({\mathbb {R}}^N)$ and $v_n\ge 1$ a.e. on ${\mathcal {K}}_n^+$. Hence,

$$\begin{aligned} {\textrm{Cap}}_{m,\,{\mathbb {R}}^{N}}^{\ge }({\mathcal {K}}_n^+)\le \int _{{\mathbb {R}}^N}|\nabla ^mv_n|^2 =\frac{1}{\left( c_n^+\right) ^2}\int _{{\mathbb {R}}^N}|\nabla ^mu_n|^2. \end{aligned}$$

By arbitrariness of $u_n$ this yields $\left( c_n^+\right) ^2{\textrm{Cap}}_{m,\,{\mathbb {R}}^{N}}^{\ge }({\mathcal {K}}_n^+)\le {\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}}_n^+,f)\le {\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},f)$, since ${\mathcal {K}}_n^+\subset {\mathcal {K}}$ for all $n\in {\mathbb {N}}$. Using now the equivalence of the capacities in ${\mathbb {R}}^N$ stated in Lemma 2.7, one infers that

$$\begin{aligned} {\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},f)\ge c\left( c_n^+\right) ^2{\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}}_n^+)>0 \end{aligned}$$

by (4.19). This concludes the proof. $\square$

Remark 9

In view of Remark 4 it is immediate to see that, if ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}})>0$ and $Z_{U_0}^{{\mathcal {K}}}$ has dimension $d\le N-2m$, then the assumptions of Proposition 4.10 are fulfilled, thus ensuring that ${\textrm{cap}}_{m,{\mathbb {R}}^N}({\mathcal {K}},U_0)>0$.

5 Open problems

We finally discuss possible generalizations and questions which are left open by our analysis and which we believe of interest.

Higher-order Navier setting. The results in Sect. 3 for the Navier setting are obtained in the general case $m\ge 2$. On the other hand, Theorem 4.5 and its consequent Theorem 4.7 are established only for $m=2$. The main difficulty in their extension to higher orders relies in the characterization of homogeneous Sobolev spaces via Hardy–Rellich inequalities. In our argument this was necessary to compensate for the lack of a trivial extension, which is instead available in the Dirichlet setting. Although we envision that a generalization of the Hardy–Rellich inequality of Proposition 2.4 is reachable, the extension of the characterization contained in Proposition 2.5 seems to be a non trivial problem. Indeed, for $m=2$ the only intermediate derivative is the gradient and $\nabla u=Du$; on the other hand for $m\ge 3$ the Hardy–Rellich inequality would provide a weighted estimate on the derivatives $\nabla ^ku$, $k\in \{1,\dots ,m-1\}$, while one would need to estimate the full tensor of the derivatives $D^ku$ to be able to conclude that $u\in D^{m,2}_0({\mathbb {R}}^N)$.

Small dimensions. Most of our results deal with the high dimensional case $N\ge 2m$, because the concentration of the family of sets $\{K_\varepsilon \}_{\varepsilon >0}$ to a zero $V^m$-capacity compact set was needed. Recall that for $N<2m$ all compact sets are of positive capacity, see Proposition 2.3. Nevertheless, in order to prove that the asymptotic expansions given by Theorem 1.2 are sharp, in Theorems 1.3 and 1.4 we have to restrict to $N>2m$. It seems that the conformal case $N = 2m$ cannot be treated by blow-up analysis, not only due to the different characterization of the spaces $D^{m,2}_0({\mathbb {R}}^N)$ and the use of Hardy–Rellich inequalities, but also because the m-capacity ${\textrm{cap}}_{m,{\mathbb {R}}^{2m}}(K)$ of any compact set K in ${\mathbb {R}}^{2m}$ is null (see [23]). A different approach for conformal (and smaller!) dimensions should be in fact developed and we expect that the expansion involves the logarithm of the diameter of the shrinking sets, in analogy with the results in [2] for the case $m=1$. We remark in particular that, when our results are applied to the biharmonic operator, i.e. $m=2$, we cover the case $N\ge 5$, while the two-dimensional case, from a completely different point of view, is studied in [8, 21, 22]. The case $N=3$ is left open and $N=4$ only partially answered by Theorem 1.2.

Equivalent definitions of capacities. As described in Sect. 2, both ${\textrm{cap}}_{m,{\mathbb {R}}^N}$ in (2.6) and ${\textrm{Cap}}_{m,\,{\mathbb {R}}^{N}}^{\ge }$ in (2.8) are good definitions of capacity, and they are also equivalent for $N>2m$, see Lemma 2.7. In the second order case, it is not difficult to prove that the two coincide, while—up to our knowledge—this is still unknown in the higher-order setting. A weaker question would be to ask whether the two capacities are asymptotic for families of shrinking domains, e.g. for $K_\varepsilon =\varepsilon {\mathcal {K}}$ as considered in Sect. 4.1. It would be also interesting to understand whether the equivalence remains true for the weighted capacities ${\textrm{cap}}_{m,{\mathbb {R}}^N}(\cdot ,h)$ and the analogue ${\textrm{Cap}}_{m,\,{\mathbb {R}}^{N}}^{\ge }(\cdot ,h)$ for some class of nonconstant functions h.

Boundary conditions. As mentioned in the introduction, it would be interesting to investigate the complementary cases of prescribing Navier BCs on the removed set and either Navier or Dirichlet BCs on the external boundary ${\partial \Omega }$. Because of the lack of an extension by zero in case of Navier BCs, which has consequences on the mutual relations between the spaces $V^m(\Omega \setminus K)$, a different argument would be needed. More in general, it would be challenging to consider more general types of BCs, which yield a different quadratic form associated to the polyharmonic operator, involving possibly also boundary integrals. An interesting case in the biharmonic setting, related to the physical model of hinged thin plates, is for example given by Steklov BCs $u=\Delta u-d\kappa \partial _nu=0$, where $d\in {\mathbb {R}}$ and $\kappa$ is the signed curvature of the boundary.