Boundary value problems on non-Lipschitz uniform domains: stability,compactness and the existence of optimal shapes

Abstract

We study boundary value problems for bounded uniform domains in $R^{n}$ , $n ⩾ 2$ , with non-Lipschitz, and possibly fractal, boundaries. We prove Poincaré inequalities with uniform constants and trace terms for $(ε, \infty)$ -domains contained in a fixed bounded Lipschitz domain. We introduce generalized Dirichlet, Neumann, and Robin problems for Poisson-type equations and prove the Mosco convergence of the associated energy functionals along sequences of suitably converging domains. This implies a stability result for weak solutions, the norm convergence of the associated resolvents, and the convergence of the corresponding eigenvalues and eigenfunctions. We provide compactness results for parametrized classes of admissible domains, energy functionals, and weak solutions. Using these results, we can then prove the existence of optimal shapes in these classes in the sense that they minimize the initially given energy functionals. For the Robin boundary problems, this result is new.

Keywords

Fractal boundaries Poincaré inequalities Robin problems Mosco convergence norm resolvent convergence optimal shapes

1. Introduction

This article deals with boundary value problems for elliptic equations that involve rough shapes. The broad aim of our research is to understand the behavior of solutions to a wide variety of problems for sequences of domains that converge to an optimal, possibly rough, shape. In particular, we consider the limiting solutions for sequences of Lipschitz or non-Lipschitz domains converging to a non-Lipschitz domain.

To focus on the central issues, we write our results for Poisson-type equations; generalizations to other elliptic equations are straightforward. We consider domains $Ω \subset R^{n}$ , $n ⩾ 2$ , not necessarily Lipschitz, together with closed sets $Γ \subset \overline{Ω}$ that support a nonzero finite Borel measure μ of a certain upper regularity. The set Γ is treated as a generalized boundary on which boundary conditions are imposed. Boundary value problems in the more narrow sense that $Γ = \partial Ω$ are included as a special case. We study the behavior of weak, i.e., variational, solutions for varying domains Ω and measures μ, and we address stability, compactness, and the existence of optimal (energy-minimizing) shapes. We are interested in the case of the Robin problem, for which the literature on the existence of optimal shapes in admissible classes of rough domains is still sparse. The problem is quite nontrivial because it involves integrals on Γ with respect to a measure μ, which may be fractal. Since there is no need to exclude Dirichlet and homogeneous Neumann problems from our discussion, we also comment on them, although they can be dealt with by more straightforward methods, [11,29,31]. For instance, the homogeneous Dirichlet boundary condition allows working with a more general class of domains, sometimes even arbitrary domains [19].

This article is an important part of a series of papers. In the papers [21,30,31] we develop general techniques for how to deal with boundary value problems in non-Lipschitz uniform domains whose boundaries are not d-sets but satisfy only weaker upper and lower Ahlfors regularity estimates of type $μ (B (x, r)) ⩽ c_{d} r^{d}$ respectively $μ (\overline{B (x, r)}) ⩾ c_{s} r^{s}$ , $x \in supp μ$ , $0 < r ⩽ 1$ , with possibly different lower and upper exponents $d ⩽ s$ and no doubling property. In [18] we apply these results to the study of the nonlinear Westervelt equation [20,58] (see (1)) modeling the optimal absorption of ultrasound waves by a reflective or isolating boundary. The previous results were obtained using the assumption of having a nontrivial part of the boundary with a Dirichlet boundary condition. We now prove general results without this extra condition and, in particular, treat pure Robin boundary problems.

In this paper, we show four main results. The first and central main result gives families of Poincaré inequalities with uniform constants, Theorem 3.1. There we assume that $n ⩾ 2$ and that $D \subset R^{n}$ is a bounded Lipschitz domain; we refer to it as a bounded confinement. The theorem states that if $ε > 0$ , $0 ⩽ d ⩽ n$ , $n - p < d$ and $c_{d} > 0$ are given, then for any $(ε, \infty)$ -domain $Ω \subset D$ and any Borel probability measure μ supported in $\overline{Ω}$ and upper d-regular (cf. (3)) with constant $c_{d}$ , a Poincaré inequality for $W^{1, p} (Ω)$ -functions holds with a uniform constant not depending on the choice of Ω or μ. The conditions on d ensure the existence of a well-defined trace on Γ in the sense of [1, Theorem 6.2.1 and Section 7], see Corollary 2.2. The trick in the proof of Theorem 3.1 is to consider the cone of all $u \in W^{1, p} (Ω)$ -functions for which we can find a probability measure annihilating the trace of u on its support. Theorem 3.1 is new and is the key to further results; it provides uniform bounds which enable us to prove the existence of optimal shapes without prescribing Dirichlet conditions on any part of the boundary.

For $p = 2$ , we formulate abstract boundary value problems of Dirichlet, Robin, and Neumann type for the linear Poisson equation on $W^{1, 2}$ -extension domains, that is, domains $Ω \subset R^{n}$ for which there is a bounded linear extension operator extending elements of $W^{1, 2} (Ω)$ to elements of $W^{1, 2} (R^{n})$ . By [32, Theorem 1] each $(ε, \infty)$ -domain is a $W^{1, 2}$ -extension domain. We then consider the situation that ${(Ω_{m})}_{m} \subset D$ is a sequence of $(ε, \infty)$ -domains $Ω_{m}$ converging to some domain Ω in the Hausdorff sense and in the sense of characteristic functions. In the Robin case, we additionally assume that ${(μ_{m})}_{m}$ is a sequence of measures $μ_{m}$ supported inside ${\overline{Ω}}_{m}$ , respectively, all upper d-regular with the same constant $c_{d}$ , and weakly converging to some measure μ. Theorem 5.1 then tells that the energy functionals associated with the respective type of boundary value problem converge in the sense of Mosco. Together with Theorem 3.1 (respectively its Corollary 3.2), this leads to our second main result, Theorem 5.2. It is a stability result. It states that in the homogeneous case, a suitable subsequence of weak solutions converges in $L^{2} (D)$ to the weak solution to a problem of the same type on the limit domain Ω. We also prove norm resolvent convergence and the convergence of the corresponding spectral projectors, eigenvalues, and eigenfunctions, Theorem 6.1.

To proceed further, we introduce parametrized classes $U_{ad} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ of what we call shape admissible triples $(Ω, ν, μ)$ , Definition 7.1. This definition generalizes [31, Definition 2]. The parameters D, ε, d and $c_{d}$ are as above. Any triple $(Ω, ν, μ)$ in the class consists of an $(ε, \infty)$ -domain Ω contained in D and containing a prescribed fixed open set $D_{0}$ , and two measures μ and ν. As before, the measure μ is supported in $\overline{Ω}$ and upper d-regular as defined in (3), with constant $c_{d}$ . The second measure ν has support $\partial Ω$ , has total mass bounded above by $\bar{c} > 0$ and is lower s-regular in the sense of (54), with fixed exponent $n - 1 ⩽ s < n$ and constant ${\bar{c}}_{s} > 0$ . These measures ν are introduced as a topological tool, Remark 7.1. Within this setup, an improvement of a compactness result from [31, Theorem 3] holds, Theorem 7.1. It tells that, given such a parametrized class of shape admissible triples, any sequence $({(Ω_{m}, ν_{m}, μ_{m})}_{m})$ it contains must have a subsequence that converges to a shape admissible triple in the same class in the Hausdorff sense, the sense of characteristic functions, the sense of compacts and the sense of weak convergence of the measures $ν_{m}$ and the measures $μ_{m}$ . This enables our third main result, Theorem 7.2, which tells that the corresponding energy forms inherit this compactness. Theorem 7.1 and Theorem 5.2 together also entail that weak solutions to the respective problems have convergent subsequences, Corollary 7.1. This fact then implies our fourth main result, namely the existence of optimal shapes within a given parametrized class, formulated in Theorems 8.1 and 8.2.

These results are key contributions to a larger research program on shape optimization involving rough shapes with boundaries that may be fractal. In the long run, a theory of shape optimization robust enough to discuss rough shapes with fractal boundaries could significantly impact fields like acoustical engineering, biomimetic architecture, and surface design. In [37], mixed boundary value problems for the Helmholtz equation had been considered. It had been observed that since integrals over the boundary are involved, classes of Lipschitz admissible shapes may not contain the shape realizing the infimum energy over the class, see also [30, Section 5]. In [31, Definition 2] we therefore proposed larger classes of admissible shapes involving bounded $(ε, \infty)$ -domains and proved that in these classes optimal, i.e., energy minimizing, shapes exist, [31, Section 7]. Note that also [8, Theorem 3.2] used the idea to enlarge the classes of admissible shapes beyond Lipschitz, although in a methodologically different context. The equations discussed here and in [30,31,37] are linear. In [19,21] the authors proved well-posedness results for damped linear wave equations and for the nonlinear Westervelt equation $\begin{matrix} (1) & \partial_{t}^{2} u - c^{2} Δ u - ν Δ \partial_{t} u = α u \partial_{t}^{2} u + α {(\partial_{t} u)}^{2} + f on (0, T) \times Ω, \end{matrix}$ with constant coefficients; it is a model for the propagation of ultrasound waves. Here $c > 0$ is the sound speed in the unperturbed homogeneous medium, $ν > 0$ is the viscosity of the medium, and $α > 0$ is a nonlinearity constant. The Westervelt model arises as an approximation of the compressible Navier–Stokes system if one performs small perturbations of a constant medium state under the assumptions that the velocity field is conservative and the viscosity of the medium is small, see [20]. Related Robin problems are particularly important in applications because they model partial sound absorption and reflection at absorbent boundaries made of porous material. Theorems 3.1 and 7.1 now open the way to prove the existence of Robin optimal shapes for the nonlinear Westervelt equation within parametrized classes of bounded uniform domains with fractal boundaries. This is done in [18]. We point out that in our preceding articles [21,31] uniform Poincaré inequalities were consequences of Dirichlet conditions on some part of the boundary. For the present results and those in [18], Dirichlet conditions are not necessary. We can instead rely on Theorem 3.1.

There is a huge amount of literature on Poincaré inequalities. We mention only a few sources related to our text. A general presentation can be found in [23, Section 7]. Poincaré inequalities with uniform constants for classes of bounded and uniformly Lipschitz domains were discussed in [5], their Theorem 2 may be seen as a Lipschitz predecessor of our Theorem 3.1. Poincaré inequalities with uniform constants for domains satisfying certain cone conditions were proved in [47] and [49]. Optimal constants (in the sense of isoperimetry) for Poincaré inequalities involving trace terms on bounded Lipschitz domains were determined in [10].

Robin boundary value problems on arbitrary domains were studied in [16], based on former ideas of Maz’ya, [39, Section 4.11]. Within this approach, traces on the boundary are defined somewhat abstractly and always considered with respect to the $(n - 1)$ -dimensional Hausdorff measure. The traces can ‘disconnect’ from the function on the domain, and Robin conditions can reduce to Dirichlet conditions, [16, Remarks 3.5 (a) and (d)]. Based on this approach, Robin problems for linear and nonlinear equations on varying domains were investigated in [15], where it was shown that in the limit problem, a different type of boundary condition could appear. Within our ‘parametrized’ setup, such effects are excluded. An abstract potential theoretic approach to Robin problems on arbitrary domains was introduced in [2]. Using variational methods, [6,11,14], Robin problems on sequences of confined Lipschitz domains were studied in the article [9]. Under the assumptions that the domains converge in the Hausdorff sense, their total volumes converge, and the measures on the boundary are diffuse enough and bounded below by a uniform constant times the $(n - 1)$ -dimensional Hausdorff measure, it was shown that the weak solutions of the individual problems converge to the weak solution of the limit problem, which is again Robin, [9, Theorems 3.1 and 3.4]. Also, norm resolvent convergence and the convergence of the corresponding eigenvalues were proved, [9, Theorem 3.2 and Corollary 3.5]. No uniform (in this case, ‘equi-Lipschitz’) condition was imposed on the geometry of the domains. This is different in the present article, where the fixed parameter ε forces a uniform ‘worst case’ control, although within the more general class of bounded uniform domains. The stability results in [9] were established using the fact that the indicators of Lipschitz domains with a finite measure on the boundary are $BV$ -functions; extension operators are not needed. Their proof also uses a Faber–Krahn inequality for Robin problems established in [17]; nonlinear generalizations of this inequality were provided in [7] via regularity results for free discontinuity problems. A free discontinuity approach to shape optimization for Robin problems on varying domains was proposed in [8]. The original problem the authors addressed involves a very general nonlinear energy integral, from which the energy functional to be minimized is obtained by taking an infimum of $W^{1, p}$ -functions on the respective domain, subject to a pinning (‘obstacle type’) condition on a given fixed open subset inside the domain. Instead of this original problem, they then studied a relaxed one, where the original energy integral was replaced by a free discontinuity functional involving integrals of (possibly different) one-sided traces on the boundary in the sense of $BV$ -functions, and the class of admissible shapes was enlarged to a collection of open domains Ω with countably rectifiable boundaries $\partial Ω$ of finite $(n - 1)$ -dimensional Hausdorff measure. The existence of optimal shapes for the relaxed problem was then proved in [8, Theorem 3.2]. In the present article, we do not consider a relaxed problem but work with the initially given energy functional, and it is not our goal to study boundary discontinuities. We do not insist on any rectifiability properties; the boundaries of optimal shapes may have any Hausdorff dimension in $[n - 1, n)$ , see Remark 8.1.

In Section 2 we collect preliminary material about traces of Sobolev functions on closed subsets. In Section 3 we prove the Poincaré inequalities with uniform constants. Section 4 discusses generalized boundary value problems, energy functionals, and a first compactness lemma. The Mosco convergence of energy functionals along varying domains is proved in Section 5; also, the stability result for weak solutions is shown there. Norm resolvent convergence and spectral convergence are stated and proved in Section 6. In Section 7 we define the parametrized classes of admissible triples and state the compactness results. The existence of optimal shapes is shown in Section 8.

2. Traces on closed subsets

We collect preliminary facts on traces of Sobolev functions on closed subsets of $R^{n}$ . For any $1 < p < + \infty$ and $β > 0$ let $H^{β, p} (R^{n})$ denote the $L^{p}$ -Bessel-potential space of order β, [1,50,51], that is, the space of all $f \in S^{'} (R^{n})$ such that ${({(1 + | ξ |^{2})}^{β / 2} \hat{f})}^{\lor} \in L^{p} (R^{n})$ . Here $f \mapsto \hat{f}$ denotes the Fourier transform and $f \mapsto \overset{ˇ}{f}$ its inverse, and $S^{'} (R^{n})$ is the space of tempered distributions. Endowed with the norm $\begin{matrix} ‖ f ‖_{H^{β, p} (R^{n})} : = {‖ {({(1 + | ξ |^{2})}^{β / 2} \hat{f})}^{\lor} ‖}_{L^{p} (R^{n})} \end{matrix}$ the space $H^{β, p} (R^{n})$ is Banach, and Hilbert if $p = 2$ .

We write $λ^{n}$ to denote the n-dimensional Lebesgue measure and $B (x, r)$ for the open ball in $R^{n}$ of radius $r > 0$ centered at x. Given $f \in L_{loc}^{1} (R^{n})$ , the limit $\begin{matrix} (2) & \tilde{f} (x) = lim_{r \to 0} \frac{1}{λ^{n} (B (x, r))} \int_{B (x, r)} f (y) d y \end{matrix}$ exists at $λ^{n}$ -a.e. $x \in R^{n}$ . Suppose that $1 < p < + \infty$ and $f \in H^{β, p} (R^{n})$ . If $β > n / p$ , then the function f is Hölder continuous, [50, 2.8.1], the limit (2) exists at any point x and equals $f (x)$ . If $0 < β ⩽ n / p$ , then it exists at $H^{β, p} (R^{n})$ -quasi every $x \in R^{n}$ and $\tilde{f}$ defines a $H^{β, p} (R^{n})$ -quasi-continuous version of f, [1, Theorem 6.2.1]. At later occasions we abbreviate ‘for $H^{β, p} (R^{n})$ -quasi every $x \in R^{n}$ ’ by $H^{β, p} (R^{n})$ -q.e. Recall that this means ‘for all $x \in R^{n}$ outside a set of zero $H^{β, p} (R^{n})$ -capacity’. Any set of zero $H^{β, p} (R^{n})$ -capacity is also a set of zero n-dimensional Lebesgue measure, and any property valid $H^{β, p} (R^{n})$ -q.e. is valid $λ^{n}$ -a.e.

Now let μ be a Borel measure with support $Γ = supp μ$ and satisfying the (local) upper regularity condition $\begin{matrix} (3) & μ (B (x, r)) ⩽ c_{d} r^{d}, x \in Γ, 0 < r ⩽ 1, \end{matrix}$ with some $0 < d ⩽ n$ . This implies that the Hausdorff dimension of Γ is at least d, [25,38], it also implies that μ is a locally finite measure. If condition (3) holds and β is large enough, then we can conclude that each set of zero $H^{β, p} (R^{n})$ -capacity is a μ-null set and consequently the limit in (2) exists for μ-a.e. $x \in Γ$ . The following is a version of [31, Theorem 4]. We use the convention $1 / 0 : = + \infty$ , and we write $L^{0} (Γ, μ)$ for the space of μ-equivalence classes of Borel functions on Γ.

Theorem 2.1.
Let $0 ⩽ d ⩽ n$ , $(n - d) / p < β$ and suppose that μ is a Borel measure with support $Γ = supp μ$ and satisfying ( 3 ). Then $\begin{matrix} (4) & {Tr}_{Γ} f : = \tilde{f} \end{matrix}$ defines a linear map ${Tr}_{Γ}$ from $H^{β, p} (R^{n})$ into $L^{0} (Γ, μ)$ .
If $β ⩽ n / p$ and $p < q < p d / (n - p β)$ , then ${Tr}_{Γ} : H^{β, p} (R^{n}) \to L^{q} (Γ, μ)$ is a bounded linear operator and $\begin{matrix} ‖ {Tr}_{Γ} f ‖_{L^{q} (Γ, μ)} ⩽ c_{Tr} ‖ f ‖_{H^{β, p} (R^{n})}, f \in H^{β, p} (R^{n}), \end{matrix}$ with a constant $c_{Tr} > 0$ depending only on n, p, β, d, $c_{d}$ and q. If Γ is compact, then this operator ${Tr}_{Γ}$ is compact.

If $β > n / p$ , then ${Tr}_{Γ} : H^{β, p} (R^{n}) \to L^{\infty} (Γ, μ)$ is a bounded linear operator and $\begin{matrix} ‖ {Tr}_{Γ} f ‖_{L^{\infty} (Γ, μ)} ⩽ c_{Tr} ‖ f ‖_{H^{β, p} (R^{n})}, f \in H^{β, p} (R^{n}), \end{matrix}$ with a constant $c_{Tr} > 0$ depending only on n, p and β. If Γ is compact, then this operator ${Tr}_{Γ}$ is compact.

Proof.
Part (i) follows from [1, Theorems 7.2.2 and 7.3.2 and Propositions 5.1.2, 5.1.3 and 5.1.4]. Part (ii) follows from the Sobolev embedding of $H^{β, p} (R^{n})$ into a space of bounded Hölder continuous functions, the compactness part is due to the Arzela–Ascoli Theorem. □

We write ${Tr}_{Γ} (H^{β, p} (R^{n}))$ for the image of $H^{β, p} (R^{n})$ under ${Tr}_{Γ}$ in $L^{0} (Γ, μ)$ . The following is immediate from the general theory.
Corollary 2.1.
Let $0 ⩽ d ⩽ n$ and $(n - d) / p < β$ . Suppose that μ is a Borel measure with support $Γ = supp μ$ and satisfying ( 3 ). The map $\begin{matrix} (5) & φ \mapsto ‖ φ ‖_{{Tr}_{Γ} (H^{β, p} (R^{n}))} : = inf {‖ g ‖_{H^{β, p} (R^{n})} : g \in H^{β, p} (R^{n}), {Tr}_{Γ} g = φ μ -a.e.} \end{matrix}$ is a norm that makes ${Tr}_{Γ} (H^{β, p} (R^{n}))$ a Banach space, and for $p = 2$ Hilbert.

We briefly point out that the trace space ${Tr}_{Γ} (H^{β, p} (R^{n}))$ does not depend on the choice of an individual measure μ but only on the choice of an equivalence class of measures. Recall that if μ is a Borel measure charging no set of zero $H^{β, p} (R^{n})$ -capacity, then a set $Γ_{q} \subset R^{n}$ is called an $H^{β, p} (R^{n})$ -quasi support of μ if (i) we have $μ (R^{n} ∖ Γ_{q}) = 0$ , (ii) the set $Γ_{q}$ is $H^{β, p} (R^{n})$ -quasi closed and (iii) if ${\tilde{Γ}}_{q}$ is another set with these properties, then $Γ_{q} \subset {\tilde{Γ}}_{q}$ $H^{β, p} (R^{n})$ -quasi everywhere. See [26, Section 2.9]. Two $H^{β, p} (R^{n})$ -quasi supports of μ can differ only by a set of zero $H^{β, p} (R^{n})$ -capacity. Since any closed set is $H^{β, p} (R^{n})$ -quasi closed, we may always assume that an $H^{β, p} (R^{n})$ -quasi support $Γ_{q}$ of μ satisfies $Γ_{q} \subset Γ = supp μ$ . See [27, Section 4.6, p. 168] for the case $p = 2$ . If $β > n / p$ , then point sets have positive $H^{β, p} (R^{n})$ -capacity and Γ is the only quasi-support of μ.
Proposition 2.1.
Let $0 ⩽ d ⩽ n$ and $(n - d) / p < β$ . Suppose that μ is a Borel measure with support $Γ = supp μ$ and satisfying ( 3 ). If $Γ_{q}$ is an $H^{β, p} (R^{n})$ -quasi support of μ, then $\begin{array}{l} ‖ φ ‖_{{Tr}_{Γ} (H^{β, p} (R^{n}))} \\ = inf {‖ g ‖_{H^{β, p} (R^{n})} : g \in H^{β, p} (R^{n}), \tilde{g} = φ H^{β, p} (R^{n}) -q.e. on Γ_{q}}, φ \in H^{β, p} (R^{n}) . \end{array}$ If $μ^{'}$ is another Borel measure satisfying ( 3 ) (with a possibly different constant $c_{d}^{'} > 0$ ) and having $Γ_{q}$ as a quasi support, then the trace space $({Tr}_{Γ} (H^{β, p} (R^{n})), ‖ \cdot ‖_{{Tr}_{Γ} (H^{β, p} (R^{n}))})$ defined using $μ^{'}$ is the same as the one defined using μ.
Remark 2.1.
Suppose that μ and $μ^{'}$ are two Borel measures equivalent to each other and both satisfying (3), possibly with different multiplicative constants. Then any $H^{β, p} (R^{n})$ -quasi support of μ is also one for $μ^{'}$ and vice versa.
Proof.
For the first statement, one can follow [27, Theorem 4.6.2], the arguments remain valid for $p \neq 2$ . The second is an immediate consequence. □

In the case of finite measures, we can conclude further properties.
Corollary 2.2.
Let $0 ⩽ d ⩽ n$ and $(n - d) / p < β$ . Suppose that μ is a finite Borel measure with support $Γ = supp μ$ and satisfying ( 3 ). Then ${Tr}_{Γ}$ as in ( 4 ) defines a bounded linear operator from $H^{β, p} (R^{n})$ into $L^{p} (Γ, μ)$ with norm depending only on n, p, β, a nonnegative power of $μ (Γ)$ and in the case $β ⩽ n / p$ also on d and $c_{d}$ . This operator is compact if Γ is. Seen as an operator from $H^{β, p} (R^{n})$ onto ${Tr}_{Γ} (H^{β, p} (R^{n}))$ normed by ( 5 ), the operator ${Tr}_{Γ}$ is a linear contraction: $‖ {Tr}_{Γ} ‖ ⩽ 1$ . The space ${Tr}_{Γ} (H^{β, p} (R^{n}))$ is dense in $L^{p} (Γ, μ)$ .
Proof.
The first statements are clear from Theorem 2.1, the finiteness of μ and Hölder’s inequality. The stated contractivity is clear from (5). The claimed density follows from the fact that $C_{c}^{\infty} (R^{n})$ is dense in $H^{β, p} (R^{n})$ together with the Stone–Weierstrass theorem. □
Remark 2.2.
It is well-known that under stronger assumptions on μ one can identify the trace space ${Tr}_{Γ} (H^{β, p} (R^{n}))$ explicitly as a Besov space, [33–35], see also [31, Section 5.1].

For later use, we fix a simple observation on the consistency of trace procedures.
Corollary 2.3.
Let $0 ⩽ d ⩽ n$ and $(n - d) / p < β$ . Suppose that μ is a finite Borel measure with support $Γ = supp μ$ and satisfying ( 3 ). If $Γ^{'}$ is a closed subset of Γ with $μ (Γ^{'}) > 0$ then ${Tr}_{Γ^{'}}$ is a bounded linear operator from $H^{β, p} (R^{n})$ into $L^{p} (Γ^{'}, μ)$ , and for any $f \in H^{β, p} (R^{n})$ we have ${Tr}_{Γ^{'}} f = {Tr}_{Γ} f |_{Γ^{'}}$ μ-a.e. on $Γ^{'}$ .
Proof.
Since the closed set $Γ^{'}$ is the support of the measure $μ^{'} : = μ |_{Γ^{'}}$ , which satisfies (3) at all $x \in Γ^{'}$ , the claims follow from Corollary 2.2 and (4). □

Now let $Ω \subset R^{n}$ be an arbitrary domain. For $1 < p < + \infty$ and $β > 0$ we write $\begin{matrix} H^{β, p} (Ω) = {f \in D^{'} (Ω) : f = g |_{Ω} for some g \in H^{β, p} (R^{n})}, \end{matrix}$ where $D^{'} (Ω)$ denotes the space of Schwartz distributions on Ω. Endowed with the norm $\begin{matrix} ‖ f ‖_{H^{β, p} (Ω)} : = inf {‖ g ‖_{H^{β, p} (R^{n})} : g \in H^{β, p} (R^{n}), f = g |_{Ω} in D^{'} (Ω)} . \end{matrix}$ it becomes a Banach space (and Hilbert for $p = 2$ ). See for instance [52, Section 2.2].

Suppose that μ is a finite Borel measure as specified in Corollary 2.2 and that $Γ = supp μ \subset \overline{Ω}$ . Given $f \in H^{β, p} (Ω)$ , we set $\begin{matrix} {Tr}_{Ω, Γ} f : = {Tr}_{Γ} g, \end{matrix}$ where g is an element of $H^{β, p} (R^{n})$ such that $f = g |_{Ω}$ in $D^{'} (Ω)$ . By the arguments used to prove [4, Theorem 6.1] one can see that ${Tr}_{Ω, Γ}$ is a well-defined linear map ${Tr}_{Ω, Γ} : H^{β, p} (Ω) \to L^{p} (Γ, μ)$ in the sense that if $g, h \in H^{β, p} (R^{n})$ satisfy $g = h$ $λ^{n}$ -a.e. in Ω, then ${Tr}_{Γ} g = {Tr}_{Γ} h$ μ-a.e. on Γ. It is a bounded linear operator, and the image ${Tr}_{Ω, Γ} (H^{β, p} (Ω))$ agrees with ${Tr}_{Γ} (H^{β, p} (R^{n}))$ .

Later we will make use of the following variant of [31, Theorem 5] on the convergence of integrals of traces in bounded confinement D. In contrast to the result there, we now only assume weak convergence of functions.
Lemma 2.1.
Let $D \subset R^{n}$ be a bounded Lipschitz domain, $0 ⩽ d ⩽ n$ and $(n - d) / p < β$ . Let ${(μ_{m})}_{m}$ be a sequence of Borel measures with supports $Γ_{m} = supp μ_{m}$ contained in $\overline{D}$ and such that ( 3 ) holds for all m with the same constant $c_{d}$ . Suppose that ${(μ_{m})}_{m}$ converges weakly to a Borel measure μ. Then $Γ : = supp μ$ is contained in $\overline{D}$ , and if ${(v_{m})}_{m} \subset H^{β, p} (D)$ converges to some v weakly in $H^{β, p} (D)$ , then $\begin{matrix} lim_{m \to \infty} \int_{Γ_{m}} {Tr}_{D, Γ_{m}} v_{m} d μ_{m} = \int_{Γ} {Tr}_{D, Γ} v d μ \end{matrix}$ and $\begin{matrix} lim_{m \to \infty} \int_{Γ_{m}} | {Tr}_{D, Γ_{m}} v_{m} |^{p} d μ_{m} = \int_{Γ} | {Tr}_{D, Γ} v |^{p} d μ . \end{matrix}$
Proof.
Note first that since all $μ_{m}$ satisfy (3) and are supported in the compact set $\overline{D}$ , we have ${sup}_{m} μ_{m} (R^{n}) < + \infty$ . The fact that $Γ \subset \overline{D}$ is clear from weak convergence. To see the stated limit relations, let ${(φ_{j})}_{j} \subset C_{c}^{\infty} (\overline{D})$ be a sequence converging to v in $H^{β, p} (D)$ . Similarly, as in the proof of [12, Theorem 5.2], we observe that $\begin{array}{l} | \int_{Γ_{m}} {Tr}_{D, Γ_{m}} v_{m} d μ_{m} - \int_{Γ} {Tr}_{D, Γ} v d μ | \\ ⩽ | \int_{Γ_{m}} {Tr}_{D, Γ_{m}} v_{m} d μ_{m} - \int_{Γ_{m}} {Tr}_{D, Γ_{m}} v d μ_{m} | + | \int_{Γ_{m}} {Tr}_{D, Γ_{m}} v d μ_{m} - \int_{Γ_{m}} φ_{j} d μ_{m} | \\ (6) & + | \int_{Γ_{m}} φ_{j} d μ_{m} - \int_{Γ} φ_{j} d μ | + | \int_{Γ} φ_{j} d μ - \int_{Γ} {Tr}_{Γ} v d μ | . \end{array}$ Now let $(n - d) / p < β^{'} < β$ . By Hölder’s inequality and Corollary 2.2, the first term on the right-hand side of (6) is bounded by $\begin{array}{l} \int_{Γ_{m}} | {Tr}_{D, Γ_{m}} (v_{m} - v) | d μ_{m} & ⩽ μ_{m} {(R^{n})}^{1 / q} {‖ {Tr}_{D, Γ_{m}} (v_{m} - v) ‖}_{L^{p} (Γ_{m}, μ_{m})} \\ ⩽ c_{Tr} μ_{m} {(R^{n})}^{1 / q} ‖ v_{m} - v ‖_{H^{β^{'}, p} (D)} \end{array}$ with $c_{Tr} > 0$ independent of m. Since the embedding $H^{β, p} (D) \subset H^{β^{'}, p} (D)$ is compact, [52, Theorem 2.7], this goes to zero as $m \to \infty$ . When $j \to \infty$ , the second term and the last in (6) converge to zero uniformly in m. The third term converges to zero as $m \to \infty$ by weak convergence of measures.

The second statement follows from a slight modification of the proof of [31, Theorem 5]: We have an estimate analogous to (6) but with $| {Tr}_{D, Γ_{m}} v_{m} |^{p}$ , $| {Tr}_{D, Γ_{m}} v |^{p}$ and $| φ_{j} |^{p}$ in place of ${Tr}_{D, Γ_{m}} v_{m}$ , ${Tr}_{D, Γ_{m}} v$ and $φ_{j}$ . The convergence of the last three summands on the right-hand side of this inequality follows from the trace operator’s uniform norm and the measures’ weak convergence. For the first summand, we can use the mean value theorem for $s \mapsto s^{p}$ , the reverse triangle inequality, and Hölder’s inequality w.r.t. $μ_{m}$ to see that $\begin{array}{l} | \int_{Γ_{m}} | {Tr}_{D, Γ_{m}} v_{m} |^{p} d μ_{m} - \int_{Γ_{m}} | {Tr}_{D, Γ_{m}} v |^{p} d μ_{m} | \\ ⩽ p \int_{Γ_{m}} | | {Tr}_{D, Γ_{m}} v_{m} | - | {Tr}_{D, Γ_{m}} v_{m} | | {(| {Tr}_{D, Γ_{m}} v_{m} | + | {Tr}_{D, Γ_{m}} v |)}^{p - 1} d μ_{m} \\ (7) & ⩽ p {(\int_{Γ_{m}} {| {Tr}_{D, Γ_{m}} (v_{m} - v) |}^{p} d μ_{m})}^{1 / p} {(\int_{Γ_{m}} {(| {Tr}_{D, Γ_{m}} v_{m} | + | {Tr}_{D, Γ_{m}} v |)}^{p} d μ_{m})}^{(p - 1) / p} . \end{array}$

Again by Corollary 2.2 we have $\begin{matrix} max {sup_{m} ‖ {Tr}_{D, Γ_{m}} v_{m} ‖_{L^{p} (Γ_{m}, μ_{m})}, sup_{m} ‖ {Tr}_{D, Γ_{m}} v ‖_{L^{p} (Γ_{m}, μ_{m})}} ⩽ c_{Tr}^{'} sup_{m} ‖ v_{m} ‖_{H^{β, p} (D)} \end{matrix}$ with $c_{Tr}^{'} > 0$ independent of m. Using the same compact embedding as above, we see that $\begin{matrix} lim_{m \to \infty} {‖ {Tr}_{D, Γ_{m}} (v_{m} - v) ‖}_{L^{p} (Γ_{m}, μ_{m})} = 0 . \end{matrix}$ Consequently, (7) also converges to zero. □

We briefly discuss linear extension operators for the case $p = 2$ . This discussion will not be needed in later sections, but it is closely related to the trace operators discussed above. Let μ be a nonzero Borel measure on $R^{n}$ satisfying (3) whose support $Γ = supp μ$ is a subset of $R^{n}$ . Given $β > 0$ let ${\overset{˚}{H}}^{β, 2} (R^{n} ∖ Γ)$ be the closure of $C_{c}^{\infty} (R^{n} ∖ Γ)$ in the Hilbert space $H^{β, 2} (R^{n})$ . It coincides with the subspace of all elements v such that $\tilde{v} = 0$ $H^{β, 2} (R^{n})$ -q.e. on Γ, [27, Corollary 2.3.1]. Suppose that $0 ⩽ d ⩽ n$ , $(n - d) / 2 < β$ and that $φ \in {Tr}_{Γ} (H^{β, 2} (R^{n}))$ . For any $α ⩾ 0$ , the generalized Dirichlet problem, formally stated as $\begin{matrix} (8) & \{\begin{matrix} {(α - Δ)}^{β} u = 0 & on R^{n} ∖ Γ \\ {Tr}_{Γ} u = φ & on Γ, \end{matrix} \end{matrix}$ can be written rigorously by saying that $u \in H^{β, 2} (R^{n})$ is a weak solution to (8) if ${Tr}_{Γ} u = φ$ $H^{β, 2} (R^{n})$ -q.e. and $\begin{matrix} (9) & {⟨ {(α - Δ)}^{β / 2} u, {(α - Δ)}^{β / 2} v ⟩}_{L^{2} (R^{n})} = 0, v \in {\overset{˚}{H}}^{β, 2} (R^{n} ∖ Γ) . \end{matrix}$ Here Δ is the standard self-adjoint Laplacian on $R^{n}$ , the operators ${(α - Δ)}^{β}$ and ${(α - Δ)}^{β / 2}$ are the fractional powers of order β and $β / 2$ of the nonnegative self-adjoint operator $α - Δ$ , defined through spectral theory, [44, Section VIII.3]. The next two statements are a version of [31, Corollary 1].
Proposition 2.2.
Let $0 ⩽ d ⩽ n$ and $(n - d) / 2 < β$ . Suppose that μ is a nonzero finite Borel measure whose support $Γ = supp μ$ is a subset of $R^{n}$ and which satisfies ( 3 ). For any $α > 0$ and $φ \in {Tr}_{Γ} (H^{β, 2} (R^{n}))$ there is a unique weak solution $H_{Γ}^{α} φ$ to ( 8 ).
Proof.
Let ${⟨ \cdot, \cdot ⟩}_{α}$ denote the bilinear form on the left-hand side of (9). Let $w \in H^{β, 2} (R^{n})$ be such that ${Tr}_{Γ} w = φ$ . By the Riesz representation theorem there is a unique element $\tilde{u}$ of ${\overset{˚}{H}}^{β, 2} (R^{n} ∖ Γ)$ such that ${⟨ \tilde{u}, v ⟩}_{α} = - {⟨ w, v ⟩}_{α}$ , $v \in {\overset{˚}{H}}^{β, 2} (R^{n} ∖ Γ)$ . Then $u : = \tilde{u} + w$ has the desired properties. □

We use the shortcut notation $\begin{matrix} (10) & H_{Γ} φ : = H_{Γ}^{1} φ . \end{matrix}$ It is common to refer to $H_{Γ}$ as the 1-harmonic extension operator. Corollary 2.4.
Let $0 ⩽ d ⩽ n$ and $(n - d) / 2 < β$ . Suppose that μ is a nonzero finite Borel measure whose support $Γ = supp μ$ is a subset of $R^{n}$ and which satisfies ( 3 ). The map $\begin{matrix} H_{Γ} : {Tr}_{Γ} (H^{β, 2} (R^{n})) \to H^{β, 2} (R^{n}), φ \mapsto H_{Γ} φ, \end{matrix}$ is a linear extension operator of norm one, and ${Tr}_{Γ} (H_{Γ} φ) = φ$ , $φ \in {Tr}_{Γ} (H^{β, 2} (R^{n}))$ .

If $0 < β ⩽ 1$ and φ is a bounded Borel function on Γ such that $φ \in {Tr}_{Γ} (H^{β, 2} (R^{n}))$ in the sense of μ-equivalence classes, then $H_{Γ} φ \in H^{β, 2} (R^{n}) \cap L^{\infty} (R^{n})$ with $‖ H_{Γ} φ ‖_{L^{\infty} (R^{n})} ⩽ {sup}_{x \in Γ} | φ (x) |$ .
Proof.
The first statements follow from the preceding. The last one follows from [27, Theorem 4.3.1] because for $0 < β ⩽ 1$ the pair $(‖ \cdot ‖_{H^{β, 2} (R^{n})}^{2}, H^{β, 2} (R^{n}))$ is a regular Dirichlet form. □
Remark 2.3.
The same strategy, based on a nonlinear version of (8), yields extension operators also for the case $p \neq 2$ , but they may not be linear. See [34,51].

3. Poincaré inequalities with uniform constants

We turn to Poincaré inequalities. Let $Ω \subset R^{n}$ be a domain and let $1 < p < + \infty$ . The Sobolev space $W^{1, p} (Ω)$ is defined as the space of all $f \in L^{p} (Ω)$ whose gradient $\nabla f$ , defined in distributional sense, belongs to $L^{p} (Ω, R^{n})$ . Endowed with the norm $\begin{matrix} ‖ f ‖_{W^{1, p} (Ω)} : = {(‖ f ‖_{L^{p} (Ω)}^{p} + ‖ \nabla f ‖_{L^{p} (Ω, R^{n})}^{p})}^{1 / p} \end{matrix}$ it is Banach, and for $p = 2$ Hilbert. It is well-known that $W^{1, p} (R^{n}) = H^{1, p} (R^{n})$ in the sense of equivalently normed vector spaces.

Obviously the restriction ${Tr}_{Ω} f = f |_{Ω}$ of an element $f \in W^{1, p} (R^{n})$ to a domain Ω is an element of $W^{1, p} (Ω)$ , the operator ${Tr}_{Ω}$ is a linear contraction, and it is compact if Ω is bounded. The domain Ω is said to be a $W^{1, p}$ -extension domain if there is a bounded linear extension operator $E_{Ω} : W^{1, p} (Ω) \to W^{1, p} (R^{n})$ . In this case we have $(E f) |_{Ω} = f$ for all $f \in W^{1, p} (Ω)$ . Note that trivially, $Ω = R^{n}$ itself is an extension domain, and we may choose $E_{R^{n}}$ to be the identity.

Now suppose that Ω is a $W^{1, p}$ -extension domain and that μ is a nonzero finite Borel measure whose support $supp μ = Γ$ is a subset of $\overline{Ω}$ and which satisfies (3) with some $0 ⩽ d ⩽ n$ such that $n - p < d$ . Similarly, as in the last section, [4, Theorem 6.1] guarantees that $\begin{matrix} {Tr}_{Ω, Γ} : = {Tr}_{Γ} \circ E_{Ω} : W^{1, p} (Ω) \to L^{p} (Γ, μ) \end{matrix}$ is well defined in the sense that if $u, v \in W^{1, p} (R^{n})$ are such that $u = v$ $λ^{n}$ -a.e. in Ω, then ${Tr}_{Γ} u = {Tr}_{Γ} v$ μ-a.e. on Γ. See also [55, Theorem 1]. We have ${Tr}_{Ω, Γ} (W^{1, p} (Ω)) = {Tr}_{Γ} (W^{1, p} (R^{n}))$ . Corollary 2.2 implies that ${Tr}_{Ω, Γ}$ is a bounded linear operator, and it is compact if Γ is. Seen as a linear operator ${Tr}_{Ω, Γ} : W^{1, p} (Ω) \to {Tr}_{Γ} (W^{1, p} (R^{n}))$ it is a contraction. In the case that $Ω = R^{n}$ and $E_{R^{n}}$ is the identity, we have ${Tr}_{R^{n}, Γ} = {Tr}_{Γ}$ .

Recall notation (10). We briefly record the following immediate consequence of Corollary 2.4.

Corollary 3.1.
Let Ω be a $W^{1, 2}$ -extension domain and suppose that μ is a nonzero finite Borel measure satisfying ( 3 ) with some $0 ⩽ d ⩽ n$ such that $n - 2 < d$ and whose support $supp μ = Γ$ is a subset of $\overline{Ω}$ . The map $\begin{matrix} (11) & H_{Γ, Ω} : = {Tr}_{Ω} \circ H_{Γ} : {Tr}_{Γ} (W^{1, 2} (R^{n})) \to W^{1, 2} (Ω) . \end{matrix}$ is a linear extension operator of norm one, and ${Tr}_{Ω, Γ} (H_{Γ, Ω} φ) = φ$ , $φ \in {Tr}_{Γ} (W^{1, 2} (R^{n}))$ . If φ is a bounded Borel function on Γ such that $φ \in {Tr}_{Γ} (W^{1, 2} (R^{n}))$ in the sense of μ-equivalence classes, then $H_{Γ, Ω} φ \in W^{1, 2} (Ω) \cap L^{\infty} (Ω)$ with $‖ H_{Γ, Ω} φ ‖_{L^{\infty} (Ω)} ⩽ {sup}_{x \in Γ} | φ (x) |$ .

Throughout the remaining section, we assume $n ⩾ 2$ . We recall the following classical definition, [32,55].
Definition 3.1.
Let $ε > 0$ . A bounded domain $Ω \subset R^{n}$ is called an $(ε, \infty)$ -domain if for all $x, y \in Ω$ there is a rectifiable curve $γ \subset Ω$ with length $ℓ (γ)$ joining x to y and satisfying
$ℓ (γ) ⩽ \frac{| x - y |}{ε}$ and

$d (z, \partial Ω) ⩾ ε | x - z | \frac{| y - z |}{| x - y |}$ for $z \in γ$ .

A bounded $(ε, \infty)$ -domain is a uniform domain in the sense of [53]. By [32, Theorem 1] any bounded $(ε, \infty)$ -domain $Ω \subset R^{n}$ is a $W^{1, p}$ -extension domain, and there is an extension operator $E_{Ω}$ whose operator norm depends only on n, p and ε, see also [3,46]. Theorem 3.1 and Lemma 3.3 below will also make use of a homogeneous version of this statement, [32, Theorem 2]. In both cases, the parameter ε provides a quantitative control on the norm of extension operators.

For $(ε, \infty)$ -domains Ω inside bounded confinement D and Borel measures μ supported inside $\overline{Ω}$ and satisfying (3) with the same constant, we can obtain Poincaré inequalities with constants that are ‘uniform’, that is, independent of the particular shape of the domain Ω or the set Γ. A related result for the case that $Γ \subset \partial Ω$ and the functions in question have vanishing trace on Γ had been shown in [21, Theorem 10].
Theorem 3.1.
Let $D \subset R^{n}$ be a bounded Lipschitz domain, $1 < p < + \infty$ , $0 ⩽ d ⩽ n$ , $n - p < d$ and $ε > 0$ . Suppose that $Ω \subset D$ is a $(ε, \infty)$ -domain and that μ is a Borel probability measure on $R^{n}$ with $Γ = supp μ \subset \overline{Ω}$ and such that ( 3 ) holds. Then we have $\begin{matrix} {‖ u - \int_{Γ} {Tr}_{Ω, Γ} u d μ ‖}_{L^{p} (Ω)} ⩽ C_{P} ‖ \nabla u ‖_{L^{p} (Ω, R^{n})}, u \in W^{1, p} (Ω), \end{matrix}$ with a constant $C_{P} > 0$ depending only on n, D, p, ε, d and $c_{d}$ .
Remark 3.1.

The constant $C_{P}$ does not depend on the particular choice of μ.

The requirement that μ is a probability measure is made for notational convenience only and no restriction: If μ is a nonzero Borel measure on $R^{n}$ with compact support $Γ = supp μ \subset \overline{Ω}$ such that (3) holds, then the Borel probability measure $μ^{'} : = μ / μ (Γ)$ satisfies (3) with $c_{d} / μ (Γ)$ in place of $c_{d}$ .

Theorem 3.1 yields equivalent norms on $W^{1, p} (Ω)$ with uniform constants in the norm comparison.
Corollary 3.2.
Let the hypotheses of Theorem 3.1 be in force and let $a > 0$ . Then $\begin{matrix} ‖ u ‖_{W^{1, p} (Ω), μ, a} : = {(‖ \nabla u ‖_{L^{p} (Ω, R^{n})}^{p} + a ‖ {Tr}_{Ω, Γ} u ‖_{L^{p} (Γ, μ)}^{p})}^{1 / p} \end{matrix}$ defines an equivalent norm on $W^{1, p} (Ω)$ , and we have $\begin{matrix} (12) & c_{1}^{- 1} ‖ u ‖_{W^{1, p} (Ω)} ⩽ ‖ u ‖_{W^{1, p} (Ω), μ, a} ⩽ c_{2} ‖ u ‖_{W^{1, p} (Ω)}, u \in W^{1, p} (Ω), \end{matrix}$ with constants $\begin{matrix} c_{1} = max (1 + C_{P}, \frac{λ^{n} {(Ω)}^{1 / p}}{a^{1 / p}}) and c_{2} = max (1, a^{1 / p} c_{Tr} c_{E, 1}) \end{matrix}$ depending only on n, D, p, ε, d, $c_{d}$ and $λ^{n} (Ω)$ . Here $C_{P}$ is as in Theorem 3.1 , $c_{Tr}$ is a norm bound for the trace operator from $W^{1, p} (R^{n})$ into $L^{p} (Γ, μ)$ in Corollary 2.2 and $c_{E, 1}$ is the bound for the extension operator $E_{Ω}$ from [ 32 , Theorem 1].
Remark 3.2.
Inequalities as in Theorem 3.1 and (12) hold for any bounded $W^{1, p}$ -extension domain Ω and measure μ as specified. The message of Theorem 3.1 and Corollary 3.2 is that under more specific hypotheses, the constants depend only on the stated parameters, and this is the key to proving the existence of optimal shapes for the Robin problem in Section 8.

We formulate proofs of Theorem 3.1 and Corollary 3.2 using three intermediate statements. In the sequel let n, p, d and $c_{d}$ be as specified in Theorem 3.1.

For any $W^{1, p}$ -extension domain $U \subset R^{n}$ we write $\begin{array}{rcl} X (U) & : = & {u \in W^{1, p} (U) : there is a Borel probability measure μ on R^{n} \\ (13) & with Γ : = supp μ \subset \overline{U}, satisfying (3) and such that \int_{Γ} {Tr}_{U, Γ} u d μ = 0} . \end{array}$

The set $X (U)$ is a cone in the sense that for any $u \in X (U)$ and $λ ⩾ 0$ we have $λ u \in X (U)$ . Moreover, $X (U)$ does not contain any constant except zero.
Remark 3.3.
In general $X (U)$ is not convex: For $U = {(0, 1)}^{2}$ , $u (x_{1}, x_{2}) = x_{1}$ and $u (x_{1}, x_{2}) = 1 - x_{1}$ we have $u_{1}, u_{2} \in X (U)$ : Clearly $u_{1}, u_{2} \in W^{1, p} (U)$ , and $\int u_{1} d μ_{1} = \int u_{2} d μ_{2} = 0$ for the probability measures $μ_{1} : = H^{1} |_{{0} \times (0, 1)}$ and $μ_{2} : = H^{1} |_{{1} \times (0, 1)}$ . However, $\frac{1}{2} (u_{1} + u_{2}) = \frac{1}{2}$ is not in $X (U)$ .
Lemma 3.1.
Let $D \subset R^{n}$ be a bounded Lipschitz domain. Then the set $X (D)$ is weakly closed in $W^{1, p} (D)$ .
Proof.
Suppose that ${(u_{k})}_{k} \subset X (D)$ converges to some u weakly in $W^{1, p} (D)$ . For each k let $μ_{k}$ be a probability measure corresponding to $u_{k}$ as in (13). Since by the Riesz–Markov and Banach–Alaoglu theorems, [44, Theorems IV.18 and IV.21], the unit ball in the space of (signed) Borel regular measures on $\overline{D}$ of finite total variation is compact with respect to weak convergence, there is a sequence ${(k_{j})}_{j}$ of indices such that ${(μ_{k_{j}})}_{j}$ converges to some probability measure μ weakly on $\overline{D}$ . By weak convergence $Γ : = supp μ$ is contained in $\overline{D}$ , and by [31, Proposition 2 (i)] the measure μ satisfies (3). Writing $Γ_{k} : = supp μ_{k}$ we obtain $\begin{matrix} \int_{Γ} {Tr}_{D, Γ} u d μ = lim_{j} \int_{Γ_{k_{j}}} {Tr}_{D, Γ_{k_{j}}} u_{k_{j}} d μ = 0 \end{matrix}$ from Lemma 2.1. Consequently $u \in X (D)$ . □

We can now conclude by a Poincaré inequality for elements of $X (D)$ .
Lemma 3.2.
Let $D \subset R^{n}$ be a bounded Lipschitz domain. Then we have $\begin{matrix} (14) & ‖ u ‖_{L^{p} (D)} ⩽ c (n, D, p, d, c_{d}) ‖ \nabla u ‖_{L^{p} (D, R^{n})}, u \in X (D), \end{matrix}$ with $c (n, D, p, d, c_{d}) > 0$ depending only on n, D, p, d and $c_{d}$ .

Apart from the use of $X (D)$ the argument is standard, see for instance [23, Proposition 7.1] or [24, Section 5.8, Theorem 1].
Proof.
Suppose that (14) does not hold. Then there is a sequence ${(u_{k})}_{k} \subset X (D)$ such that $\frac{1}{k} ‖ u_{k} ‖_{L^{p} (D)} ⩾ ‖ \nabla u_{k} ‖_{L^{p} (D)}$ . Since $X (D)$ is a cone, we may divide each $u_{k}$ by its norm and therefore assume that $\begin{matrix} (15) & ‖ u_{k} ‖_{L^{p} (D)} = 1 for all k . \end{matrix}$ Then obviously ${lim}_{k} \nabla u_{k} = 0$ in $L^{p} (D, R^{n})$ , and since ${sup}_{k} ‖ u_{k} ‖_{W^{1, p} (D)} < + \infty$ and $W^{1, p} (D)$ is reflexive, there is a subsequence ${(u_{k_{j}})}_{j}$ that converges to some u weakly in $W^{1, p} (D)$ . By Lemma 3.1 we have $u \in X (D)$ , and since $\nabla : W^{1, p} (D) \to L^{p} (D, R^{n})$ is continuous, ${(\nabla u_{k_{j}})}_{j}$ converges to $\nabla u$ weakly in $L^{p} (D, R^{n})$ . But this means that $\nabla u = 0$ , so that $u \in X (D)$ is constant, hence $u = 0$ , and we see that ${lim}_{j} u_{k_{j}} = 0$ weakly in $W^{1, p} (D)$ . By the Rellich–Kondrachov theorem for D it now follows that ${lim}_{j} u_{k_{j}} = 0$ strongly in $L^{p} (D)$ , what contradicts (15). □

The preceding gives a Poincaré inequality valid simultaneously for any $(ε, \infty)$ -domain Ω contained in D with a uniform constant.
Lemma 3.3.
Let $D \subset R^{n}$ be a bounded Lipschitz domain and $ε > 0$ . There is a constant $C (n, D, p, ε, d, c_{d}) > 0$ depending only on n, D, p, ε, d and $c_{d}$ such that for any $(ε, \infty)$ -domain $Ω \subset D$ we have $\begin{matrix} (16) & ‖ v ‖_{L^{p} (Ω)} ⩽ C (n, D, p, ε, d, c_{d}) ‖ \nabla v ‖_{L^{p} (Ω, R^{n})}, v \in X (Ω) . \end{matrix}$
Proof.
Suppose first that $v \in X (Ω) \cap L^{\infty} (Ω)$ and set $S : = ‖ v ‖_{L^{\infty} (Ω)}$ . Let ${\dot{E}}_{Ω}$ denote the extension operator introduced in [32, Theorem 2] and $c_{E, 2} > 0$ a bound for its norm (and depending only on n, p, ε). Then this theorem gives $\begin{matrix} {‖ \nabla {({\dot{E}}_{Ω} v)}_{S} ‖}_{L^{p} (D, R^{n})} ⩽ {‖ \nabla ({\dot{E}}_{Ω} v) ‖}_{L^{p} (R^{n}, R^{n})} ⩽ c_{E, 2} ‖ \nabla v ‖_{L^{p} (Ω, R^{n})}, \end{matrix}$ where we use the notation ${(f)}_{S} : = min (max (f, - S), S)$ . Since $| {({\dot{E}}_{Ω} v)}_{S} | ⩽ S$ and D is bounded, it follows that ${({\dot{E}}_{Ω} v)}_{S} \in W^{1, p} (D)$ . If μ is a Borel probability measure with support $Γ \subset \overline{Ω}$ with respect to which ${Tr}_{Ω, Γ} v$ has integral zero, then also ${Tr}_{D, Γ} {({\dot{E}}_{Ω} v)}_{S}$ has integral zero with respect to μ because ${Tr}_{Ω, Γ} v = {Tr}_{D, Γ} {({\dot{E}}_{Ω} v)}_{S}$ μ-a.e. on Γ by [4, Theorem 6.1]. Consequently ${({\dot{E}}_{Ω} v)}_{S} \in X (D)$ , and combining with Lemma 3.2 we obtain $\begin{array}{l} ‖ v ‖_{L^{p} (Ω)} & ⩽ {‖ {({\dot{E}}_{Ω} v)}_{S} ‖}_{L^{p} (D)} \\ ⩽ c (n, D, p, d, c_{d}) {‖ \nabla {({\dot{E}}_{Ω} v)}_{S} ‖}_{L^{p} (D, R^{n})} \\ ⩽ c (n, D, p, d, c_{d}) c_{E, 2} ‖ \nabla v ‖_{L^{p} (Ω, R^{n})} . \end{array}$ Given general $v \in X (Ω)$ , the preceding shows that $\begin{matrix} {‖ {(v)}_{N} ‖}_{L^{p} (Ω)} ⩽ c (n, D, p, d, c_{d}) c_{E, 2} {‖ \nabla {(v)}_{N} ‖}_{L^{p} (Ω, R^{n})} ⩽ c (n, D, p, d, c_{d}) c_{E, 2} ‖ \nabla v ‖_{L^{p} (Ω, R^{n})} \end{matrix}$ for each $N ⩾ 1$ , and the result follows for $N \to \infty$ . □

Now Theorem 3.1 and Corollary 3.2 are easily obtained.
Proof of Theorem 3.1.
For any $u \in W^{1, p} (Ω)$ the function $v : = u - \int_{Γ} {Tr}_{Ω, Γ} u d μ$ is in $X (Ω)$ , and Lemma 3.3 yields the desired inequality. □
Proof of Corollary 3.2.
With $p < q < p d / (n - p)$ Hölder’s inequality, Theorem 2.1 and [32, Theorem 1] yield $\begin{matrix} ‖ {Tr}_{Ω, Γ} u ‖_{L^{p} (Γ, μ)} ⩽ ‖ {Tr}_{Ω, Γ} u ‖_{L^{q} (Γ, μ)} ⩽ c_{Tr} c_{E, 1} ‖ u ‖_{W^{1, p} (Ω)} . \end{matrix}$ On the other hand, Theorem 3.1 implies $\begin{matrix} ‖ u ‖_{L^{p} (Ω)} ⩽ C_{P} ‖ \nabla u ‖_{L^{p} (Ω, R^{n})} + λ^{n} {(Ω)}^{1 / p} ‖ {Tr}_{Ω, Γ} u ‖_{L^{p} (Γ, μ)} . \end{matrix}$ □

As an auxiliary observation, we state the following version of Lemma 3.2, it is similar to the classical Poincaré inequality for balls, [24, Section 5.8, Theorem 2]. Corollary 3.3.
For each $x \in R^{n}$ and $r > 0$ we have $\begin{matrix} ‖ u ‖_{L^{p} (B (x, r))} ⩽ c (n, p, d, c_{d}) r ‖ \nabla u ‖_{L^{p} (B (x, r), R^{n})}, u \in X (B (x, r)) . \end{matrix}$
Proof.
We use the shortcut notation $B : = B (x, r)$ and set $ℓ (y) : = x + r y$ , $y \in R^{n}$ . Suppose that $u \in W^{1, p} (B)$ and μ is a probability measure with $Γ = supp μ \subset \overline{B}$ such that $\int_{Γ} u d μ = 0$ . Then $v : = u \circ ℓ$ is an element of $W^{1, p} (B (0, 1))$ . Clearly $w : = E_{B} u \circ ℓ$ is an extension of v to an element of $W^{1, p} (R^{n})$ , and the pointwise redefinition $\tilde{w}$ of w as in (2) is ${(E_{B} u)}^{\sim} \circ ℓ$ , note that $\begin{matrix} (17) & \frac{1}{λ^{n} (B (0, ε))} \int_{B (y, ε)} w (z) d z = \frac{1}{λ^{n} (B (0, r ε))} \int_{B (x + r y, r ε)} (E_{B} u) (ζ) d ζ \end{matrix}$ for any $ε > 0$ . The push-forward $ν : = {(ℓ^{- 1})}_{*} μ$ of μ under $ℓ^{- 1}$ has support $Γ^{'} : = supp ν \subset \overline{B (0, 1)}$ , and since by [4, Theorem 6.1] (or [55, Theorem 1]) we have ${Tr}_{B (0, 1), Γ^{'}} v = {Tr}_{Γ^{'}} w$ , equality (17) implies $\begin{matrix} \int_{Γ^{'}} {Tr}_{B (0, 1), Γ^{'}} v d ν = \int_{Γ^{'}} \tilde{w} d ν = \int_{Γ^{'}} {(E_{B} u)}^{\sim} d μ = \int_{Γ} {Tr}_{B, Γ} u d μ = 0 . \end{matrix}$ Consequently $v \in X (B (0, 1))$ , and by Lemma 3.2, $‖ v ‖_{L^{p} (B (0, 1))} ⩽ c (n, p, d, c_{d}) ‖ \nabla v ‖_{L^{p} (B (0, 1)), R^{n})}$ , what yields the statement. □

4. Generalized boundary value problems

To discuss generalized boundary value problems, we specialize again in the case $p = 2$ . Let Ω be a $W^{1, 2}$ -extension domain (not necessarily bounded). Suppose that μ is a nonzero finite Borel measure whose support $supp μ = Γ$ is a subset of $\overline{Ω}$ and which satisfies (3) with some $0 ⩽ d ⩽ n$ , $n - 2 < d$ . We write $\begin{matrix} (18) & V (Ω, Γ) : = {v \in W^{1, 2} (Ω) : {(E_{Ω} v)}^{\sim} = 0 W^{1, 2} (R^{n}) -q.e. on Γ} \end{matrix}$ for the closed subspace of $W^{1, 2} (Ω)$ consisting of elements with zero trace on Γ in the $W^{1, 2} (R^{n})$ -q.e. sense. Let $α ⩾ 0$ , $f \in L^{2} (Ω)$ and $φ \in {Tr}_{Γ} (W^{1, 2} (R^{n}))$ . We make the generalized Dirichlet problem $\begin{matrix} (19) & \{\begin{matrix} (α - Δ) u = f & on Ω ∖ Γ \\ {Tr}_{Ω, Γ} u = φ & on Γ \end{matrix} \end{matrix}$ rigorous by saying that $u \in W^{1, 2} (Ω)$ is a weak solution to (19) if ${Tr}_{Ω, Γ} u = φ$ holds $W^{1, 2} (R^{n})$ -q.e. on Γ and we have $\begin{matrix} (20) & \int_{Ω} \nabla u \nabla v d x + α \int_{Ω} u v d x = \int_{Ω} f v d x, v \in V (Ω, Γ) . \end{matrix}$

Remark 4.1.

For $Ω = R^{n}$ and $f \equiv 0$ problem (19) is just problem (8) with $β = 1$ .

A function $u \in W^{1, 2} (Ω)$ is a weak solution to (19) if and only if the first equation in (19) holds in $D^{'} (Ω ∖ Γ)$ and the second holds $W^{1, 2} (R^{n})$ -q.e. If u is a weak solution, then by the fact that $f \in L^{2} (Ω ∖ Γ)$ the first equation holds in $L^{2} (Ω ∖ Γ)$ and since $φ \in L^{2} (Γ, μ)$ the second holds in $L^{2} (Γ, μ)$ .

Suppose that u is a weak solution to (19) and that $μ^{'}$ is a finite Borel measure equivalent to μ and satisfying (3) (possibly with a different constant $c_{d}^{'} > 0$ ). Then, trivially, u is also a weak solution to (19) based on $μ^{'}$ in place of μ.

Proposition 4.1.
Let Ω be a $W^{1, 2}$ -extension domain. Let $0 ⩽ d ⩽ n$ , $n - 2 < d$ and suppose that μ is a nonzero finite Borel measure satisfying ( 3 ) whose support $Γ = supp μ$ is a subset of $\overline{Ω}$ . Suppose that $α ⩾ 0$ , $f \in L^{2} (Ω)$ and $φ \in {Tr}_{Γ} (W^{1, 2} (R^{n}))$ .

If $α > 0$ or Ω is bounded, then there is a unique weak solution $u \in W^{1, 2} (Ω)$ to ( 19 ), it satisfies $\begin{matrix} ‖ u ‖_{W^{1, 2} (Ω)} ⩽ c ‖ f ‖_{L^{2} (Ω)} + ‖ φ ‖_{{Tr}_{Γ} (W^{1, 2} (R^{n}))} \end{matrix}$ with $c = max (1, 1 / α)$ in the first case and $c = 1 + C_{P}^{2}$ in the second; here $C_{P}$ is the constant in the Poincaré inequality for Ω.
Proof.
One can proceed by superposition as usual: If $w \in W^{1, 2} (Ω)$ is such that ${Tr}_{Ω, Γ} w = φ$ and $\tilde{u}$ is the unique element of $V (Ω, Γ)$ that satisfies (20) in place of u and with right hand side replaced by the dual pairing $⟨ f + Δ w - α w, v ⟩$ , then $u = \tilde{u} + w$ is as desired. Existence and uniqueness of $\tilde{u}$ follow using the Riesz representation theorem in $V (Ω, Γ)$ , normed by $v \mapsto {(‖ \nabla v ‖_{L^{2} (Ω, R^{n})}^{2} + α ‖ v ‖_{L^{2} (Ω)}^{2})}^{1 / 2}$ ; in the case that $α = 0$ and Ω is bounded, the classical zero trace Poincaré inequality ensures this is a norm. See [28, Section 1.3] or [59, Section 22.2a]. □

To discuss other types of boundary value problems, we introduce generalized normal derivatives by a variant of a standard procedure. We say that $u \in W^{1, 2} (Ω)$ is in $D_{Ω, Γ} (Δ)$ if there is some $f \in L^{2} (Ω)$ such that $\begin{matrix} \int_{Ω} \nabla u \nabla v d x = - \int_{Ω} f v d x, v \in V (Ω, Γ) . \end{matrix}$ For such u we set $Δ u : = f$ . For all $u \in D_{Ω, Γ} (Δ)$ we can define a bounded linear functional $\frac{\partial u}{\partial n_{Γ}} \in {({Tr}_{Γ} (W^{1, 2} (R^{n})))}^{'}$ by the identity $\begin{array}{l} {⟨ \frac{\partial u}{\partial n_{Γ}}, ψ ⟩}_{{({Tr}_{Γ} (W^{1, 2} (R^{n})))}^{'}, {Tr}_{Γ} (W^{1, 2} (R^{n}))} \\ (21) & = \int_{Ω} H_{Γ, Ω} ψ Δ u d x + \int_{Ω} \nabla H_{Γ, Ω} ψ \nabla u d x, ψ \in {Tr}_{Γ} (W^{1, 2} (R^{n})), \end{array}$ here $H_{Γ, Ω}$ is defined as in (11). We refer to $\frac{\partial u}{\partial n_{Γ}}$ as the generalized normal derivative of u on Γ.
Remark 4.2.

The right hand side of equality (21) remains unchanged if the function $H_{Γ, Ω} ψ$ is replaced by an arbitrary element w of $W^{1, 2} (Ω)$ satisfying ${Tr}_{Ω, Γ} w = ψ$ . This is due to the orthogonal decomposition of $W^{1, 2} (Ω)$ into $V (Ω, Γ)$ plus ${H_{Γ, Ω} ψ : ψ \in {Tr}_{Γ} (W^{1, 2} (R^{n}))}$ , which follows from (20), see [27, Section 2.3].

If Ω is bounded and $Γ = \partial Ω$ , then $\frac{\partial u}{\partial n_{Γ}}$ is the generalized normal derivative $\frac{\partial u}{\partial n}$ on $\partial Ω$ as discussed in [31, Theorem 7].

If $μ^{'}$ is another Borel measure satisfying (3) (with a possibly different constant $c_{d}^{'} > 0$ ) and $μ^{'}$ is equivalent to μ, then by Proposition 2.1 the generalized normal derivative $\frac{\partial u}{\partial n_{Γ}}$ of u on Γ based on $μ^{'}$ is the same as the one based on μ.

Now let $α ⩾ 0$ , let $γ \in L^{\infty} (Γ, μ)$ be a Borel function, $f \in L^{2} (Ω)$ and $φ \in {Tr}_{Γ} (W^{1, 2} (R^{n}))$ . We make the generalized Robin problem $\begin{matrix} (22) & \{\begin{matrix} (α - Δ) u = f & in Ω ∖ Γ \\ \frac{\partial u}{\partial n_{Γ}} + γ {Tr}_{Ω, Γ} u = φ & on Γ . \end{matrix} \end{matrix}$ rigorous by saying that $u \in W^{1, 2} (Ω)$ is a weak solution to (22) if u solves $\begin{array}{l} \int_{Ω} \nabla u \nabla v d x + α \int_{Ω} u v d x + \int_{Γ} γ {Tr}_{Ω, Γ} u {Tr}_{Ω, Γ} v d μ \\ (23) & = \int_{Ω} f v d x + \int_{Γ} φ {Tr}_{Ω, Γ} v d μ, v \in W^{1, 2} (Ω) . \end{array}$ Note that the special case $γ \equiv 0$ makes (22) an generalized Neumann problem.
Remark 4.3.
If $α > 0$ or $γ ≢ 0$ , then we can observe the following.
A function $W^{1, 2} (Ω)$ is a weak solution to (22) if and only if it satisfies the first equation in (22) in $D^{'} (Ω ∖ Γ)$ and the second in ${({Tr}_{Γ} (W^{1, 2} (R^{n})))}^{'}$ . If u is a weak solution, then by the fact that $f \in L^{2} (Ω ∖ Γ)$ the first equation holds in $L^{2} (Ω ∖ Γ)$ and since $φ \in L^{2} (Γ, μ)$ the second equation holds in $L^{2} (Γ, μ)$ and in particular, $\frac{\partial u}{\partial n_{Γ}} \in L^{2} (Γ, μ)$ .

Suppose that u is a weak solution to (22) and that $μ^{'}$ is a finite Borel measure equivalent to μ and satisfying (3) (possibly with a different constant $c_{d}^{'} > 0$ ). Then u is also a weak solution to (22) based on $μ^{'}$ in place of μ.

Remark 4.4.
Suppose that $α = 0$ , $γ \equiv 0$ and Ω is bounded. If f and φ satisfy the condition $\begin{matrix} (24) & \int_{Ω} f d x + \int_{Γ} φ d μ = 0, \end{matrix}$ then similar observations as in Remark 4.3(i) remain true. If (24) is not satisfied, one can investigate the variational problem (23) with $W^{1, 2} (Ω)$ replaced by $\begin{matrix} V (Ω) = {u \in W^{1, 2} (Ω) : \int_{Ω} u d x = 0}, \end{matrix}$ which, endowed with the norm $‖ u ‖_{V (Ω)} : = ‖ \nabla u ‖_{L^{2} (Ω, R^{n})}$ , is a Hilbert space. However, solutions of this variational problem may not directly correspond to (22). An analog of Remark 4.3(ii) remains true in either case.
Proposition 4.2.
Let Ω be a $W^{1, 2}$ -extension domain. Let $0 ⩽ d ⩽ n$ , $n - 2 < d$ and suppose that μ is a nonzero finite Borel measure satisfying ( 3 ) whose support $Γ = supp μ$ is a subset of $\overline{Ω}$ . Suppose that $α ⩾ 0$ , $γ \in L^{\infty} (Γ, μ)$ is nonnegative, $f \in L^{2} (Ω)$ and $φ \in L^{2} (Γ, μ)$ .

If $α > 0$ , then there is a unique weak solution u to ( 22 ) and it satifies $\begin{matrix} ‖ u ‖_{W^{1, 2} (Ω)} ⩽ c ‖ f ‖_{L^{2} (Ω)} + c c_{Tr} ‖ φ ‖_{L^{2} (Γ, μ)} \end{matrix}$ with $c = max (1, 1 / α)$ and where $c_{Tr}$ is a norm bound for the trace operator from $W^{1, 2} (R^{n})$ into $L^{2} (Γ, μ)$ as in Corollary 2.2 . If $α = 0$ , Ω is bounded and $0 < γ_{0} : = {essinf}_{Γ} γ$ , then there is a unique weak solution u to ( 22 ), it satisfies an estimate of the same form with $c = max ({(1 + C_{P})}^{2}, λ^{n} (Ω) / γ_{0})$ and with $\sqrt{c} / \sqrt{γ_{0}}$ in place of $c c_{Tr}$ . If $α = 0$ , Ω is bounded, $γ \equiv 0$ and ( 24 ) holds, then there is a weak solution u to ( 22 ), it is unique in $V (Ω)$ and it satisfies $\begin{matrix} ‖ \nabla u ‖_{L^{2} (Ω, R^{n})} ⩽ C_{P} ‖ f ‖_{L^{2} (Ω)} + c_{Tr} (1 + C_{P}) ‖ φ ‖_{L^{2} (Γ, μ)} . \end{matrix}$
Proof.
For $α > 0$ , one can apply the Riesz representation theorem in $W^{1, 2} (Ω)$ directly. For $α \equiv 0$ , Ω bounded and γ bounded away from zero, Corollary 3.2, applied with D being a large enough ball, ensures the coercivity of $u \mapsto \int_{Ω} | \nabla u |^{2} d x + \int_{Γ} γ {({Tr}_{Ω, Γ} u)}^{2} d μ$ . See, for instance, [59, Section 22.2g]. For $α \equiv 0$ , Ω bounded and $γ \equiv 0$ we can again find a large ball D containing Ω, and the Rellich–Kondrachov theorem applied for D implies the same for Ω and therefore a Poincaré inequality for Ω by the standard argument, see [23, Proposition 7.1] or [24, Section 5.8, Theorem 1]. In particular, we have $‖ u ‖_{L^{2} (Ω)} ⩽ C_{P} ‖ \nabla u ‖_{L^{2} (Ω, R^{n})}$ for all u from $V (Ω)$ . Using the Riesz theorem on $V (Ω)$ , we arrive at the desired result, see, for instance, [28, Section 1.4], or [59, Section 22.2f]. □
Remark 4.5.
In the above problems, the respective condition is imposed on the generalized boundary Γ, while $\partial Ω ∖ Γ$ is subject to a homogeneous Neumann condition.
Remark 4.6.
For the Dirichlet problem and the homogeneous Neumann problem, it is clear that no measure μ on Γ is needed; we use it only to ensure that Γ has positive $W^{1, 2} (R^{n})$ -capacity and because we perceive the Neumann problem as a particular case of the Robin problem.
Remark 4.7.
Regularity results for solutions to elliptic problems on domains with fractal boundaries can be found in [41,42,54].

Given $α ⩾ 0$ and nonnegative $γ \in L^{\infty} (Γ, μ)$ we consider the quadratic (energy) form $\begin{matrix} (25) & E_{α, γ} (u) = \int_{Ω} | \nabla u |^{2} d x + α \int_{Ω} u^{2} d x + \int_{Γ} γ {({Tr}_{Ω, Γ} u)}^{2} d μ, u \in W^{1, 2} (Ω), \end{matrix}$ it corresponds to the choice $v = u$ in the left hand side of (20) respectively (23). The following is a special case of a well-known fact; see, for instance, [28, Corollary 1.2] or [59, Section 22.2].
Proposition 4.3.
Let Ω be a $W^{1, 2}$ -extension domain. Let $0 ⩽ d ⩽ n$ , $n - 2 < d$ and suppose that μ is a nonzero finite Borel measure satisfying ( 3 ) whose support $Γ = supp μ$ is a subset $\overline{Ω}$ . Suppose that $α ⩾ 0$ , $γ \in L^{\infty} (Γ, μ)$ is nonnegative, $f \in L^{2} (Ω)$ and $φ \in {Tr}_{Γ} (W^{1, 2} (R^{n}))$ .
If $α > 0$ or Ω is bounded, then $u \in V (Ω, Γ)$ is a weak solution to ( 19 ) with $φ \equiv 0$ if and only if it minimizes the functional $v \mapsto \frac{1}{2} E_{α, 0} (v) - \int_{Ω} f v d x$ on $V (Ω, Γ)$ .

Suppose that γ is bounded away from zero. If $α > 0$ or Ω is bounded, then $u \in W^{1, 2} (Ω)$ is a weak solution to ( 22 ) if and only if it minimizes the functional $v \mapsto \frac{1}{2} E_{α, γ} (v) - \int_{Ω} f v d x - \int_{Γ} φ {Tr}_{Ω, Γ} v d μ$ on $W^{1, 2} (Ω)$ .

Suppose that $γ \equiv 0$ . If $α > 0$ , then $u \in W^{1, 2} (Ω)$ is a weak solution to ( 22 ) if and only if it minimizes the functional $v \mapsto \frac{1}{2} E_{α, 0} (v) - \int_{Ω} f v d x - \int_{Γ} φ {Tr}_{Ω, Γ} v d μ$ on $W^{1, 2} (Ω)$ . If $α = 0$ , Ω is bounded and ( 24 ) holds, then a member u of the space $V (Ω)$ is a weak solution to ( 22 ) if and only if it minimizes the same functional on $V (Ω)$ .

Partially following [44, Section VIII.6], we refer to a closed, densely defined, and nonnegative definite symmetric bilinear form on a Hilbert space as a closed quadratic form. Recall (18).
Corollary 4.1.
Under the respective hypotheses on Ω, μ, α and γ formulated in Proposition 4.3 (i), (ii) and (iii), respectively, the forms $\begin{matrix} (E_{α, 0}, V (Ω, Γ)), (E_{α, γ}, W^{1, 2} (Ω)) and (E_{α, 0}, W^{1, 2} (Ω)) \end{matrix}$ are closed quadratic forms on $L^{2} (Ω)$ .

Let $D \subset R^{n}$ be a domain. Given a subdomain $Ω \subset D$ we may view $L^{2} (Ω)$ as a closed subspace of $L^{2} (D)$ by continuation by zero. Then $\begin{matrix} P_{Ω} f : = 1_{Ω} f \end{matrix}$ is the orthogonal projection from $L^{2} (D)$ onto $L^{2} (Ω)$ . We denote the spaces of all $f \in L^{2} (D)$ such that $P_{Ω} f \in W^{1, 2} (Ω)$ respectively $P_{Ω} f \in V (Ω, Γ)$ by $W^{1, 2} {(Ω)}_{D}$ respectively $V {(Ω, Γ)}_{D}$ . If it is a priori clear that we are talking about an element f of $L^{2} (D)$ , then we also just write $W^{1, 2} (Ω)$ respectively $V (Ω, Γ)$ . This is in line with our general policy to understand the notations $v \in W^{1, 2} (Ω)$ and $E (v)$ as $v |_{Ω} \in W^{1, 2} (Ω)$ and $E (v |_{Ω})$ if $E$ is defined on $W^{1, 2} (Ω)$ and v is defined on a larger set containing Ω.

To point out the dependency of the quadratic forms defined in (25) on the domain $Ω \subset D$ and the measure μ, we also write $\begin{matrix} E_{α, γ}^{Ω, μ} : = E_{α, γ} . \end{matrix}$ It is straightforward to see that Corollary 4.1 implies the following.
Corollary 4.2.
Let $D \subset R^{n}$ be a domain and $Ω \subset D$ a $W^{1, 2}$ -extension domain. Under the respective hypotheses on Ω, μ, α and γ formulated in Proposition 4.3 (i), (ii) and (iii), the forms $\begin{matrix} (E_{α, 0}^{Ω, μ} \circ P_{Ω}, V {(Ω, Γ)}_{D}), (E_{α, γ}^{Ω, μ} \circ P_{Ω}, W^{1, 2} {(Ω)}_{D}) and (E_{α, 0}^{Ω, μ} \circ P_{Ω}, W^{1, 2} {(Ω)}_{D}) \end{matrix}$ are closed quadratic forms on $L^{2} (D)$ .

We absorb the projection operators into the notation and set, under the respective hypotheses stated in Corollary 4.2, $\begin{matrix} (26) & {\hat{E}}_{α, γ}^{Ω, μ} : = E_{α, γ} \circ P_{Ω} . \end{matrix}$

For any $α > 0$ we can define a resolvent operator ${\hat{G}}_{α, 0}^{Ω, μ, D} : L^{2} (D) \to V {(Ω, Γ)}_{D}$ by the identity $\begin{matrix} (27) & {\hat{E}}_{α, 0}^{Ω, μ} ({\hat{G}}_{α, 0}^{Ω, μ, D} f, v) = \int_{D} f v d x, f \in L^{2} (D), v \in V {(Ω, Γ)}_{D}; \end{matrix}$ if Ω is bounded, then also $α = 0$ may be chosen. If γ is bounded away from zero then for any $α > 0$ we can define a resolvent operator ${\hat{G}}_{α, γ}^{Ω, μ, R} : L^{2} (D) \to W^{1, 2} {(Ω)}_{D}$ by the identity $\begin{matrix} (28) & {\hat{E}}_{α, γ}^{Ω, μ} ({\hat{G}}_{α, γ}^{Ω, μ, R} f, v) = \int_{D} f v d x, f \in L^{2} (D), v \in W^{1, 2} {(Ω)}_{D}; \end{matrix}$ again we can permit $α = 0$ if Ω is bounded. If $γ \equiv 0$ , then for any $α > 0$ we can define a resolvent operator ${\hat{G}}_{α, 0}^{Ω, μ, N} : L^{2} (D) \to W^{1, 2} {(Ω)}_{D}$ by the identity $\begin{matrix} (29) & {\hat{E}}_{α, 0}^{Ω, μ} ({\hat{G}}_{α, 0}^{Ω, μ, N} f, v) = \int_{D} f v d x, f \in L^{2} (D), v \in W^{1, 2} {(Ω)}_{D} . \end{matrix}$

In the following, we will also use the shortcut notation ${\hat{G}}_{α, γ}^{Ω, μ, }$ , where ∗ stands for D, R or N, the meaning will be clear from the context.

To clarify the relationship of the operators ${\hat{G}}_{α, γ}^{Ω, μ, }$ and the original problems (19) and (22) on the domain Ω, let $G_{α, γ}^{Ω, μ, }$ denote the resolvent operator with parameter $α > 0$ uniquely associated with $E_{α, γ}^{Ω, μ}$ on $L^{2} (Ω)$ .
Proposition 4.4.
Let* $D \subset R^{n}$ be a domain and $Ω \subset D$ a $W^{1, 2}$ -extension domain. Suppose that $0 ⩽ d ⩽ n$ , $n - 2 < d$ and that μ is a nonzero finite Borel measure satisfying ( 3 ) whose support $Γ = supp μ$ is a subset $\overline{Ω}$ . Let α and γ be as specified in ( 27 ), ( 28 ) or ( 29 ).

Then we have $\begin{matrix} (30) & P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } = G_{α, γ}^{Ω, μ, } \circ P_{Ω} . \end{matrix}$

In particular, if $f \in L^{2} (D)$ , then $\begin{matrix} P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, D} f, P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, R} f and P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, N} f \end{matrix}$ are the continuations by zero of the unique weak solutions for the case of zero boundary data $φ \equiv 0$ to ( 19 ), to ( 22 ) with γ bounded away from zero and to ( 22 ) with $γ \equiv 0$ and $α > 0$ , respectively.
Proof.
Let H be a Hilbert space, $K \subset H$ a closed subspace, and P the orthogonal projection onto K. Suppose that $(Q, D (Q))$ is a closed quadratic form on K and $G_{α}$ the associated α-resolvent operator, defined by the identity $Q_{α} (G_{α} f, g) = {⟨ f, g ⟩}_{K}$ , $f \in K$ , $g \in D (Q)$ . Let $(\hat{Q}, D (\hat{Q}))$ be defined by $D (\hat{Q}) : = {f \in H : P f \in D (Q)}$ and $\hat{Q} : = Q \circ P$ . It is straightforward to see that $(\hat{Q}, D (\hat{Q}))$ is a closed quadratic form on H. Let ${\hat{G}}_{α}$ be its α-resolvent operator, that is, ${\hat{Q}}_{α} ({\hat{G}}_{α} f, g) = {⟨ f, g ⟩}_{H}$ , $f \in H$ , $g \in D (\hat{Q})$ . Then $\begin{array}{rcl} Q_{α} (P ({\hat{G}}_{α} f), P g) & = & {\hat{Q}}_{α} ({\hat{G}}_{α} f, P g) \\ = & {⟨ f, P g ⟩}_{H} = {⟨ P f, P g ⟩}_{H} = {⟨ P f, P g ⟩}_{K} = Q_{α} (G_{α} (P f), P g) \end{array}$ for all $f \in H$ and $g \in D (\hat{Q})$ . This implies that $P ({\hat{G}}_{α} f) = G_{α} (P f)$ in $D (Q)$ for all $f \in H$ and therefore (30). The remaining statements are a consequence. □
Remark 4.8.

Let $(L_{γ}^{Ω, μ, }, D (L_{γ}^{Ω, μ, }))$ be the generator of $E_{0, γ}^{Ω, μ}$ with the respective boundary conditions, that is, the unique non-positive definite self-adjoint operator on $L^{2} (Ω)$ satisfying $\begin{matrix} E_{0, γ}^{Ω, μ} (u, ψ) = - {⟨ L_{γ}^{Ω, μ, } u, ψ ⟩}_{L^{2} (Ω)} \end{matrix}$ for all $u \in D (L_{γ}^{Ω, μ, })$ and all ψ from $V (Ω, Γ)$ (if $* = D$ ), $W^{1, 2} (Ω)$ (if $* = R$ or N). The operator in (30) is the quasi-inverse of $(α - L_{γ}^{Ω, μ, }) \circ P_{Ω}$ in the sense of [56].

Since further projections have no effect, the operator in (30) can conveniently be written as $\begin{matrix} (31) & P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } \circ P_{Ω} = P_{Ω} \circ G_{α, γ}^{Ω, μ, } \circ P_{Ω} . \end{matrix}$

The operator $P_{Ω, 0} f : = 1_{Ω} f - 1_{Ω} \int_{Ω} f d x$ is the orthogonal projection from $L^{2} (Ω)$ (or $L^{2} (D)$ ) onto $L^{2} {(Ω)}_{0} : = {g \in L^{2} (Ω) : \int_{Ω} g d x = 0}$ . The restriction of $L_{0}^{Ω, μ, N}$ to $L^{2} {(Ω)}_{0}$ has a bounded inverse ${\overset{˚}{G}}_{0, 0}^{Ω, μ, N}$ , and $P_{Ω, 0} \circ {\overset{˚}{G}}_{0, 0}^{Ω, μ, N} \circ P_{Ω, 0}$ is a quasi-inverse of $(- L_{0}^{Ω, μ, N}) \circ P_{Ω, 0}$ .

In the case of a bounded confinement D, we can observe the following compactness property of quasi-inverses; the statement is similar to [9, Lemma 4.7]. Lemma 4.1.
Let* $D \subset R^{n}$ be a bounded Lipschitz domain and $Ω \subset D$ a $W^{1, 2}$ -extension domain. Suppose that $0 ⩽ d ⩽ n$ , $n - 2 < d$ and that μ is a nonzero Borel measure satisfying ( 3 ) whose support $Γ = supp μ$ is a subset $\overline{Ω}$ . Suppose further that $γ \equiv 0$ , $α ⩾ 0$ and $* = D$ , or that γ is bounded away from zero, $α ⩾ 0$ and $* = R$ , or that $γ \equiv 0$ , $α > 0$ and $* = N$ .

If ${(g_{m})}_{m = 1}^{\infty} \subset L^{2} (D)$ converges weakly in $L^{2} (D)$ to some g, then there is a sequence ${(m_{k})}_{k = 1}^{\infty}$ with $m_{k} ↑ + \infty$ such that $\begin{matrix} (32) & lim_{k \to \infty} P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } g_{m_{k}} = P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } g in L^{2} (D) . \end{matrix}$
Proof.
Let $\begin{matrix} v_{m} : = P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } g_{m} and v : = P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } g . \end{matrix}$ Then, as observed in Proposition 4.4, $u_{m} = v_{m} |_{Ω_{m}}$ and $u = v |_{Ω}$ are the unique weak solutions to (19) or (22) with zero boundary data on $Ω_{m}$ with $f = g_{m}$ and on Ω with $f = g$ . By the symmetry and boundedness of ${\hat{G}}_{α, γ}^{Ω, μ, }$ we see that ${lim}_{m \to \infty} v_{m} = v$ weakly in $L^{2} (D)$ and weakly in $L^{2} (Ω)$ . On the other hand, we can again use the fact that they may also serve as a test function in (20) respectively (23). This implies that $\begin{matrix} ‖ v_{m} ‖_{W^{1, 2} (Ω)}^{2} ⩽ c E_{α, γ}^{Ω, μ} (v_{m}) = c \int_{Ω} g_{m} v_{m} d x ⩽ c (sup_{m} ‖ g_{m} ‖_{L^{2} (D)}) ‖ v_{m} ‖_{L^{2} (Ω)} \end{matrix}$ with a constant $c > 0$ independent of m and therefore ${sup}_{m} ‖ v_{m} ‖_{W^{1, 2} (Ω)} < + \infty$ . Since Ω is a $W^{1, 2}$ -extension domain, it follows that ${sup}_{m} ‖ E_{Ω} (v_{m} |_{Ω}) ‖_{W^{1, 2} (D)} < + \infty$ , and by Rellich–Kondrachov, applied to D, there is some $v^{} \in L^{2} (Ω)$ such that along a sequence ${(m_{k})}_{k}$ with $m_{k} ↑ + \infty$ we have ${lim}_{k \to \infty} E_{Ω} (v_{m_{k}} |_{Ω}) = v^{}$ in $L^{2} (D)$ . Consequently ${lim}_{k \to \infty} v_{m_{k}} = v^{} = v$ in $L^{2} (Ω)$ , and since v and all $v_{m_{k}}$ vanish outside Ω, the result follows. □
Remark 4.9.
A similar statement could also be shown for the quasi-inverses in Remark 4.8 (iii).

5. Stability under convergence of domains and measures

In this section, we assume that $n ⩾ 2$ , and $D \subset R^{n}$ is a bounded Lipschitz domain. Suppose that $Ω \subset D$ is a $W^{1, 2}$ -extension domain and μ is a nonzero Borel measure satisfying (3) whose support $Γ = supp μ$ is a subset of $\overline{Ω}$ . Let $α ⩾ 0$ and let $γ : \overline{D} \to [0, + \infty)$ be a bounded Borel function. By $E_{α, γ}^{Ω, μ}$ we denote the quadratic form $E_{α, γ}$ defined in (25) for this fixed domain Ω and measure μ. We define an energy functional $J_{α, γ} (Ω, μ)$ on $L^{2} (D)$ by $\begin{matrix} (33) & J_{α, γ} (Ω, μ) (v) = \{\begin{matrix} E_{α, γ}^{Ω, μ} (v), & v \in W^{1, 2} (Ω), \\ + \infty, & v \notin W^{1, 2} (Ω) . \end{matrix} \end{matrix}$ In the case that $γ \equiv 0$ the space $W^{1, 2} (Ω)$ can also be replaced by $V (Ω, Γ)$ .

If we are given sequences of such domains and measures, and these sequences converge in suitable ways, then we can observe the convergence of the associated energy functionals. To formulate the result, we recall the corresponding notions of convergence.

For any closed set $K \subset R^{n}$ and $ε > 0$ we write ${(K)}_{ε} = {x \in R^{n} : d (x, K) ⩽ ε}$ for its closed (outer) ε-parallel set. Recall that the Hausdorff distance between two closed sets $K_{1}, K_{2} \subset R^{n}$ is defined as $\begin{matrix} d^{H} (K_{1}, K_{2}) : = inf {ε > 0 : K_{1} \subset {(K_{2})}_{ε} and K_{2} \subset {(K_{1})}_{ε}} . \end{matrix}$ A sequence ${(K_{m})}_{m}$ of closed sets $K_{m} \subset R^{n}$ is said to converge to a closed set $K \subset R^{n}$ in the Hausdorff sense if ${lim}_{m \to \infty} d^{H} (K_{m}, K) = 0$ . A sequence ${(Ω_{m})}_{m}$ of open sets $Ω_{m} \subset D$ is said to converge to an open set $Ω \subset D$ in the Hausdorff sense if $\begin{matrix} d^{H} (\overline{D} ∖ Ω_{m}, \overline{D} ∖ Ω) \to 0 as m \to \infty, \end{matrix}$ [29, Definition 2.2.8]. This definition does not depend on the choice of D, [29, Remark 2.2.11]. (See [48] for related recent work.)

A sequence ${(Ω_{m})}_{m}$ of open sets $Ω \subset R^{n}$ is said to converge to an open Ω in the sense of characteristic functions if $\begin{matrix} lim_{m \to \infty} 1_{Ω_{m}} = 1_{Ω} in L_{loc}^{p} (R^{n}) for all p \in [1, \infty), \end{matrix}$ [29, Definition 2.2.3], in other words, if locally the Lebesgue measure of the symmetric differences of the domains converges to zero.

A sequence ${(I_{m})}_{m}$ of quadratic functionals $I_{m} : L^{2} (D) \to [0, + \infty]$ converges to a quadratic functional $I : L^{2} (D) \to [0, + \infty]$ in the sense of Mosco, [40, Definition 2.1.1], if

we have ${lim_{_}}_{m \to \infty} I_{m} (u_{m}) ⩾ I (u)$ for every seqence ${(u_{m})}_{m}$ converging weakly to u in $L^{2} (D)$ ,

for every $u \in L^{2} (D)$ there exists a sequence ${(u_{m})}_{m}$ converging strongly in $L^{2} (D)$ such that ${lim^{‾}}_{m \to \infty} I_{m} (u_{m}) ⩽ I (u)$ .

A sequence

{(I_{m})}_{m}

of quadratic functionals

I_{m} : L^{2} (D) \to [0, + \infty]

converges to a quadratic functional

I : L^{2} (D) \to [0, + \infty]

in the Gamma-sense if (ii) above holds and (i) above holds with weak convergence replaced by strong convergence, see [9, Section 2], [40, Definition 2.2.1] or [6,14]. Obviously, convergence in the Mosco sense implies convergence in the Gamma sense.

Theorem 5.1.
Let $D \subset R^{n}$ be a bounded Lipschitz domain, $α ⩾ 0$ , $γ \in C (\overline{D})$ strictly positive or identically zero, and $ε > 0$ , $0 ⩽ d ⩽ n$ , $n - 2 < d$ . Let ${(Ω_{m})}_{m}$ be a sequence of $(ε, \infty)$ -domains $Ω_{m} \subset D$ and ${(μ_{m})}_{m}$ a sequence of nonzero Borel measures $μ_{m}$ , all satisfying ( 3 ) with the same constant $c_{d}$ and with $Γ_{m} = supp μ_{m} \subset {\overline{Ω}}_{m}$ , respectively. For each m, let $J_{α, γ} (Ω_{m}, μ_{m})$ be as in ( 33 ) but with $Ω_{m}$ , $μ_{m}$ in place of Ω, μ.

If ${lim}_{m} Ω_{m} = Ω$ in the Hausdorff sense and in the sense of characteristic functions and ${lim}_{m} μ_{m} = μ$ weakly, then Ω is a $(ε, \infty)$ -domain contained in D, μ is a Borel measure satisfying ( 3 ) and with $Γ = supp μ \subset \overline{Ω}$ , and we have $\begin{matrix} lim_{m} J_{α, γ} (Ω_{m}, μ_{m}) = J_{α, γ} (Ω, μ) \end{matrix}$ in the sense of Mosco and hence in the Gamma-sense. In the special case that $γ > 0$ on $\overline{D}$ and $α ⩾ 0$ , this means that the energy forms associated with the homogeneous (generalized) Robin problem converge in the sense of Mosco and hence in the Gamma-sense.
Remark 5.1.
Measures are not needed to prove the convergence in the homogeneous Dirichlet and Neumann cases (see [19] for weaker sufficient conditions in the homogeneous Dirichlet case). As already pointed out in Remark 4.6, their optional involvement is only due to our concentration on the Robin case.

Theorem 5.1 follows by similar arguments as used for [31, Theorem 6.3], we provide the necessary modifications.
Proof.
That $Ω \subset D$ is an $(ε, \infty)$ -domain follows from [31, Theorem 2.3]. The support Γ of μ is contained in the Hausdorff limit ${lim}_{m} Γ_{m}$ , and by [29, 2.2.3.2 and Theorem 2.2.25] the latter is a subset of $\overline{Ω}$ .

Let ${(u_{m})}_{m} \subset L^{2} (D)$ be a sequence converging to u weakly in $L^{2} (D)$ and ${(u_{m_{k}})}_{k} \subset {(u_{m})}_{m}$ such that ${lim_{_}}_{m} J_{α, γ} (Ω_{m}, μ_{m}) (u_{m}) = {lim}_{k} J_{α, γ} (Ω_{m_{k}}, μ_{m_{k}}) (u_{m_{k}})$ . We will show that $\begin{matrix} (34) & lim_{k} J_{α, γ} (Ω_{m_{k}}, μ_{m_{k}}) (u_{m_{k}}) ⩾ J_{α, γ} (Ω, μ) (u), \end{matrix}$ which then implies condition (i). We may assume the left hand side of (34) is finite, so that $u_{m_{k}} \in W^{1, 2} (Ω_{m_{k}})$ for all k and ${sup}_{k} J_{α, γ} (Ω_{m_{k}}, μ_{m_{k}}) (u_{m_{k}}) < + \infty$ .

Suppose first that $α > 0$ or γ is bounded away from zero. Then the preceding implies that ${sup}_{k} ‖ u_{m_{k}} ‖_{W^{1, 2} (Ω_{m_{k}})} < + \infty$ , in the first case this is immediate, in the second it is due to Corollary 3.2 and the weak convergence of the measures. (Note that since ${lim}_{m} μ_{m} (R^{n}) = μ (R^{n})$ we may assume that the renormalized measures $μ_{m} / μ_{m} (R^{n})$ satisfy (3) with uniform constant $2 c_{d} / μ (R^{n})$ .) By [32, Theorem 1] there exist extensions $E_{Ω_{m_{k}}} u_{m_{k}} \in W^{1, 2} (D)$ and a constant $c_{E} > 0$ such that $‖ E_{Ω_{m_{k}}} u_{m_{k}} ‖_{W^{1, 2} (D)} ⩽ c_{E} ‖ u_{m_{k}} ‖_{W^{1, 2} (Ω_{m_{k}})}$ for all k. Passing to subsequences, we may assume that ${(E_{Ω_{m_{k}}} u_{m_{k}})}_{k}$ converges to some $u^{}$ weakly in $L^{2} (D)$ and that the sequence of its convex combinations strongly converges to $u^{}$ . (Recall that by the Banach–Saks theorem, [45, Section 38], each weakly convergent sequence has a subsequence whose arithmetic means converge strongly.) The hypotheses then imply that $u^{} |_{Ω} = u$ and that ${lim}_{k} 1_{Ω_{m_{k}}} E_{Ω_{m_{k}}} u_{m_{k}} = 1_{Ω} u$ weakly in $L^{2} (D)$ and hence ${lim_{_}}_{k} \int_{Ω_{m_{k}}} u_{m_{k}}^{2} d x ⩾ \int_{Ω} u^{2} d x$ . We may also assume that ${(\nabla E_{Ω_{m_{k}}} u_{m_{k}})}_{k}$ converges to some $w^{}$ weakly in $L^{2} (D, R^{n})$ , with convex combinations converging strongly. Since the convex combinations of ${(E_{Ω_{m_{k}}} u_{m_{k}})}_{k}$ also converge strongly in $W^{1, 2} (D)$ , they necessarily converge to $u^{}$ , which therefore is in $W^{1, 2} (D)$ , and we have $w^{} = \nabla u^{}$ . Consequently ${lim}_{k} 1_{Ω_{m_{k}}} \nabla E_{Ω_{m_{k}}} u_{m_{k}} = 1_{Ω} \nabla u$ weakly in $L^{2} (D, R^{n})$ and therefore $\begin{matrix} (35) & {lim_{_}}_{k} \int_{Ω_{m_{k}}} | \nabla u_{m_{k}} |^{2} d x ⩾ \int_{Ω} | \nabla u |^{2} d x . \end{matrix}$ Combining and using the superadditivity of $lim_{_}$ , we arrive at $\begin{matrix} (36) & {lim_{_}}_{k} {\int_{Ω_{m_{k}}} | \nabla u_{m_{k}} |^{2} d x + α \int_{Ω_{m_{k}}} u_{m_{k}}^{2} d x} ⩾ \int_{Ω} | \nabla u |^{2} d x + α \int_{Ω} u^{2} d x . \end{matrix}$ Using Lemma 2.1 and proceeding as in the proof of [31, Theorem 6.3] we can obtain the limit relation $\begin{matrix} lim_{k} \int_{Γ_{m_{k}}} {({Tr}_{Ω_{m_{k}}, Γ_{m_{k}}} u_{m_{k}})}^{2} d μ_{m_{k}} = \int_{Γ} {({Tr}_{Ω, Γ} u)}^{2} d μ, \end{matrix}$ and together with (36) this shows (34) and therefore (i).

In the case that $α = 0$ and $γ \equiv 0$ we have ${sup}_{k} ‖ \nabla u_{m_{k}} ‖_{L^{2} (Ω_{m_{k}})} < + \infty$ and there are extensions $E_{Ω_{m_{k}}} u_{m_{k}} \in L^{0} (D)$ and a constant $c_{E} > 0$ such that $‖ \nabla E_{Ω_{m_{k}}} u_{m_{k}} ‖_{L^{2} (D)} ⩽ c_{E} ‖ \nabla u_{m_{k}} ‖_{L^{2} (Ω_{m_{k}})}$ for all k, [32, Theorem 2]. By Poincaré’s inequality for D we may assume that the sequence ${((E_{Ω_{m_{k}}} u_{m_{k}} - c_{k}))}_{k}$ , where $c_{k} = λ^{n} {(D)}^{- 1} \int_{D} E_{Ω_{m_{k}}} u_{m_{k}} d x$ , converges to some $u^{}$ weakly in $L^{2} (D)$ , and the sequence of its convex combinations converges strongly. Together with the hypothesis this shows that $1_{Ω} u - 1_{Ω} u^{} = {lim}_{k} 1_{Ω_{m_{k}}} c_{k}$ must equal $c^{} 1_{Ω}$ with a constant $c^{} \in R$ . On the other hand we may also assume that ${(\nabla E_{Ω_{m_{k}}} u_{m_{k}})}_{k}$ converges to some $w^{}$ weakly in $L^{2} (D, R^{n})$ , with its convex combinations converging strongly. Each single function $E_{Ω_{m_{k}}} u_{m_{k}}$ is in $W^{1, 2} (D)$ , and the image of $W^{1, 2} (D)$ under ∇ is a closed subspace of $L^{2} (D, R^{n})$ , see for instance [22, Chapter XIX, Section 1.3, Theorem 4]. Since it is then also weakly closed, we must have $w^{} = \nabla g$ with some $g \in W^{1, 2} (D)$ , and without loss of generality $\int_{Ω} g d x = 0$ . Since the convex combinations of the gradients $\nabla E_{Ω_{m_{k}}} u_{m_{k}}$ converge to $\nabla g$ strongly in $L^{2} (D, R^{n})$ , another application of Poincaré’s inequality shows that $g = u^{}$ . Using the convergence in the sense of characteristic functions, we can again conclude that ${lim}_{k} 1_{Ω_{m_{k}}} \nabla E_{Ω_{m_{k}}} u_{m_{k}} = 1_{Ω} \nabla u$ weakly in $L^{2} (D, R^{n})$ , so that (35) holds, which in the present case shows (34) and therefore (i).

Condition (ii) can be verified as in [31, Theorem 6.3]. □

We record the following simple observation.
Proposition 5.1.
If $D \subset R^{n}$ is a bounded domain, $Ω_{m}, Ω \subset D$ are subdomains and $Ω_{m} \to Ω$ in the sense of characteristic functions, then ${lim}_{m \to \infty} P_{Ω_{m}} = P_{Ω}$ in the strong sense on $L^{2} (D)$ .
Proof.
Given $ψ \in L^{2} (D)$ , we have $‖ P_{Ω_{m}} ψ - P_{Ω} ψ ‖_{L^{2} (D)}^{2} = \int_{D} | 1_{Ω_{m}} - 1_{Ω} |^{2} ψ^{2} d x$ . This goes to zero by Vitali’s convergence theorem, note that the sequence ${(1_{Ω_{m}})}_{m}$ is $L^{2}$ -uniformly integrable w.r.t. the finite measure $ψ^{2} λ^{n}$ , and the convergence of the domains in the sense of characteristic functions implies that $1_{Ω_{m}} \to 1_{Ω}$ in $λ^{n}$ -measure and by absolute continuity then also in $ψ^{2} λ^{n}$ -measure. □

The following preliminary convergence results for the resolvents (and quasi-inverses) is closely related to [56, Theorem 3.1], see also [57, p. 155/156].
Corollary 5.1.
Let n, D, α, γ, ε, d, ${(Ω_{m})}_{m}$ and ${(μ_{m})}_{m}$ be as in Theorem 5.1 . Suppose that ${lim}_{m} Ω_{m} = Ω$ in the Hausdorff sense and in the sense of characteristic functions and ${lim}_{m} μ_{m} = μ$ weakly. For any $α > 0$ we have both ${lim}_{m \to \infty} {\hat{G}}_{α, γ}^{Ω_{m}, μ_{m}, } = {\hat{G}}_{α, γ}^{Ω, μ, }$ and $\begin{matrix} (37) & lim_{m \to \infty} P_{Ω_{m}} \circ {\hat{G}}_{α, γ}^{Ω_{m}, μ_{m}, } = P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, }, \end{matrix}$ in the strong sense on $L^{2} (D)$ . If $γ \equiv 0$ , $α = 0$ and $* = D$ or γ is bounded away from zero, $α = 0$ and $* = R$ , then ( 37 ) still holds.

We write $‖ \cdot ‖$ to denote the operator norm for operators from $L^{2} (D)$ to itself.
Proof.
Suppose $α > 0$ . Then the strong convergence of the resolvents is immediate from Theorem 5.1 and [40, Theorem 2.4.1]. Identity (37) follows using $\begin{array}{l} {‖ P_{Ω_{m}} \circ {\hat{G}}_{α, γ}^{Ω_{m}, μ_{m}, } f - P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } f ‖}_{L^{2} (D)} \\ ⩽ {‖ P_{Ω_{m}} \circ {\hat{G}}_{α, γ}^{Ω_{m}, μ_{m}, } f - P_{Ω_{m}} \circ {\hat{G}}_{α, γ}^{Ω, μ, } f ‖}_{L^{2} (D)} + {‖ (P_{Ω_{m}} - P_{Ω}) \circ {\hat{G}}_{α, γ}^{Ω, μ, } f ‖}_{L^{2} (D)}, \end{array}$ valid for all $f \in L^{2} (D)$ , together with Proposition 5.1. A key ingredient for this argument is the trivial uniform bound on the resolvents, ${sup}_{m} ‖ {\hat{G}}_{α, γ}^{Ω_{m}, μ_{m}, } ‖ ⩽ \frac{1}{α}$ . Under hypotheses stated for $α = 0$ , we still have $\begin{matrix} (38) & sup_{m} ‖ P_{Ω_{m}} \circ {\hat{G}}_{0, γ}^{Ω_{m}, μ_{m}, } ‖ < + \infty, \end{matrix}$ note that by Theorem 3.1 and Corollary 3.2, combined with (27) respectively (28), we have $\begin{matrix} {‖ {\hat{G}}_{0, γ}^{Ω_{m}, μ_{m}, } f ‖}_{L^{2} (Ω_{m})}^{2} ⩽ c E_{0, γ}^{Ω_{m}, μ_{m}, } ({\hat{G}}_{0, γ}^{Ω_{m}, μ_{m}, } f) ⩽ c ‖ f ‖_{L^{2} (D)} {‖ {\hat{G}}_{0, γ}^{Ω_{m}, μ_{m}, } f ‖}_{L^{2} (Ω_{m})}, f \in L^{2} (D), \end{matrix}$ with a constant $c > 0$ independent of m. To discuss the case $α = 0$ under the stated hypotheses, note that Remark 4.8(ii) we have $P_{Ω} \circ {\hat{G}}_{β, γ}^{Ω, μ, } \circ {\hat{G}}_{α, γ}^{Ω, μ, } \circ P_{Ω} = P_{Ω} \circ {\hat{G}}_{β, γ}^{Ω, μ, } \circ P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } \circ P_{Ω}$ , and similarly for $Ω_{m}$ and $μ_{m}$ in place of Ω and μ. Therefore the operators $\begin{matrix} P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } \circ P_{Ω}, α ⩾ 0, and P_{Ω_{m}} \circ {\hat{G}}_{α, γ}^{Ω_{m}, μ_{m}, } \circ P_{Ω_{m}}, α ⩾ 0, \end{matrix}$ are quickly seen to satisfy the resolvent equation. As a consequence, $\begin{array}{l} P_{Ω_{m}} \circ {\hat{G}}_{α, γ}^{Ω_{m}, μ_{m}, } \circ P_{Ω_{m}} - P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } \circ P_{Ω} \\ = (1 + (α - β) P_{Ω_{m}} \circ {\hat{G}}_{α, γ}^{Ω_{m}, μ_{m}, } \circ P_{Ω_{m}}) (P_{Ω_{m}} \circ {\hat{G}}_{β, γ}^{Ω_{m}, μ_{m}, } \circ P_{Ω_{m}} - P_{Ω} \circ {\hat{G}}_{β, γ}^{Ω, μ, } \circ P_{Ω}) \\ (39) & \times (1 + (α - β) P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } \circ P_{Ω}) \end{array}$ for all $α, β ⩾ 0$ , and combining this with the uniform bound (38) we can follow the proof of [36, Theorem VIII.1.3] to see that (37) actually remains valid for $α = 0$ . □
Remark 5.2.

In Theorem 6.1 below, we upgrade this convergence statement to convergence in operator norm. The convergence for the case $α = 0$ under the stated hypotheses could then also be concluded from [36, Chapter IV, Section 2.6, Theorem 2.25].

Using methods similarly as in [43], one could try to bypass Mosco convergence and provide direct proof of a generalized form of norm resolvent convergence.

We formulate an assumption to simplify the statement of further results on the convergence of resolvents.
Assumption 5.1.
The parameters α, γ, and ∗ satisfy one of the following:
$γ \equiv 0$ , $α ⩾ 0$ and $ = D$ , or

γ is bounded away from zero, $α ⩾ 0$ and $* = R$ , or

$γ \equiv 0$ , $α > 0$ and $* = N$ .

The following theorem may be seen as a ‘joint’ upgrade of Lemma 4.1 and Corollary 5.1. The first statement is similar to [9, Lemma 4.8]. The second part relies on Corollary 3.2. Theorem 5.2.
Let n, D, α, γ, ε, d, ${(Ω_{m})}_{m}$ and ${(μ_{m})}_{m}$ be as in Theorem 5.1 and let Assumption 5.1 be satisfied. Suppose that ${lim}_{m} Ω_{m} = Ω$ in the Hausdorff sense and in the sense of characteristic functions and ${lim}_{m} μ_{m} = μ$ weakly. Let ${(g_{m})}_{m = 1}^{\infty} \subset L^{2} (D)$ be a sequence that converges weakly in $L^{2} (D)$ to some g. Then there is a sequence ${(m_{k})}_{k = 1}^{\infty}$ with $m_{k} ↑ + \infty$ such that the following hold:
We have $\begin{matrix} lim_{k \to \infty} P_{Ω_{m_{k}}} \circ {\hat{G}}_{α, γ}^{Ω_{m_{k}}, μ_{m_{k}}, } g_{m_{k}} = P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } g in L^{2} (D) . \end{matrix}$

If $u_{m}$ and u be the unique weak solutions to ( 19 ) or ( 22 ) with zero boundary data on $Ω_{m}$ with $f = g_{m}$ and on Ω with $f = g$ , then there is some $u^{} \in W^{1, 2} (D)$ with* $u^{} |_{Ω} = u$ such that* $\begin{matrix} (40) & lim_{k \to \infty} E_{Ω_{m_{k}}} u_{m_{k}} = u^{} weakly in W^{1, 2} (D) . \end{matrix}$ Moreover, we have* $\begin{matrix} (41) & lim_{k \to \infty} J_{α, γ} (Ω_{m_{k}}, μ_{m_{k}}) (u_{m_{k}}) = J_{α, γ} (Ω, μ) (u) \end{matrix}$ and $\begin{matrix} (42) & lim_{k \to \infty} E_{Ω_{m_{k}}} u_{m_{k}} = u in W^{1, 2} (Ω) . \end{matrix}$

Remark 5.3.
Corollary 5.1 and Theorem 5.2 (applied with $g = g_{m} = f$ ) constitute a stability result for the respective boundary value problem along a sequence of varying domains in bounded confinement.
Proof.
Let $v_{m} : = P_{Ω_{m}} \circ {\hat{G}}_{α, γ}^{Ω_{m}, μ_{m}, } g_{m}$ and $v : = P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } g$ and recall $v_{m}$ and v the continuations by zero of the unique weak solutions $u_{m}$ and u to (19) or (22) as stated. We have $\begin{matrix} (43) & lim_{m \to \infty} v_{m} = v weakly in L^{2} (D), \end{matrix}$ note that $\begin{array}{rcl} lim_{m \to \infty} {⟨ v_{m}, w ⟩}_{L^{2} (D)} & = & lim_{m \to \infty} {⟨ g_{m}, P_{Ω_{m}} \circ {\hat{G}}_{α, γ}^{Ω_{m}, μ_{m}, } P_{Ω_{m}} w ⟩}_{L^{2} (D)} \\ (44) & = & {⟨ g, P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } P_{Ω} w ⟩}_{L^{2} (D)} = {⟨ v, w ⟩}_{L^{2} (D)} \end{array}$ for any $w \in L^{2} (D)$ by Remark 4.8(ii) and Corollary 5.1.

Testing the unique weak solution of the respective boundary value problem (Dirichlet, Robin or Neumann) against itself, we observe that $\begin{matrix} ‖ v_{m} ‖_{W^{1, 2} (Ω_{m})}^{2} ⩽ c E_{α, γ}^{Ω_{m}, μ_{m}} (v_{m}) = c \int_{Ω_{m}} g_{m} v_{m} d x ⩽ c (sup_{m} ‖ g_{m} ‖_{L^{2} (D)}) ‖ v_{m} ‖_{L^{2} (Ω_{m})} \end{matrix}$ with a constant $c > 0$ independent of m. The independence of c of m is obvious if $α > 0$ , it follows from Corollary 3.2 if $α = 0$ and $* = D$ and or $* = R$ . (By weak convergence, we may assume that the probability measures $μ_{m} / μ_{m} (R^{n})$ satisfy (3) with uniform constant $2 c_{d} / μ (R^{n})$ .) Consequently $\begin{matrix} sup_{m} ‖ v_{m} ‖_{W^{1, 2} (Ω_{m})} < + \infty . \end{matrix}$

Since all $Ω_{m}$ are $(ε, \infty)$ -domains with the same ε, we can use [32, Theorems 1] to see that there are extensions $E_{Ω_{m}} (v_{m} |_{Ω_{m}}) \in W^{1, 2} (D)$ of $u_{m} = v_{m} |_{Ω_{m}}$ such that $\begin{matrix} (45) & sup_{m} {‖ E_{Ω_{m}} (v_{m} |_{Ω_{m}}) ‖}_{W^{1, 2} (D)} < + \infty . \end{matrix}$

As a consequence there are a sequence ${(m_{k})}_{k}$ with $m_{k} ↑ + \infty$ and a function $u^{} \in W^{1, 2} (D)$ such that (40) holds, and by the Banach–Saks theorem, [45, Section 38], we may assume that the convex combinations of the functions $E_{Ω_{m_{k}}} u_{m_{k}}$ converge to $u^{}$ in $W^{1, 2} (D)$ . By Rellich–Kondrachov we may assume that also ${lim}_{k \to \infty} E_{Ω_{m_{k}}} u_{m_{k}} = w$ in $L^{2} (D)$ with some suitable $w \in L^{2} (D)$ , and by the preceding we then must have $w = u^{}$ . Using Proposition 5.1 it follows that $\begin{matrix} lim_{k \to \infty} v_{m_{k}} = lim_{k \to \infty} P_{Ω_{m_{k}}} v_{m_{k}} = lim_{k \to \infty} P_{Ω_{m_{k}}} E_{Ω_{m_{k}}} u_{m_{k}} = P_{Ω} u^{} \end{matrix}$ in $L^{2} (D)$ , so that $P_{Ω} u^{} = v$ by (43). This proves (i) and (40).

To prove (41), we can now proceed as in [13, Theorem 5.3]. By (i) we have $\begin{matrix} (46) & lim_{k \to \infty} \int_{Ω_{m_{k}}} u_{m_{k}} g_{m_{k}} d x = \int_{Ω} u g d x . \end{matrix}$ The first condition of the convergence in the sense of Mosco implies that $\begin{matrix} (47) & \frac{1}{2} J_{α, γ} (Ω, μ) (u) - \int_{Ω} u g d x ⩽ {lim_{_}}_{k \to \infty} {\frac{1}{2} J_{α, γ} (Ω_{m_{k}}, μ_{m_{k}}) (u_{m_{k}}) - \int_{Ω_{m_{k}}} u_{m_{k}} g_{m_{k}} d x} . \end{matrix}$ By the second condition there is a sequence ${(w_{k})}_{k} \subset L^{2} (D)$ such that ${lim}_{k \to \infty} w_{k} = u^{}$ in $L^{2} (D)$ and $\begin{matrix} (48) & {lim^{‾}}_{k \to \infty} {\frac{1}{2} J_{α, γ} (Ω_{m_{k}}, μ_{m_{k}}) (w_{k}) - \int_{Ω_{m_{k}}} w_{k} g_{m_{k}} d x} ⩽ \frac{1}{2} J_{α, γ} (Ω, μ) (u) - \int_{Ω} u g d x . \end{matrix}$ Since by Proposition 4.3 the function $u_{m_{k}}$ minimizes $w \mapsto \frac{1}{2} J_{α, γ} (Ω_{m_{k}}, μ_{m_{k}}) (w) - \int_{Ω_{m_{k}}} w g_{m_{k}} d x$ , it follows that (48) dominates $\begin{matrix} {lim^{‾}}_{k \to \infty} {\frac{1}{2} J_{α, γ} (Ω_{m_{k}}, μ_{m_{k}}) (u_{m_{k}}) - \int_{Ω_{m_{k}}} u_{m_{k}} g_{m_{k}} d x} . \end{matrix}$ Combining with (47) and taking into account (46), we obtain (41).

To see (42), note that by (i) we have ${lim}_{k \to \infty} \int_{Ω_{m_{k}}} u_{m_{k}}^{2} d x = \int_{Ω} u^{2} d x$ , and since ${lim}_{m \to \infty} γ \cdot μ_{m} = γ \cdot μ$ weakly, Lemma 2.1 and (40) imply that $\begin{matrix} lim_{k \to \infty} \int_{Γ_{m_{k}}} γ {({Tr}_{Ω_{m_{k}}, Γ_{m_{k}}} u_{m_{k}})}^{2} d μ_{m_{k}} = \int_{Γ} γ {({Tr}_{Ω, Γ} u)}^{2} d μ . \end{matrix}$ Together with (41) it follows that $\begin{matrix} (49) & lim_{k \to \infty} ‖ u_{m_{k}} ‖_{W^{1, 2} (Ω_{m_{k}})} = ‖ u ‖_{W^{1, 2} (Ω)} . \end{matrix}$ By the above and by the Helmholtz decomposition on D, [22, Chapter XIX, Section 1.3, Theorem 4], the sequence ${(\nabla E_{Ω_{m_{k}}} u_{m_{k}})}_{k}$ converges to $\nabla u^{*}$ weakly in $L^{2} (D, R^{n})$ , and by the convergence in the sense of characteristic functions we obtain $\begin{matrix} lim_{k \to \infty} \int_{D} (1_{Ω_{m_{k}}} - 1_{Ω}) {(\nabla E_{Ω_{m_{k}}} u_{m_{k}})}^{2} d x = 0 . \end{matrix}$ Similarly, we have $\begin{matrix} lim_{k \to \infty} \int_{D} (1_{Ω_{m_{k}}} - 1_{Ω}) {(E_{Ω_{m_{k}}} u_{m_{k}})}^{2} d x = 0, \end{matrix}$ and combining with (49), it follows that $\begin{matrix} lim_{k \to \infty} ‖ E_{Ω_{m_{k}}} u_{m_{k}} ‖_{W^{1, 2} (Ω)} = ‖ u ‖_{W^{1, 2} (Ω)} . \end{matrix}$ Since by (45) we may assume that ${lim}_{k \to \infty} E_{Ω_{m_{k}}} u_{m_{k}} = u$ weakly in $W^{1, 2} (Ω)$ , we arrive at (42). □

6. Norm resolvent convergence and eigenvalues

As before, let $D \subset R^{n}$ , $n ⩾ 2$ , and suppose that $Ω \subset D$ is a $W^{1, 2}$ -extension domain. Let $γ : \overline{D} \to [0, + \infty)$ be a bounded Borel function. Assume that $γ \equiv 0$ and $* = D$ , that γ is bounded away from zero and $* = R$ , or that $γ \equiv 0$ and $* = N$ . Recall that $L_{γ}^{Ω, μ, *}$ denotes the generator of $E_{0, γ}^{Ω, μ}$ , Remark 4.8(i). The eigenvalues $\begin{matrix} 0 ⩽ λ_{1} (Ω, μ, *) ⩽ λ_{2} (Ω, μ, *) ⩽ \cdot \end{matrix}$ of $- L_{γ}^{Ω, μ, *}$ have finite multiplicity and accumulate only at infinity, and the eigenfunctions $φ_{n} (Ω, μ, *)$ corresponding to the eigenvalues $λ_{n} (Ω, μ, *)$ , respectively, form a complete orthonormal system ${(φ_{n} (Ω, μ, *))}_{n}$ of $L^{2} (Ω)$ . Given $a < b$ , let $π_{(a, b)} (Ω, μ, *)$ denote the spectral projector associated with the interval $(a, b)$ .

We denote the continuation of $φ_{n} (Ω, μ, *)$ by zero to D by the same symbol. Let ${(χ_{n} (D ∖ Ω, μ, *))}_{n}$ be a complete orthonormal system in $L^{2} (D ∖ Ω)$ and denote the extensions by zero of its elements again by the same symbols. Then ${(φ_{n} (Ω, μ, *))}_{n} \cup {(χ_{n} (D ∖ Ω, μ, *))}_{n}$ is a complete orthonormal system in $L^{2} (D)$ , the $λ_{n} (Ω, μ, *)$ are also eigenvalues of ${\hat{E}}_{0, γ}^{Ω, μ}$ on $L^{2} (D)$ , in the sense that we have ${\hat{E}}_{0, γ}^{Ω, μ} (φ_{n} (Ω, μ, *), ψ) = λ_{n} (Ω, μ, *) \int_{D} φ_{n} (Ω, μ, *) ψ d x$ for all ψ from $V {(Ω, Γ)}_{D}$ respectively $W^{1, 2} {(Ω)}_{D}$ , and all $χ_{n} (D ∖ Ω, μ, *)$ are eigenfunctions corresponding to the eigenvalue zero.

Suppose we are in the situation of Theorem 5.1. Theorem 5.2 allows passing from the strong convergence of the resolvents observed in Corollary 5.1 and Theorem 5.2 to norm convergence. We can then conclude convergence properties for spectral projectors, eigenvalues, and eigenfunctions.

Theorem 6.1.
Let n, D, α, γ, ε, d, ${(Ω_{m})}_{m}$ and ${(μ_{m})}_{m}$ be as in Theorem 5.1 and let Assumption 5.1 be satisfied. Suppose that ${lim}_{m} Ω_{m} = Ω$ in the Hausdorff sense and in the sense of characteristic functions and ${lim}_{m} μ_{m} = μ$ weakly. There is a sequence ${(m_{k})}_{k = 1}^{\infty}$ with $m_{k} ↑ + \infty$ such that the following hold.
We have ${lim}_{k \to \infty} P_{Ω_{m_{k}}} \circ {\hat{G}}_{α, γ}^{Ω_{m_{k}}, μ_{m_{k}}, } = P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, }$ in operator norm.

If $0 < a < b$ are in the resolvent set of $- L_{γ}^{Ω, μ, }$ , then* ${lim}_{k \to \infty} π_{(a, b)} (Ω_{m_{k}}, μ_{m_{k}}, ) = π_{(a, b)} (Ω, μ, )$ in operator norm.

The spectra of the operators $- L_{γ}^{Ω_{m_{k}}, μ_{m_{k}}, }$ converge to the spectrum of* $- L_{γ}^{Ω, μ, }$ in the Hausdorff sense. The eigenvalues* $λ_{n} (Ω, μ, )$ of the operator* $- L_{γ}^{Ω, μ, }$ are exactly the limits as* $k \to \infty$ of sequences of the eigenvalues of the operators $- L_{γ}^{Ω_{m_{k}}, μ_{m_{k}}, }$ , $\begin{matrix} (50) & λ_{n} (Ω, μ, ) = lim_{k \to \infty} λ_{n} (Ω_{m_{k}}, μ_{m_{k}}, ) . \end{matrix}$

For each n we can find a sequence of normalized eigenfunctions* $φ_{n}^{(m_{k})}$ of the operators $- L_{γ}^{Ω_{m_{k}}, μ_{m_{k}}, }$ that converges to* $φ_{n} (Ω, μ, )$ in $L^{2} (D)$ . If all eigenvalues* $λ_{n} (Ω_{m_{k}}, μ_{m_{k}}, )$ are simple, then the* $φ_{n}^{(m_{k})}$ equal $φ_{n} (Ω_{m_{k}}, μ_{m_{k}}, )$ , $k ⩾ 1$ , up to signs.*

For any n the sequence ${(φ_{n} (Ω_{m_{k}}, μ_{m_{k}}, ))}_{k}$ has a subsequence converging in* $L^{2} (D)$ to a normalized eigenfunction $φ_{n}$ of $- L_{γ}^{Ω, μ, }$ corresponding to* $λ_{n} (Ω, μ, )$ . If* $λ_{n} (Ω, μ, )$ is a simple eigenvalue, then the full sequence converges to* $φ_{n} = \pm φ_{n} (Ω, μ, )$ .

Proof.
To see (i), we can follow the proof of [9, Theorem 3.2]. It suffices to verify that along a sequence ${(m_{k})}_{k}$ as stated, we have $\begin{matrix} (51) & lim_{k \to \infty} sup_{g \in L^{2} (D), ‖ g ‖_{L^{2} (D)} ⩽ 1} {‖ P_{Ω_{m_{k}}} \circ {\hat{G}}_{α, γ}^{Ω_{m_{k}}, μ_{m_{k}}, } g - P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } g ‖}_{L^{2} (D)} = 0 . \end{matrix}$ For each m we can, by Lemma 4.1, find a function $g_{m} \in L^{2} (D)$ with $‖ g_{m} ‖_{L^{2} (D)} ⩽ 1$ such that $\begin{array}{l} {‖ P_{Ω_{m}} \circ {\hat{G}}_{α, γ}^{Ω_{m}, μ_{m}, } g_{m} - P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } g_{m} ‖}_{L^{2} (D)} \\ (52) & = sup_{g \in L^{2} (D), ‖ g ‖_{L^{2} (D)} ⩽ 1} {‖ P_{Ω_{m}} \circ {\hat{G}}_{α, γ}^{Ω_{m}, μ_{m}, } g - P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } g ‖}_{L^{2} (D)} . \end{array}$ Since ${(g_{m})}_{m} \subset L^{2} (D)$ is bounded, we can find a sequence ${(m_{k})}_{k}$ with $m_{k} ↑ + \infty$ such that ${lim}_{k \to \infty} g_{m_{k}} = g$ weakly in $L^{2} (D)$ . Another application of Lemma 4.1 gives $\begin{matrix} lim_{k \to \infty} {‖ P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } g_{m_{k}} - P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } g ‖}_{L^{2} (D)} = 0, \end{matrix}$ and from Theorem 5.2 we obtain $\begin{matrix} lim_{k \to \infty} {‖ P_{Ω_{m_{k}}} \circ {\hat{G}}_{α, γ}^{Ω_{m_{k}}, μ_{m_{k}}, } g_{m_{k}} - P_{Ω} \circ {\hat{G}}_{α, γ}^{Ω, μ, } g ‖}_{L^{2} (D)} = 0 . \end{matrix}$ Combining, using the triangle inequality and taking into account (52), we arrive at (51).

The projection $P_{Ω}$ does not affect the complement of the eigenspace of $- L_{γ}^{Ω, μ, }$ for the eigenvalue zero. Therefore statement (ii) follows using well-known arguments, [44, Theorem VIII.23]. See also [57, p. 154 and the related Theorem 2].

The first part in (iii) follows from [36, Chapter IV, Section 3.1, Theorem 3.1 and Remark 3.3], applied to the resolvents, together with [36, Chapter VIII, Section 1.2, Theorem 1.14].

Next, note that the first eigenvalue $λ_{1} (Ω, μ, )$ of $- L_{γ}^{Ω, μ, }$ is zero if and only $* = N$ and $γ \equiv 0$ . In this case also $λ_{1} (Ω_{m_{k}}, μ_{m_{k}}, ) = 0$ for all k. The corresponding normalized eigenfunctions are $φ_{1} (Ω, μ, N) = {(λ^{n} (Ω))}^{- 1} 1_{Ω}$ and $φ_{1} (Ω_{m_{k}}, μ_{m_{k}}, N) = {(λ^{n} (Ω_{m_{k}}))}^{- 1} 1_{Ω_{m_{k}}}$ , and by the convergence of the domains in the sense of characteristic functions (and the bounded convergence theorem) we have ${lim}_{k \to \infty} φ_{1} (Ω_{m_{k}}, μ_{m_{k}}, N) = φ_{1} (Ω, μ, N)$ in $L^{2} (D)$ .

Now consider the smallest nonzero eigenvalue $λ_{+} (Ω, μ, )$ and let $(a, b)$ , with $0 < a < b$ from the resolvent set of $L_{γ}^{Ω, μ, }$ , be a small interval around it and containing no other eigenvalue of $- L_{γ}^{Ω, μ, }$ . Then for large enough k the number a is in the resolvent set of $- L_{γ}^{Ω_{m_{k}}, μ_{m_{k}}, }$ , the interval $(a, b)$ also contains its smallest nonzero eigenvalue $λ_{+} (Ω_{m_{k}}, μ_{m_{k}}, )$ , but no other of its eigenvalues; moreover, $λ_{+} (Ω_{m_{k}}, μ_{m_{k}}, )$ and $λ_{+} (Ω, μ, )$ have the same multiplicity, [36, Chapter IV, Section 3.4, Theorem 3.16, and Section 3.5]. We have $\begin{matrix} | λ_{+} (Ω_{m_{k}}, μ_{m_{k}}, ) - λ_{+} (Ω, μ, ) | ⩽ {(b - a)}^{2} | {(λ_{+} (Ω_{m_{k}}, μ_{m_{k}}, ) - a)}^{- 1} - {(λ_{+} (Ω, μ, ) - a)}^{- 1} | . \end{matrix}$ By the preceding $π_{(a, b)} (Ω, μ, ) = P_{Ω} \circ π_{(a, b)} (Ω, μ, )$ is the orthogonal projection onto the eigenspace for $λ_{+} (Ω, μ, )$ , we therefore have $‖ {\hat{G}}_{a, γ}^{Ω, μ, } \circ π_{(a, b)} (Ω, μ, ) ‖ = 1 / (λ_{+} (Ω, μ, ) - a)$ , and similarly for $Ω_{m_{k}}$ and $μ_{m_{k}}$ in place of Ω and μ. Using the reverse triangle inequality, we therefore have $\begin{array}{l} | {(λ_{+} (Ω_{m_{k}}, μ_{m_{k}}, ) - a)}^{- 1} - {(λ_{+} (Ω, μ, ) - a)}^{- 1} | \\ (53) & ⩽ ‖ {\hat{G}}_{a, γ}^{Ω_{m_{k}}, μ_{m_{k}}, } \circ π_{(a, b)} (Ω_{m_{k}}, μ_{m_{k}}, ) - {\hat{G}}_{a, γ}^{Ω, μ, } \circ π_{(a, b)} (Ω, μ, ) ‖, \end{array}$ and by the triangle inequality which is bounded by $\begin{matrix} ‖ {\hat{G}}_{a, γ}^{Ω_{m_{k}}, μ_{m_{k}}, } \circ P_{Ω_{m_{k}}} - {\hat{G}}_{a, γ}^{Ω, μ, } \circ P_{Ω} ‖ + a^{- 1} ‖ π_{(a, b)} (Ω_{m_{k}}, μ_{m_{k}}, ) - π_{(a, b)} (Ω, μ, ) ‖ . \end{matrix}$ Since by (i) and (ii) this converges to zero as $k \to \infty$ , we obtain (50) for $λ_{+} (Ω, μ, )$ . We can now move on to the next eigenvalue of $- L_{γ}^{Ω, μ, }$ strictly larger than $λ_{+} (Ω, μ, )$ and apply the same arguments. Proceeding inductively, we obtain (50) for all nonzero eigenvalues.

To see (iv) for eigenfunctions corresponding to a nonzero eigenvalue $λ_{n} (Ω, μ, )$ , let $(a, b)$ be a small interval as above containing this but no other eigenvalue of $- L_{γ}^{Ω, μ, }$ . Statement (ii) implies that for any corresponding normalized eigenfunction $φ_{n} (Ω, μ, )$ , we have $\begin{matrix} lim_{k \to \infty} π_{(a, b)} (Ω_{m_{k}}, μ_{m_{k}}, ) φ_{n} (Ω, μ, ) = π_{(a, b)} (Ω, μ, ) φ_{n} (Ω, μ, ) = φ_{n} (Ω, μ, ) in L^{2} (D) \end{matrix}$ and in particular, $π_{(a, b)} (Ω_{m_{k}}, μ_{m_{k}}, ) φ_{n} (Ω, μ, ) \neq 0$ for large enough k. The functions $\begin{matrix} φ_{n}^{(m_{k})} : = π_{(a, b)} (Ω_{m_{k}}, μ_{m_{k}}, ) φ_{n} (Ω, μ, ) {‖ π_{(a, b)} (Ω_{m_{k}}, μ_{m_{k}}, ) φ_{n} (Ω, μ, ) ‖}_{L^{2} (D)}^{- 1} \end{matrix}$ converge to $φ_{n} (Ω, μ, )$ in $L^{2} (D)$ as $k \to \infty$ . Since, again by [36, Chapter IV, Section 3.4, Theorem 3.16 and Section 3.5], the images of the $π_{(a, b)} (Ω_{m_{k}}, μ_{m_{k}}, )$ are eigenspaces of the $- L_{γ}^{Ω_{m_{k}}, μ_{m_{k}}, }$ for a single eigenvalue $λ_{n} (Ω_{m_{k}}, μ_{m_{k}}, )$ , respectively, the functions $φ_{n}^{(m_{k})}$ are normalized eigenfunctions for these eigenvalues. If all $λ_{n} (Ω_{m_{k}}, μ_{m_{k}}, )$ are simple, then $φ_{n}^{(m_{k})} = \pm φ_{n} (Ω_{m_{k}}, μ_{m_{k}}, )$ by normalization. This shows the first part in (iv). To see the second, let ${(φ_{n}^{(m_{k})})}_{k}$ now denote a sequence of normalized eigenfunctions $φ_{n}^{(m_{k})}$ corresponding to $λ_{n} (Ω_{m_{k}}, μ_{m_{k}}, )$ , respectively. By (ii), the triangle inequality and the reverse triangle inequality, we have ${lim}_{k \to \infty} ‖ π_{(a, b)} (Ω, μ, ) φ_{n}^{(m_{k})} ‖ = 1$ , so that for large enough k, we can define $\begin{matrix} φ_{n, k} : = π_{(a, b)} (Ω, μ, ) φ_{n}^{(m_{k})} {‖ π_{(a, b)} (Ω, μ, ) φ_{n}^{(m_{k})} ‖}_{L^{2} (D)}^{- 1} . \end{matrix}$ Again using (ii), we can conclude that $\begin{array}{l} lim_{k \to \infty} {‖ φ_{n, k} - φ_{n}^{(m_{k})} ‖}_{L^{2} (D)} ⩽ & lim_{k \to \infty} | {‖ π_{(a, b)} (Ω, μ, ) φ_{n}^{(m_{k})} ‖}_{L^{2} (D)}^{- 1} - 1 | {‖ π_{(a, b)} (Ω, μ, ) φ_{n}^{(m_{k})} ‖}_{L^{2} (D)} \\ + lim_{k \to \infty} {‖ π_{(a, b)} (Ω, μ, ) φ_{n}^{(m_{k})} - π_{(a, b)} (Ω_{m_{k}}, μ_{m_{k}}, ) φ_{n}^{(m_{k})} ‖}_{L^{2} (D)} = 0 . \end{array}$ Since the eigenspace of $λ_{n} (Ω, μ, )$ is finite-dimensional, the sequence ${(φ_{n, k})}_{k}$ has a subsequence converging in $L^{2} (D)$ to a limit $φ_{n}$ which is a normalized eigenfunction of $L_{γ}^{Ω, μ, }$ corresponding to $λ_{n} (Ω, μ, )$ . If the eigenspace is one-dimensional, this limit $φ_{n}$ must be $\pm φ_{n} (Ω, μ, *)$ . □

7. Shape admissible domains and compactness

We define classes of admissible domains and prove their compactness. As before, we assume $n ⩾ 2$ . The following definition is a generalization of [31, Definition 2].

Definition 7.1.
Let $D_{0} \subset D \subset R^{n}$ be non-empty bounded Lipschitz domains. A triple $(Ω, ν, μ)$ is called shape admissible with parameters D, $D_{0}$ , $ε > 0$ , $n - 1 ⩽ s < n$ , $\bar{c} ⩾ {\bar{c}}_{s} > 0$ , $0 ⩽ d ⩽ n$ and $c_{d} > 0$ , if
Ω is an $(ε, \infty)$ -domain such that $D_{0} \subset Ω \subset D$ ,

ν is a finite Borel measure ν with $supp ν = \partial Ω$ satisfying the (weak, local) lower regularity condition $\begin{matrix} (54) & ν (\overline{B (x, r)}) ⩾ {\bar{c}}_{s} r^{s}, x \in \partial Ω, 0 < r ⩽ 1, \end{matrix}$ and the total mass bound $\begin{matrix} (55) & ν (\partial Ω) ⩽ \bar{c} . \end{matrix}$

μ is a nonzero Borel measure satisfying (3) and $Γ = supp μ \subset \overline{Ω}$ .
The set of such triples is denoted by $U_{ad} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ . We refer to the measures ν as in (ii) as boundary volumes and the measures μ in (iii) as trace volumes.

Condition (54) implies that the Hausdorff dimension of $\partial Ω$ is less or equal s, see, for instance, [25,38].

A sequence $(Ω_{m})$ of open sets $Ω_{m} \subset R^{n}$ is said to converge to an open set $Ω \subset R^{n}$ in the sense of compacts if for any compact $K \subset Ω$ we have $K \subset Ω_{m}$ for all sufficiently large m and for any compact $K \subset R^{n} ∖ \overline{Ω}$ we have $K \subset R^{n} ∖ {\overline{Ω}}_{m}$ for all sufficiently large m. See [29, Definition 2.2.21].

We state a corresponding compactness result which had basically been shown in [31, Theorems 3 and Remark 6].
Theorem 7.1.
Suppose that the parameters D, $D_{0}$ , ε, s, ${\bar{c}}_{s}$ , $\bar{c}$ , d, $c_{d}$ are as in Definition 7.1 .
The class $U_{ad} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ of shape admissible triples is compact in the Hausdorff sense, in the sense of characteristic functions, in the sense of compacts, and in the sense of weak convergence of the boundary volumes and the trace volumes.

If for a sequence ${((Ω_{m}, ν_{m}, μ_{m}))}_{m} \subset U_{ad} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ the boundary volumes $ν_{m}$ converge weakly, then the domains $Ω_{m}$ converge in the Hausdorff sense, in the sense of characteristic functions, and in the sense of compacts.

[31, Theorem 3] is recovered as a special case of Theorem 7.1 in the situation that $μ_{m} = ν_{m}$ for all m. In this case, the uniform bound (55) on the total boundary volumes can be dropped because it follows from (3). Note also that (54) implies that the measures are nonzero.
Proof.
Statement (ii) had been proved in [31, Theorem 3(ii)], statement (i) follows from [31, Theorem 3(i)] and the Banach–Alaoglu theorem, applied to a (sub-)sequence of trace volumes. □
Remark 7.1.
The weak convergence of boundary volumes serves as a convenient tool: The common scaling exponent s and constant ${\bar{c}}_{s}$ guarantee that also the limit domain Ω will have a boundary $\partial Ω$ with Hausdorff dimension less or equal $s < n$ so that in particular $λ^{n} (\partial Ω) = 0$ . This is used to conclude the convergence in the sense of characteristic functions; see [31, Theorems 2 and 3]. The weak convergence of measures alone does not preserve simple bounds on the Hausdorff dimension of their supports. A bound similar to (54) was also used in [9, formula (1.6)].
Remark 7.2.
One might also want to keep the volume of the domains constant. This condition could be added without affecting the compactness result or the subsequent results below.

Combining Theorem 5.1 with Theorem 7.1, we immediately obtain a compactness result for the energy functionals defined in (33).
Theorem 7.2.
Let $D_{0} \subset D \subset R^{n}$ be non-empty bounded Lipschitz domains, $ε > 0$ , $n - 1 ⩽ s < n$ , $\bar{c} ⩾ {\bar{c}}_{s} > 0$ , $n - 2 < d ⩽ n$ and $c_{d} > 0$ . Let $α ⩾ 0$ , let $γ \in C (\overline{D})$ be nonnegative, and let Assumption 5.1 be satisfied.

Given a sequence ${((Ω_{m}, ν_{m}, μ_{m}))}_{m} \subset U_{ad} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ , there are a sequence ${(m_{k})}_{k}$ with $m_{k} ↑ + \infty$ and an admissible triple $(Ω, ν, μ) \subset U_{ad} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ such that $\begin{matrix} lim_{k \to \infty} J_{α, γ} (Ω_{m_{k}}, μ_{m_{k}}) = J_{α, γ} (Ω, μ) \end{matrix}$ in the sense of Mosco and in the Gamma-sense. The corresponding resolvent operators, spectral projectors, eigenvalues, and eigenfunctions converge as stated in Theorem 6.1 .

Theorems 5.2 and 7.1 also imply a compactness result for the unique weak solutions of boundary value problems on varying domains. Corollary 7.1.
Let the hypotheses of Theorem 7.2 be in force and let $f \in L^{2} (D)$ . Given a sequence ${((Ω_{m}, ν_{m}, μ_{m}))}_{m} \subset U_{ad} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ , there are a sequence ${(m_{k})}_{k}$ with $m_{k} ↑ + \infty$ and an admissible triple $(Ω, ν, μ) \subset U_{ad} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ such that for the continuations by zero $v_{m_{k}}$ of the unique weak solutions $u_{m_{k}}$ of ( 19 ) or ( 22 ) on $Ω_{m_{k}}$ with zero boundary values we have $\begin{matrix} lim_{k \to \infty} v_{m_{k}} = v in L^{2} (D), \end{matrix}$ where v is the continuation by zero of the corresponding unique weak solution u on Ω. There is some $u^{} \in W^{1, 2} (D)$ with* $u^{} |_{Ω} = u = v |_{Ω}$ such that* ${lim}_{k \to \infty} E_{Ω_{m_{k}}} u_{m_{k}} = u^{}$ weakly in* $W^{1, 2} (D)$ and strongly in $W^{1, 2} (Ω)$ . Moreover, we have $\begin{matrix} lim_{k \to \infty} J_{α, γ} (Ω_{m_{k}}, μ_{m_{k}}) (u_{m_{k}}) = J_{α, γ} (Ω, μ) (u) . \end{matrix}$

8. Existence of optimal shapes

From Corollary 7.1 we can conclude the existence of optimal shapes minimizing the energy (33) within a given class of shape admissible triples.

Theorem 8.1.

Let $D_{0} \subset D \subset R^{n}$ be non-empty bounded Lipschitz domains, $ε > 0$ , $n - 1 ⩽ s < n$ , $\bar{c} ⩾ {\bar{c}}_{s} > 0$ , $n - 2 < d ⩽ n$ and $c_{d} > 0$ . Let $α ⩾ 0$ , let $γ \in C (\overline{D})$ be nonnegative, and let Assumption 5.1 be satisfied.

There is a shape admissible triple $(Ω_{opt}, ν_{opt}, μ_{opt}) \in U_{ad} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ such that $\begin{matrix} J_{α, γ} (Ω_{opt}, μ_{opt}) (u (Ω_{opt}, μ_{opt})) = min_{(Ω, ν, μ) \in U_{ad} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})} J_{α, γ} (Ω, μ) (u (Ω, μ)), \end{matrix}$ where $u (Ω, μ)$ denotes the unique weak solution to ( 19 ) respectively ( 22 ) on $(Ω, ν, μ)$ with data α, γ, f and $φ \equiv 0$ .

Moreover, $(Ω_{opt}, ν_{opt}, μ_{opt})$ is the limit of a minimizing sequence $\begin{matrix} {((Ω_{m}, ν_{m}, μ_{m}))}_{m} \subset U_{ad} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d}) \end{matrix}$ in the Hausdorff sense, the sense of compacts, the sense of characteristic functions, and the sense of weak convergence of the boundary volumes and trace volumes. There is some $u^{*} \in W^{1, 2} (D)$ with $u^{*} |_{Ω} = u (Ω_{opt}, μ_{opt})$ such that ${lim}_{m \to \infty} E_{Ω_{m}} u_{m} = u^{*}$ weakly in $W^{1, 2} (D)$ and strongly in $W^{1, 2} (Ω)$ .

Proof.

Let ${((Ω_{m}, ν_{m}, μ_{m}))}_{m} \subset U_{ad} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ be a minimizing sequence for the nonnegative functional $(Ω, ν, μ) \mapsto J_{α, γ} (Ω, μ) (u (Ω, μ))$ . The result follows from an application of Corollary 7.1 to ${((Ω_{m}, ν_{m}, μ_{m}))}_{m}$ . □

We formulate the second version of this result for boundary value problems in a more narrow sense. Let $U_{ad}^{'} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ be the collection of all shape admissible triples of the form $(Ω, μ, μ)$ in $U_{ad} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ , i.e., for which boundary and trace volumes agree and in particular, $supp μ = \partial Ω$ . This is the class defined in [31, Definition 2], just with a different notation.

Theorem 8.2.

Let $D_{0} \subset D \subset R^{n}$ be non-empty bounded Lipschitz domains, $ε > 0$ , $\bar{c} ⩾ {\bar{c}}_{s} > 0$ , $\begin{matrix} (56) & n - 1 ⩽ s < n and n - 2 ⩽ d ⩽ s . \end{matrix}$ Let $α ⩾ 0$ , let $γ \in C (\overline{D})$ be nonnegative, and let Assumption 5.1 be satisfied.

There is a shape admissible triple $(Ω_{opt}, μ_{opt}, μ_{opt}) \in U_{ad}^{'} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ such that $\begin{matrix} J_{α, γ} (Ω_{opt}, μ_{opt}) (u (Ω_{opt}, μ_{opt})) = min_{(Ω, μ, μ) \in U_{ad}^{'} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})} J_{α, γ} (Ω, μ) (u (Ω, μ)), \end{matrix}$ where $u (Ω, μ)$ denotes the unique weak solution to ( 19 ) respectively ( 22 ) on $(Ω, ν, μ)$ with data α, γ, f and $φ \equiv 0$ .

Moreover, $(Ω_{opt}, μ_{opt}, μ_{opt})$ is the limit of a minimizing sequence $\begin{matrix} {((Ω_{m}, μ_{m}, μ_{m}))}_{m} \subset U_{ad}^{'} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d}) \end{matrix}$ in the Hausdorff sense, the sense of compacts, the sense of characteristic functions, and the sense of weak convergence of the boundary volumes, respectively trace volumes. There is some $u^{*} \in W^{1, 2} (D)$ with $u^{*} |_{Ω} = u (Ω_{opt}, μ_{opt})$ such that ${lim}_{m \to \infty} E_{Ω_{m}} u_{m} = u^{*}$ weakly in $W^{1, 2} (D)$ and strongly in $W^{1, 2} (Ω)$ .

Remark 8.1.

Theorem 8.2 provides a quite general result for boundary value problems in the more narrow sense: Suppose that we wish to minimize the functional defined by $\begin{matrix} J_{α, γ} (Ω, μ) (v) = \int_{Ω} | \nabla v |^{2} d x + α \int_{Ω} v^{2} d x + \int_{\partial Ω} γ {({Tr}_{Ω, Γ} v)}^{2} d μ, v \in W^{1, 2} (Ω), \end{matrix}$ and $J_{α, γ} (Ω, μ) (v) = + \infty$ if $v \in L^{2} (D) ∖ W^{1, 2} (Ω)$ . The triples in $U_{ad}^{'} (D, D_{0}, ε, s, {\bar{c}}_{s}, \bar{c}, d, c_{d})$ then are actually pairs $(Ω, μ)$ of bounded $(ε, \infty)$ -domains Ω and Borel measures μ such that we have $\begin{matrix} μ (B (x, r)) ⩽ c_{d} r^{d} and μ (\overline{B (x, r)}) ⩾ {\bar{c}}_{s} r^{s}, x \in \partial Ω, 0 < r ⩽ 1, \end{matrix}$ with (56), see [31, Definition 2]. The class of domains Ω for which this is possible is much larger than any class of domains with countably rectifiable boundaries and finite $(n - 1)$ -dimensional Hausdorff measure. The boundary $\partial Ω$ could potentially have any Hausdorff dimension in $[n - 1, n)$ .

Footnotes

Acknowledgements

We thank the anonymous referees for their careful reading and their helpful suggestions.

The first author was supported in part by the DFG IRTG 2235: ‘Searching for the regular in the irregular: Analysis of singular and random systems’.

The third author is thaknful for the generous partial support by the NSF DMS grants 1613025 and 1950543, and by the Fulbright and Simons Foundations.

The project was finished during a stay at CentraleSupélec, Univérsité Paris-Saclay, whose kind hospitality is gratefully acknowledged.

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