Optimal partition problems for the fractional Laplacian

Abstract

In this work, we prove an existence result for an optimal partition problem of the form

$$\begin{aligned} \min \{F_s(A_1,\ldots ,A_m):A_i \in {\mathcal {A}}_s, \, A_i\cap A_j =\emptyset \text{ for } i\ne j\}, \end{aligned}$$

where $F_s$ is a cost functional with suitable assumptions of monotonicity and lower semicontinuity, ${\mathcal {A}}_s$ is the class of admissible domains and the condition $A_i\cap A_j =\emptyset$ is understood in the sense of Gagliardo s-capacity, where $0<s<1$. Examples of this type of problem are related to fractional eigenvalues. As the main outcome of this article, we prove some type of convergence of the s-minimizers to the minimizer of the problem with $s=1$, studied in [5].

1 Introduction

Throughout this article, we consider $\Omega \subset {\mathbb {R}}^n$ to be a fixed Lipschitz domain, that is an open bounded subset of ${\mathbb {R}}^n$ with Lipschitz boundary. Fix $0<s<1$ and $m\in {\mathbb {N}}$. We consider optimal partition problems of the form

$$\begin{aligned} \min \left\{ F_s(A_1, \ldots , A_m):A_i \in {\mathcal {A}}_s(\Omega ), A_i\cap A_j = \emptyset \text{ for } i\ne j \right\} , \end{aligned}$$

(1.1)

where $F_s$ is a cost functional which satisfies some lower semicontinuity and monotonicity assumptions and ${\mathcal {A}}_s(\Omega )$ denotes the class of admissible domains.

Optimal partition problems were studied by several authors: Bucur, Buttazzo and Henrot [5], Bucur and Velichkov [6], Caffarelli and Lin [8], Conti, Terracini and Verzini [9, 10], Helffer, Hoffmann-Ostenhof and Terracini [19], among others.

In [8], Caffarelli and Lin established the existence of classical solutions to an optimal partition problem for the Dirichlet eigenvalue, as well as the regularity of free interfaces. One more recent work about regularity of solutions to optimal partition problems involving eigenvalues of the Laplacian is [23], where Ramos, Tavares and Terracini used the existence result of [5] and proved that the free boundary of the optimal partition is locally a $C^{1,\alpha }$-hypersurface up to a residual set.

Conti, Terracini and Verzini proved in [9] the existence of the minimal partition for a problem in N-dimensional domains related to the method of nonlinear eigenvalues introduced by Nehari in [21]. Moreover, they showed some connections between the variational problem and the behavior of competing species systems with large interaction.

Tavares and Terracini proved in [26] the existence of infinitely many sign-changing solutions for the system of m-Schrödinger equations with competition interactions and the relation between the energies associated and an optimal partition problem which involves m-eigenvalues of the Laplacian operator.

In a recent work [16], we studied a general shape optimization problem where $m=1$.

To mention some references which have to do with optimal partition problems involving fractional operators, we suggest to look through [27, 29], and references therein too.

A class of optimal partition problems involving the half-Laplacian operator and a subcritical cost functional was considered by Zilio in [29]. That work encompasses findings about optimal regularity of the density-functions which characterize the partitions, for the entire set of minimizers. Besides, a numerical-related scheme and its consequences are shown.

In [27], Terracini-Verzini-Zilio consider a class of competition-diffusion nonlinear systems involving the half-Laplacian, including the fractional Gross-Pitaevskii system.

For more references related to optimal partition problems see, for instance, [1, 2, 4, 7, 10, 18, 22, 25]

The goal of this article is to prove the existence of an optimal partition for the problem (1.1), where $F_s$ is decreasing in each coordinate and lower semicontinuous for a suitable notion of convergence in ${\mathcal {A}}_s(\Omega )$, which is the set of admissible domains. This existence result is carried out in Sect. 3. The dependence on s is related to the Gagliardo s-capacity measure and the fractional Laplacian operator $(-\Delta )^s$, and we will detail that and other preliminaries in Sect. 2.

We follow the ideas given by Bucur, Buttazzo and Henrot in [5], where the existence of solution to (1.1) in the case $s=1$ was proved.

Furthermore, we prove convergence of minimums and optimal partition shapes to those of the case $s=1$, studied in [5]. This last aim is accomplished in Sect. 4, and we consider it the most interesting contribution of this work.

At the end, we include “Appendix” with useful properties of s-capacity. Most of those results we suppose are well known. Despite that, we decided to incorporate them for completeness.

2 Preliminaries and statements

2.1 Notations and preliminaries

Given $s\in (0,1)$ we consider the fractional Laplacian, that for smooth functions u is defined as

$$\begin{aligned} (-\Delta )^s u(x)&:= c(n,s)\text{ p.v. }\int _{{\mathbb {R}}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}} \, dy\\&=-\frac{c(n,s)}{2}\int _{{\mathbb {R}}^n} \frac{u(x+z) - 2u(x) + u(x-z)}{|z|^{n+2s}}\, dz. \end{aligned}$$

where $c(n,s):= ( \int _{{\mathbb {R}}^n} \frac{1-\cos \zeta _1}{|\zeta |^{n+2s}}d\zeta )^{-1}$ is a normalization constant.

The constant c(n, s) is chosen in such a way that the following identity holds,

$$\begin{aligned} (-\Delta )^s u = {\mathcal {F}}^{-1}(|\xi |^{2s}{\mathcal {F}}(u)), \end{aligned}$$

for u in the Schwarz class of rapidly decreasing and infinitely differentiable functions, where ${\mathcal {F}}$ denotes the Fourier transform. See [14, Proposition 3.3].

The natural functional setting for this operator is the fractional Sobolev space $H^s({\mathbb {R}}^n)$ defined as

$$\begin{aligned} H^s({\mathbb {R}}^n)&:=\left\{ u\in L^2({\mathbb {R}}^n) :\frac{u(x)-u(y)}{|x-y|^{\frac{n}{2}+s}}\in L^2({\mathbb {R}}^n \times {\mathbb {R}}^n) \right\} \\&= \left\{ u\in L^2({\mathbb {R}}^n):\int _{{\mathbb {R}}^n}(1+|\xi |^{2s})|{\mathcal {F}}(u)(\xi )|^2\, d\xi <\infty \right\} \end{aligned}$$

which is a Banach space endowed with the norm $\Vert u\Vert ^2_s:= \Vert u\Vert _2^2 + [u]^2_s$, where the term

$$\begin{aligned}{}[u]^2_s:=\iint _{{\mathbb {R}}^n \times {\mathbb {R}}^n} {\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} \, dxdy} \end{aligned}$$

is the so-called Gagliardo semi-norm of u.

To contemplate the boundary condition, we work in $H_0^s(\Omega )$, which is the closure of $C_c^\infty (\Omega )$ in the norm $\Vert \cdot \Vert _s$. As we are dealing with a Lipschitz domain $\Omega$, $H^s_0(\Omega )$ coincides with the space of functions vanishing outside $\Omega$, i.e.,

$$\begin{aligned} H^s_0(\Omega )=\{u\in H^s({\mathbb {R}}^n):u=0 \text{ in } {\mathbb {R}}^n\setminus \Omega \}, \end{aligned}$$

See [17, Corollary 1.4.4.5] for a proof of the identity above.

Definition 2.1

Given $A\subset \Omega$, for any $0<s<1$, we define the Gagliardo $s-$capacity of A relative to $\Omega$ as

$$\begin{aligned} \text {cap}_s(A,\Omega )= \inf \left\{ [u]^2_s :u\in C^\infty _c(\Omega ),\ u\ge 1 \text{ in } \text{ a } \text{ neighborhood } \text{ of } A \right\} . \end{aligned}$$

We say that a subset A of $\Omega$ is an s-quasi-open subset of $\Omega$ if there exists a decreasing sequence $\{G_k\}_{k\in {\mathbb {N}}}$ of open sets such that $\lim _{k\rightarrow \infty }\text {cap}_s(G_k,\Omega )=0$ and $A\cup G_k$ is an open set.

We denote by ${\mathcal {A}}_s(\Omega )$ the class of all $s-$quasi-open subsets of $\Omega$.

In the case $s=1$ the definitions are completely analogous with $\Vert \nabla u\Vert _2^2$ instead of $[u]_s^2$.

We say that a property P(x) holds s-quasi everywhere on $E\subset \Omega$ ( s-q.e. on E), if $\text {cap}_s(\{x\in E:P(x) \text{ does } \text{ not } \text{ hold } \}, \Omega )=0$.

A function $u:{\mathbb {R}}^n \rightarrow {\mathbb {R}}$ is said s-quasi-continuous if there exists a decreasing sequence $\{G_k\}_{k\in {\mathbb {N}}}$ of open sets such that $\lim _{k\rightarrow \infty }\text {cap}_s(G_k, \Omega )= 0$ and $u|_{{\mathbb {R}}^n \setminus G_k}$ is continuous.

The following theorem allows us to work with s-quasi-continuous functions instead of the classical fractional Sobolev ones.

Theorem 2.2

(Theorem 3.7, [28]) For every function $u\in H^s_0(\Omega )$ there exist a unique $\tilde{u}:{\mathbb {R}}^n \rightarrow {\mathbb {R}}$ s-quasi-continuous function such that $u=\tilde{u}$ a.e. in ${\mathbb {R}}^n$.

From this point, we identify a function $u\in H_0^s(\Omega )$ with its s-quasi-continuous representative.

For $A \in {\mathcal {A}}_s(\Omega )$, we consider the fractional Sobolev space

$$\begin{aligned} H_0^s(A):= \{ u \in H_0^s(\Omega ) :u = 0 \ \text{ s-q.e. } \text{ in } {\mathbb {R}}^n \setminus A \}. \end{aligned}$$

To go into detail about s-capacity we refer the reader, for instance, to [24, 28].

2.2 Statements

Given $A\in {\mathcal {A}}_s(\Omega )$, we denote by $u^s_A\in H^s_0(A)$ the unique weak solution to

$$\begin{aligned} (-\Delta )^s u^s_A =1 \quad \text{ in } A, \qquad u^s_A=0 \quad \text{ in } {\mathbb {R}}^n \setminus A. \end{aligned}$$

(2.1)

With this notation, we define the following notion of set convergence.

Definition 2.3

(Strong $\gamma _s$-convergence) Let $\{A_k\}_{k\in {\mathbb {N}}}\subset {\mathcal {A}}_s(\Omega )$ and $A\in {\mathcal {A}}_s(\Omega )$. We say that $A_k\mathop {\rightarrow }\limits ^{\gamma _s} A$ if $u_{A_k}^s\rightarrow u_A^s$ strongly in $L^2(\Omega )$.

Let $m\in {\mathbb {N}}$, $\{(A_1^k,\ldots ,A_m^k)\}_{k\in {\mathbb {N}}} \subset {\mathcal {A}}_s(\Omega )^m$ and $(A_1,\ldots ,A_m) \in {\mathcal {A}}_s(\Omega )^m$. We say $(A_1^k,\ldots , A_m^k)\mathop {\rightarrow }\limits ^{\gamma _s}(A_1,\ldots ,A_m)$ if $A_i^k \mathop {\rightarrow }\limits ^{\gamma _s} A_i$ for every $i=1, \ldots , m$.

Definition 2.4

(Weak $\gamma _s$-convergence) Let $\{A_k\}_{k\in {\mathbb {N}}}\subset {\mathcal {A}}_s(\Omega )$ and $A\in {\mathcal {A}}_s(\Omega )$. We say that $A_k\mathop {\rightharpoonup }\limits ^{\gamma _s} A$ if there exists a function $u\in L^2(\Omega )$ such that $u_{A_k}^s\rightarrow u$ strongly in $L^2(\Omega )$ and $A=\{u>0\}\in {\mathcal {A}}_s(\Omega )$.

Let $m\in {\mathbb {N}}$ and $\{(A_1^k,\ldots ,A_m^k)\}_{k\in {\mathbb {N}}} \subset {\mathcal {A}}_s(\Omega )^m$ and $(A_1,\ldots ,A_m) \in {\mathcal {A}}_s(\Omega )^m$. We say $(A_1^k,\ldots , A_m^k)\mathop {\rightharpoonup }\limits ^{\gamma _s}(A_1,\ldots ,A_m)$ if $A_i^k \mathop {\rightharpoonup }\limits ^{\gamma _s} A_i$ for every $i=1, \ldots , m$.

Remark 2.5

We want to emphasize the difference between strong and weak $\gamma _s$-convergence. In the weak $\gamma _s$-convergence, the $L^2(\Omega )$-limit function u of the sequence $\{u_{A_k}^s\}_{k\in {\mathbb {N}}}$ is not required to be a solution of (2.1) in A (the weak $\gamma _s$-limit), i.e., it is not required that $u\ne u_A^s$. That is the main hassle we should get through to arrive at the compactness result on ${\mathcal {A}}_s(\Omega )$, in Sect. 3.1.

Let $m\in {\mathbb {N}}$ be fixed and $0<s\le 1$. Let $F_s :{\mathcal {A}}_s(\Omega )^m\rightarrow [0,\infty ]$ be such that

• $F_s$ is weak $\gamma _s$-lower semicontinuous, that is,

  $$\begin{aligned} F_s(A_1, \ldots , A_m)\le \liminf _{k\rightarrow \infty } F_s(A_1^k, \ldots , A_m^k), \end{aligned}$$

  for every $\{(A_1^k,\ldots , A_m^k)\}_{k\in {\mathbb {N}}} \subset A_s(\Omega )^m$ and $(A_1,\ldots ,A_m)\in {\mathcal {A}}_s(\Omega )^m$ such that $(A_1^k,\ldots , A_m^k) \mathop {\rightharpoonup }\limits ^{\gamma _s} (A_1,\ldots , A_m)$.

• $F_s$ is decreasing, that is, for every $(A_1,\ldots ,A_m), (B_1,\ldots ,B_m) \in {\mathcal {A}}_s(\Omega )^m$ such that $A_i\subset B_i$ for $i=1,\ldots , m$, we have

  $$\begin{aligned} F_s(A_1, \ldots , A_m) \ge F_s(B_1, \ldots , B_m). \end{aligned}$$

Under these assumptions, we are able to recover the existence result of [5], for the fractional case. Rigorously speaking, we have the following theorem.

Theorem 2.6

Let $F_s :{\mathcal {A}}_s(\Omega )^m\rightarrow [0,\infty ]$ be a decreasing and weak $\gamma _s$-lower semicontinuous functional. Then, there exists a solution to

$$\begin{aligned} \min \left\{ F_s(A_1, \ldots , A_m) :A_i \in {\mathcal {A}}_s(\Omega ), \ \text {cap}_s(A_i\cap A_j,\Omega )=0 \text{ for } i\ne j \right\} . \end{aligned}$$

(2.2)

The proof of Theorem 2.6 is carried out in Sect. 3 and we use ideas from [5] and [16].

Now, we present the main point of this article, that is the convergence of minimums and optimal partition shapes to those of the case $s=1$.

Once we know the existence of an optimal partition shape for each $0<s<1$, we want to analyze the limit of these minimizers and its minimum values when $s\uparrow 1$. To this aim, we need a suitable relationship between the cost functionals $F_s$, $0<s\le 1$ and a notion of set convergence.

Let us start with the notion of set convergence. For $A\in {\mathcal {A}}_1(\Omega )$, we introduce the analogous notation $u_A^1 \in H_0^1(A)$ for the unique weak solution to

$$\begin{aligned} -\Delta u_A^1 =1 \text{ in } A, \quad u_A^1=0 \text{ in } {\mathbb {R}}^n \setminus A. \end{aligned}$$

Definition 2.7

($\gamma$-convergence) Let $0<s_k\uparrow 1$, $\{A_k\}_{k\in {\mathbb {N}}}\subset {\mathcal {A}}_{s_k}(\Omega )$ and $A\in {\mathcal {A}}_1(\Omega )$. We say that $A_k\mathop {\rightarrow }\limits ^{\gamma } A$ if $u_{A_k}^{s_k}\rightarrow u_A^1$ strongly in $L^2(\Omega )$.

Let $m \in {\mathbb {N}}$, $(A_1^k,\ldots ,A_m^k)\in {\mathcal {A}}_{s_k}(\Omega )^m$ and $(A_1,\ldots ,A_m)\in {\mathcal {A}}_1(\Omega )^m$. We say that $(A_1^k,\ldots ,A_m^k) \mathop {\rightarrow }\limits ^{\gamma }(A_1,\ldots ,A_m)$ if $u_{A_i^k}^{s_k} \rightarrow u_{A_i}^1$ strongly in $L^2(\Omega )$, for every $i=1,\ldots , m$.

Let $m\in {\mathbb {N}}$ and $0<s\le 1$. Let $F_s:{\mathcal {A}}_s(\Omega )^m\rightarrow [0,\infty ]$ be decreasing and weak $\gamma _s$-lower semicontinuous functionals. Then, there exists $(A_1^s,\ldots , A_m^s)$ solution to

$$\begin{aligned} m_s:= \min \left\{ F_s(B_1,\ldots ,B_m):B_i \in {\mathcal {A}}_s(\Omega ), \, \text {cap}_s(B_i\cap B_j, \Omega )=0 \text{ for } i\ne j \right\} . \end{aligned}$$

(2.3)

The case $s=1$ was solved in [5]. For $0<s<1$, apply Theorem 2.6.

Assume the following hypotheses over the cost functionals:

• $(H_1)$ Continuity. For every $(A_1,\ldots , A_m)\in {\mathcal {A}}_1(\Omega )^m$,

  $$\begin{aligned} F_1(A_1,\ldots , A_m)=\lim _{s\uparrow 1}F_s(A_1,\ldots , A_m). \end{aligned}$$

• $(H_2)$ Liminf inequality. For every $0<s_k\uparrow 1$, $(A_1^k,\ldots , A_m^k)\in {\mathcal {A}}_{s_k}(\Omega )^m$ and $(A_1,\ldots , A_m)\in {\mathcal {A}}_1(\Omega )^m$ such that $(A_1^k,\ldots ,A_m^k) \mathop {\rightarrow }\limits ^{\gamma }(A_1,\ldots ,A_m)$,

  $$\begin{aligned} F_1(A_1,\ldots ,A_m) \le \liminf _{k\rightarrow \infty }{F_{s_k}(A_1^k,\ldots ,A_m^k)}. \end{aligned}$$

These conditions $(H_1)$-$(H_2)$ are natural and analogous to those consider in [16], where a similar shape optimization problem was studied with $m=1$.

Now, we are able to establish the main result.

Theorem 2.8

Let $m\in {\mathbb {N}}$ be fixed and $0<s\le 1$. Let $F_s:{\mathcal {A}}_{s}(\Omega )^m\rightarrow [0,\infty ]$ be a decreasing and weak $\gamma _{s}$-lower semicontinuous functional, and such that $(H_1)$-$(H_2)$ are verified. Then,

$$\begin{aligned} m_1 = \lim _{s\uparrow 1} m_s, \end{aligned}$$

(2.4)

where $m_s$ is defined in (2.3).

Moreover, if $(A_1^s,\ldots ,A_m^s)$ is a minimizer of (2.3), then, there exist a subsequence $0<s_k\uparrow 1$, $(\tilde{A}_1^{s_k},\ldots ,\tilde{A}_m^{s_k}) \in {\mathcal {A}}_{s_k}(\Omega )^m$ and $(A_1^1,\ldots ,A_m^1) \in {\mathcal {A}}_1(\Omega )^m$ such that $\tilde{A}_i^{s_k} \supset A_i^{s_k}$ and

$$\begin{aligned} (\tilde{A}_1^{s_k},\ldots ,\tilde{A}_m^{s_k}) \mathop {\rightarrow }\limits ^{\gamma } (A_1^1,\dots ,A_m^1) , \end{aligned}$$

where $(A_1^1,\dots ,A_m^1)$ is a minimizer of (2.3) with $s=1$.

The proof of Theorem 2.8 is carried out in Sect. 4, and we use again ideas from [16].

2.3 Examples

Given $A\in {\mathcal {A}}_s(\Omega )$, consider the problem

$$\begin{aligned} (-\Delta )^s u= \lambda ^s u \quad \text { in } A, \qquad u\in H_0^s(A) \end{aligned}$$

(2.5)

where $\lambda ^s\in {\mathbb {R}}$ is the eigenvalue parameter. It is well known that there exists a discrete sequence $\{\lambda _k^s(A)\}_{k\in {\mathbb {N}}}$ of positive eigenvalues of (2.5) approaching $+\infty$ whose corresponding eigenfunctions $\{u_k^s\}_{k\in {\mathbb {N}}}$ form an orthogonal basis in $L^2(A)$. Moreover, the following variational characterization holds for the eigenvalues

$$\begin{aligned} \lambda _k^s(A)=\min _{u\perp W_{k-1} }\frac{c(n,s)}{2}\frac{[u]^2_s}{\Vert u\Vert _2^2}, \end{aligned}$$

(2.6)

where $W_k$ is the space spanned by the first k eigenfunctions $u_1^s,\ldots , u_k^s$.

Due to (2.6) and the stability result proved in [3, Theorem 1.2], we know that $\lambda _k^s(A)\rightarrow \lambda _k^1(A)$, when $s\uparrow 1$, for every $k\in {\mathbb {N}}$.

Consider functionals $F_s(A_1,\dots ,A_m)= \Phi _s(\lambda _{k_1}^s(A_1),\dots , \lambda _{k_m}^s(A_m))$. Theorem 2.6 claims that for every $(k_1,\dots ,k_m)\in {\mathbb {N}}^m$, the minimum

$$\begin{aligned} \min \{ \Phi _s(\lambda _{k_1}^s(A_1),\dots , \lambda _{k_m}^s(A_m)):A_i\in {\mathcal {A}}_s(\Omega ), \, \text {cap}_s(A_i\cap A_j,\Omega ) \text{ for } i\ne j\} \end{aligned}$$

is achieved, where $\Phi _s:{\mathbb {R}}^m\rightarrow \bar{\mathbb {R}}$, is increasing in each coordinate and lower semicontinuous.

Moreover, if $\Phi _s(t_1,\dots ,t_m)\rightarrow \Phi _1(t_1,\dots ,t_m)$ for every $(t_1,\dots ,t_m)\in {\mathbb {R}}^m$ and

$$\begin{aligned} \Phi _1(t_1,\dots ,t_m)\le \liminf _{k\rightarrow \infty } \Phi _{s_k}(t_1^k,\dots ,t_m^k), \end{aligned}$$

for every $(t_1^k,\dots ,t_m^k)\rightarrow (t_1,\dots ,t_m)$, then Theorem 2.8 together with the existence result of [5] imply that

$$\begin{aligned}&\min \{ \Phi _1(\lambda _{k_1}(A_1),\dots ,\lambda _{k_m}(A_m)):A_i\in {\mathcal {A}}_1(\Omega ), \, \text {cap}_1(A_i\cap A_j,\Omega )=0 \text{ for } i\ne j\}\\&= \lim _{s\uparrow 1} \min \{ \Phi _s(\lambda _{k_1}^s(A_1),\dots ,\lambda _{k_m}^s(A_m)):A_i\in {\mathcal {A}}_s(\Omega ), \, \text {cap}_s(A_i\cap A_j,\Omega )=0 \text{ for } i\ne j\}. \end{aligned}$$

3 Proof of Theorem 2.6

In this section, we adapted the ideas from [5], where the authors consider the Laplacian operator, to recover their results for the fractional case. Despite the similarity of the proofs, we include them for the reader’s convenience and recalling that in the context of this article we need the nonlocal tools proved in [16].

3.1 Certain compactness on ${\mathcal {A}}_s(\Omega )$

Consider ${\mathcal {K}}_s$ given by

$$\begin{aligned} {\mathcal {K}}_s:=\{w\in H_0^s(\Omega ) :w\ge 0, \, (-\Delta )^s w \le 1 \text{ in } \Omega \}. \end{aligned}$$

(3.1)

Proposition 3.1

(Proposition 3.3 and Lemma 3.5, [16]) ${\mathcal {K}}_s$ is convex, closed and bounded in $H_0^s(\Omega )$. Moreover, if $u, v\in {\mathcal {K}}_s$, then, $\max \{u,v\} \in {\mathcal {K}}_s$.

Proposition 3.2

(Lemma 3.2, [16]) Given $A\in {\mathcal {A}}_s(\Omega )$, $u_A^s$ is the solution to

$$\begin{aligned} \max \left\{ w\in H_0^s(\Omega ) :w\le 0 \text{ in } {\mathbb {R}}^n \setminus A, \, (-\Delta )^s w \le 1 \text{ in } \Omega \right\} . \end{aligned}$$

Moreover, $u_A^s \in {\mathcal {K}}_s$, for every $A\in {\mathcal {A}}_s(\Omega )$.

From now on, we understand the identity $A=\{u_A^s>0\}$ in the sense of the Gagliardo s-capacity, thanks to Proposition A.5.

Remark 3.3

The class ${\mathcal {A}}_s(\Omega )$ is sequentially pre-compact with respect to the weak $\gamma _s$-convergence. Indeed, given a sequence $\{A_k\}_{k\in {\mathbb {N}}} \subset {\mathcal {A}}_s(\Omega )$, we know that $\{u_{A_k}^s\}_{k\in {\mathbb {N}}} \subset {\mathcal {K}}_s$. By Proposition 3.1, there exists a subsequence $\{u_{A_{k_j}}^s\}_{j\in {\mathbb {N}}} \subset \{u_{A_k}^s\}_{k\in {\mathbb {N}}}$ and a function $u\in {\mathcal {K}}_s$ such that $u_{A_{k_j}}^s \rightarrow u$ strongly in $L^2(\Omega )$. Denote by $A:=\{u>0\}$. Then, $A_{k_j} \mathop {\rightharpoonup }\limits ^{\gamma _s} A$.

Next proposition allows us to pass from the weak $\gamma _s$-convergence to the strong one, if we are willing to enlarge the sequence involved.

Proposition 3.4

Let $\{A_k\}_{k\in {\mathbb {N}}}\subset {\mathcal {A}}_s(\Omega )$ and $A,B\in {\mathcal {A}}_s(\Omega )$ be such that $A_k\mathop {\rightharpoonup }\limits ^{\gamma _s} A\subset B$.

Then, there exists a subsequence $\{A_{k_j}\}_{j\in {\mathbb {N}}}\subset \{A_k\}_{k\in {\mathbb {N}}}$ and a sequence $\{B_{k_j}\}_{j\in {\mathbb {N}}} \subset {\mathcal {A}}_s(\Omega )$ such that $A_{k_j}\subset B_{k_j}$ and $B_{k_j}\mathop {\rightarrow }\limits ^{\gamma _s} B$.

Proof

Since $A_k\mathop {\rightharpoonup }\limits ^{\gamma _s} A\subset B$, we know that $u_{A_k}^s\rightarrow u$ strongly in $L^2(\Omega )$, where $\{ u>0 \}=A$. As a consequence of Proposition 3.1, $u\in {\mathcal {K}}_s$. Moreover, by Proposition 3.2, $u\le u_A^s$. Since $A\subset B$, $u_A^s\le u_B^s$. Then, $u\le u_B^s$.

Denote by $B^{\varepsilon }=\{ u_B^s>\varepsilon \}$ and consider $\{u_{A_{k}\cup B^{\varepsilon }}^s\}_{k\in {\mathbb {N}}} \subset {\mathcal {K}}_s$. Again by Proposition 3.1, there exists a subsequence $\{A_{k_j}\}_{j\in {\mathbb {N}}} \subset \{A_k\}_{k\in {\mathbb {N}}}$ such that $u_{A_{k_j}\cup B^{\varepsilon }}^s \rightarrow u^{\varepsilon }$ strongly in $L^2(\Omega )$.

Due to the convergence $u_{A_{k_j}}^s\rightarrow u$ strongly in $L^2(\Omega )$ and $u\le u_B^s$, we conclude from [16, Lemma 3.6], $u^{\varepsilon }\le u_B^s$.

Inside the proof of [16, Lemma 3.7], it was shown that $(u_{B}^s-\varepsilon )^+\le u_{B^{\varepsilon }}^s$. Since $B^{\varepsilon }\subset A_{k_j} \cup B^{\varepsilon }$, it follows that $u_{B^{\varepsilon }}^s \le u_{A_{k_j} \cup B^{\varepsilon }}^s$. So, taking the limit $j\rightarrow \infty$, we obtain

$$\begin{aligned} (u_{B}^s-\varepsilon )^+\le u_{B^{\varepsilon }}^s\le u^{\varepsilon }\le u_B^s. \end{aligned}$$

The sequence $\{u^{\varepsilon }\}_{\varepsilon >0}$ is contained in ${\mathcal {K}}_s$. So, by Proposition 3.1, up to a subsequence, we know it has a weak limit in $H_0^s(\Omega )$. But, the previous inequality tells that this weak limit should be $u_B^s$. In addition, $u^{\varepsilon }\rightarrow u_B^s$ strongly in $L^2(\Omega )$.

Thus, there exists a sequence $\varepsilon _j\downarrow 0$ such that $u_{A_{k_j}\cup B^{{\varepsilon }_j}} ^s \rightarrow u_B^s$ strongly in $L^2(\Omega )$. That is, $A_{k_j}\cup B^{{\varepsilon }_j}=:B_{k_j}\mathop {\rightarrow }\limits ^{\gamma _s} B$, where $\{B_{k_j}\}_{j\in {\mathbb {N}}}$ is the enlarged sequence. $\square$

3.2 An auxiliary functional

Fix $m \in {\mathbb {N}}$ and $0<s<1$. Let $F_s:{\mathcal {A}}_s(\Omega )^m \rightarrow [0,\infty ]$ be a decreasing and strong $\gamma _s$-lower semicontinuous functional.

We define a functional $G_s:\mathcal {K}_s^m \rightarrow [0,\infty ]$

$$\begin{aligned} G_s(w_1,\dots ,w_m):=\inf \left\{ \liminf _{k\rightarrow \infty }J_s(w_1^k,\dots ,w_m^k) :w_i^k\rightarrow w_i \text{ strongly } \text{ in } L^2(\Omega ) \right\} , \end{aligned}$$

(3.2)

where $J_s:{\mathcal {K}}_s^m \rightarrow [0,\infty ]$ is defined as

$$\begin{aligned} J_s(w_1,\dots ,w_m):=\inf \left\{ F_s(A_1,\dots ,A_m):A_i\in {\mathcal {A}}_s(\Omega ), \ u_{A_i}^s\le w_i \text{ for } i=1,\dots , m \right\} \end{aligned}$$

and ${\mathcal {K}}_s$ was given in (3.1).

We will show that $G_s$ satisfies the following properties:

• ($G_1$) $G_s$ is decreasing on $\mathcal {K}_s^m$, that is $G_s(u_1,\dots , u_m)\ge G_s(v_1,\dots , v_m)$, if $u_i\le v_i$ for every $i=1,\dots ,m$.

• ($G_2$) $G_s$ is lower semicontinuous on $\mathcal {K}_s$ with respect to the strong topology on $L^2(\Omega )$,

• ($G_3$) $G_s(u_{A_1}^s,\dots , u_{A_m}^s)=F_s(A_1,\dots , A_m)$ for every $(A_1,\dots , A_m)\in {\mathcal {A}}_s(\Omega )^m$.

The conditions $(G_1)$ and $(G_2)$ are easy to check, and it is the content of next proposition.

Proposition 3.5

With the notation above, $G_s$ satisfies $(G_1)$ and $(G_2)$.

Proof

By construction, it is clear that $G_s$ verifies $(G_2)$.

To prove $(G_1)$, let $(u_1,\dots , u_m), (v_1,\dots , v_m)\in {\mathcal {K}}_s^m$ such that $u_i\le v_i$ for every $i=1,\dots ,m$.

Take $\{u_i^k\}_{k\in {\mathbb {N}}} \subset {\mathcal {K}}_s$ such that $u_i^k\rightarrow u_i$ strongly in $L^2(\Omega )$ for every $i=1,\dots ,m$ and

$$\begin{aligned} G_s(u_1,\dots , u_m)=\lim _{k\rightarrow \infty }J_s(u_1^k,\dots , u_m^k). \end{aligned}$$

Consider $v_i^k:=\max \{v_i, u_i^k\}$ for every $i=1,\dots ,m$ and $k\in {\mathbb {N}}$. By Proposition 3.1, we obtain that $v_i^k \in {\mathcal {K}}_s$. In addition, $v_i^k\rightarrow \max \{v_i, u_i\}=v_i$ strongly in $L^2(\Omega )$, for every $i=1,\dots ,m$. Thus, noticing that $J_s$ is decreasing, we have

$$\begin{aligned} G_s(v_1,\dots ,v_m)\le \liminf _{k\rightarrow \infty }J_s(v_1^k,\dots ,v_m^k)\le \lim _{k\rightarrow \infty }J_s(u_1^k,\dots ,u_m^k)=G_s(u_1,\dots ,u_m). \end{aligned}$$

$\square$

Now, we prove the most important property of $G_s$, which is the connection with the cost functional $F_s$.

Proposition 3.6

The functional $G_s$ satisfies $(G_3)$.

Proof

By definition of $G_s$ (3.2), it is clear that $G_s(u_{A_1}^s,\dots ,u_{A_m}^s)\le F_s(A_1,\dots , A_m)$, for every $(A_1,\dots , A_m)\in {\mathcal {A}}_s(\Omega )^m$.

To obtain the other inequality, it is enough to prove that for every sequence $\{u_i^k\}_{k\in {\mathbb {N}}}\subset {\mathcal {K}}_s(\Omega )$ such that $u_i^k\rightarrow u_{A_i}^s$ strongly in $L^2(\Omega )$ for $i=1,\dots , m$, we have

$$\begin{aligned} F_s(A_1,\dots ,A_m)\le \liminf _{k\rightarrow \infty }J_s(u_1^k,\dots , u_m^k). \end{aligned}$$

By definition of $J_s$, there exists $\{(A_1^k,\dots ,A_m^k)\}_{k\in {\mathbb {N}}} \subset {\mathcal {A}}_s(\Omega )^m$ such that

$$\begin{aligned} u_{A_i^k}^s\le u_i^k \text{ for } i=1,\dots , m, \, \text{ and } \, F_s(A_1^k,\dots , A_m^k)\le J_s(u_1^k,\dots ,u_m^k)+\frac{1}{k}. \end{aligned}$$

(3.3)

By Remark 3.3, there exists $v_i\in {\mathcal {K}}_s$ such that $u_{A_i^k}^s\rightarrow v_i$ strongly in $L^2(\Omega )$, up to a subsequence. That is, $A_i^k\mathop {\rightharpoonup }\limits ^{\gamma _s}B_i:=\{v_i>0\}\in {\mathcal {A}}_s(\Omega )$, for every $i=1,\dots ,m$.

Moreover, taking the limit in $u_{A_i^k}^s\le u_i^k$, we obtain that $v_i\le u_{A_i}^s$ for every $i=1,\dots ,m$. In addition, we have $B_i\subset A_i=\{u_{A_i}^s>0\}$. We are able to apply Proposition 3.4, to obtain the existence of subsequences $\{A_i^{k_j}\}_{j\in {\mathbb {N}}},\{B_i^{k_j}\}_{j\in {\mathbb {N}}} \subset {\mathcal {A}}_s(\Omega )$ such that $A_i^{k_j} \subset B_i^{k_j}$ and $B_i^{k_j}\mathop {\rightarrow }\limits ^{\gamma _s} A_i$.

Now, by using the strong $\gamma _s$-lower semicontinuity and decreasing property of $F_s$ and (3.3), we conclude

$$\begin{aligned} F_s(A_1,\dots ,A_m)&\le \liminf _{j\rightarrow \infty }{F_s(B_1^{k_j},\dots , B_m^{k_j})}\\&\le \liminf _{j\rightarrow \infty }{F_s(A_1^{k_j},\dots , A_m^{k_j})} \\&\le \liminf _{j\rightarrow \infty }{J_s(u_1^{k_j},\dots , u_m^{k_j})}, \end{aligned}$$

which implies the remaining inequality $F_s(A_1,\dots ,A_m)\le G_s(u_{A_1}^s,\dots ,u_{A_m}^s)$. $\square$

The decreasing property of a functional $F_s$ makes equivalent its weak and strong $\gamma _s$-lower semicontinuity, which is the content of next theorem.

Theorem 3.7

Let $m\in {\mathbb {N}}$ and $0<s<1$. Let $F_s:{\mathcal {A}}_s(\Omega )^m\rightarrow [0,\infty ]$ be a decreasing functional. Then, the following assertions are equivalent

• 1 $F_s$ is weakly $\gamma _s$-lower semicontinuous.

• 2 $F_s$ is strong $\gamma _s$-lower semicontinuous.

Proof

Since every strongly $\gamma _s$-convergent sequence $\{A_k\}_{k\in {\mathbb {N}}}$ is, in addition, weakly $\gamma _s$-convergent, $(1)\Rightarrow (2)$ is clear. (See definitions and Proposition A.5). Let us see the converse.

Now, suppose $F_s$ is strongly $\gamma _s$-lower semicontinuous. To arrive at the weakly $\gamma _s$-lower semicontinuity of $F_s$ from the strong one, the strategy is to take into account the auxiliary functional $G_s$ defined in (3.2) and its properties.

Fix $\{(A_1^k,\dots ,A_m^k)\}_{k\in {\mathbb {N}}}\subset {\mathcal {A}}_s(\Omega )^m$ and $(A_1,\dots ,A_m)\in {\mathcal {A}}_s(\Omega )^m$ such that

$$\begin{aligned} (A_1^k,\dots ,A_m^k)\mathop {\rightharpoonup }\limits ^{\gamma _s} (A_1,\dots ,A_m). \end{aligned}$$

That means $u_{A_i^k}^s\rightarrow u_i$ strongly in $L^2(\Omega )$ and $A_i=\{u_i>0\}$, for $i=1,\dots ,m$.

Since for every $i=1,\dots , m$, $\{u_{A_i^k}^s\}_{k\in {\mathbb {N}}}\subset {\mathcal {K}}_s$, by Proposition 3.1, $u_i \in {\mathcal {K}}_s$. Moreover, by Proposition 3.2, $u_i\le u_{A_i}^s$. Then, we can use $(G_3)$, the decreasing property of $G_s$, so that we obtain

$$\begin{aligned} G_s(u_{A_1}^s,\dots ,u_{A_m}^s)\le G_s(u_{1},\dots ,u_{m}). \end{aligned}$$

(3.4)

On the other hand, by recalling $(G_1)$, the relationship between $F_s$ and $G_s$, we get the following identities

$$\begin{aligned} F_s(A_1,\dots ,A_m)=G_s(u_{A_1}^s,\dots ,u_{A_m}^s) \quad \text { and } \quad F_s(A_1^k,\dots ,A_m^k)=G_s(u_{A_1^k}^s,\dots ,u_{A_m^k}^s), \end{aligned}$$

(3.5)

for every $k\in {\mathbb {N}}$.

Now, due to $(G_2)$ (the $L^2(\Omega )$-lower semicontinuity of $G_s$) in addition to $u_{A_i^k}^s\rightarrow u_i$ strongly in $L^2(\Omega )$ for every $i=1,\dots , m$, we connect (3.4) and (3.5) to conclude that

$$\begin{aligned} F_s(A_1,\dots ,A_m)&=G_s(u_{A_1}^s,\dots ,u_{A_m}^s)\le G_s(u_{1},\dots ,u_{m})\\&\le \liminf _{k\rightarrow \infty } G_s(u_{A_1^k}^s,\dots ,u_{A_m^k}^s)\\&=\liminf _{k\rightarrow \infty }F_s(A_1^k,\dots ,A_m^k).\\ \end{aligned}$$

Since $\{(A_1^k, \dots , A_m^k)\}_{k\in {\mathbb {N}}}$ is an arbitrary weak $\gamma _s$-convergent sequence, we get that $F_s$ is weak $\gamma _s$-lower semicontinuous, as we desired. $\square$

3.3 Existence of an optimal partition

With the help of the previous outcomes of this section, we are able to prove existence of a minimal partition shape for (2.2).

Proof of Theorem 2.6

Denote by

$$\begin{aligned} \alpha :=\inf \left\{ F_s(A_1,\dots ,A_m) :A_i\in {\mathcal {A}}_s(\Omega ), \text {cap}_s(A_i\cap A_j, \Omega )=0 \text{ for } i\ne j \right\} . \end{aligned}$$

Let $\{(A_1^k,\dots ,A_m^k)\}_{k\in {\mathbb {N}}}\subset {\mathcal {A}}_s(\Omega )^m$ be such that

$$\begin{aligned} \text {cap}_s(A_i^k\cap A_j^k,\Omega )=0 \text{ for } i\ne j, \, \text{ and } \, \lim _{k\rightarrow \infty }F_s(A_1^k,\dots ,A_m^k)=\alpha . \end{aligned}$$

By Remark 3.3, there exist $A_1\in {\mathcal {A}}_s(\Omega )$ and a subsequence $\{A_1^{k_j}\}_{j\in {\mathbb {N}}} \subset \{A_1^k\}_{k\in {\mathbb {N}}}$ such that $A_1^{k_j}\mathop {\rightharpoonup }\limits ^{\gamma _s}A_1$. Now, consider $\{A_2^{k_j}\}_{j\in {\mathbb {N}}}$ and apply again Remark 3.3. Thus, there exist $A_2\in {\mathcal {A}}_s(\Omega )$ and a subsequence $\{A_2^{k_{j_l}}\}_{l\in {\mathbb {N}}} \subset \{A_2^{k_j}\}_{j\in {\mathbb {N}}}$ such that $A_i^{k_{j_l}} \mathop {\rightharpoonup }\limits ^{\gamma _s} A_i$ for $i=1,2$. Repeating this argument, we find a sequence $\{(A_1^k,\dots , {\mathcal {A}}_m^k)\}_{k\in {\mathbb {N}}}$ and $(A_1,\dots ,A_m) \in {\mathcal {A}}_s(\Omega )$ such that $A_i^k\mathop {\rightharpoonup }\limits ^{\gamma _s} A_i$ for every $i=1,\dots ,m$.

Since $F_s$ is weak $\gamma _s$-lower semicontinuous, we obtain

$$\begin{aligned} F_s(A_1,\dots ,A_m) \le \liminf _{k\rightarrow \infty }F_s(A_1^k,\dots ,A_m^k)=\alpha . \end{aligned}$$

(3.6)

To finish the proof, let us see $\text {cap}_s(A_i\cap A_j,\Omega )=0$ for $i\ne j$ be satisfied.

Let $i,j \in \{1,\dots , m\}$ be such that $i\ne j$. Notice that this product $u_{A_i^k}^s \cdot u_{A_j^k}^s$ is an s-continuous function too, by Lemma A.1, and $u_{A_i^k}^s \cdot u_{A_j^k}^s=0$ s-q.e. in ${\mathbb {R}}^n \setminus (A_i^k\cap A_j^k)$. Moreover, since $\text {cap}_s(A_i^k \cap A_j^k,\Omega )=0$, we have $u_{A_i^k}^s \cdot u_{A_j^k}^s=0$ s-q.e. in ${\mathbb {R}}^n$.

By [28, Lemma 3.8], there exist subsequences $\{u_{A_i^k}^s\}_{k\in {\mathbb {N}}}$ and $\{u_{A_j^k}^s\}_{k\in {\mathbb {N}}}$, denoted with the same index, which converge s-q.e. to $u_i$ and $u_j$, respectively. Then, passing to the limit, we obtain $u_i \cdot u_j=0$ s-q.e. in ${\mathbb {R}}^n$. That is $\text {cap}_s(\{u_i \cdot u_j \ne 0\},\Omega )=0$. But, $\{u_i \cdot u_j \ne 0\}=A_i \cap A_j$.

We have shown that $(A_1,\dots ,A_m)$ is admissible for the minimization problem (2.2) and recalling (3.6) the result is proved. $\square$

Due to Theorems 3.7 and 2.6, we can establish the next immediate corollary.

Corollary 3.8

Let $F_s :{\mathcal {A}}_s(\Omega )^m\rightarrow [0,\infty ]$ be a decreasing and strong $\gamma _s$-lower semicontinuous functional. Then, there exists a solution to (2.2).

4 Proof of Theorem 2.8

This is the main part of the article, where we study the behavior of optimal partition shapes obtained in Sect. 3 and their minimum values. Again, we use some results from [16].

Lemma 4.1

(Lemma 4.1, [16]) Let $0<s_k\uparrow 1$ and let $u_k\in {\mathcal {K}}_{s_k}$. Then, there exists $u\in H^1_0(\Omega )$ and a subsequence $\{u_{k_j}\}_{j\in {\mathbb {N}}}\subset \{u_k\}_{k\in {\mathbb {N}}}$ such that $u_{k_j} \rightarrow u$ strongly in $L^2(\Omega )$.

Moreover, if $u_k \in {\mathcal {K}}_{s_k}$ is such that $u_k\rightarrow u$ strongly in $L^2(\Omega )$, then $u\in {\mathcal {K}}_1$.

Next proposition gives an idea of the limit behavior of $u_A^s$ when the domains also are varying with s.

Proposition 4.2

(Proposition 4.5, [16]) Let $0<s_k\uparrow 1, A^k \in {\mathcal {A}}_{s_k}(\Omega )$ be such that $u_{A^k}^{s_k}\rightarrow u$ strongly in $L^2(\Omega )$. Then, there exist $\tilde{A}^k \in {\mathcal {A}}_{s_k}(\Omega )$ such that $A^k \subset \tilde{A}^k$ and $\tilde{A}^k$ $\gamma -$converges to $A:=\{ u>0\}$.

Now we are ready to prove the main result of this article.

Proof of Theorem 2.8

First, notice that $m_1$ is achieved by [5, Theorem 3.2].

Let $0<s_k\uparrow 1$. By Theorem 2.6, there exists $(A_1^k,\dots , A_m^k)\in {\mathcal {A}}_{s_k}(\Omega )^m$ such that

$$\begin{aligned} \text {cap}_{s_k}(A_i^k\cap A_j^k,\Omega )=0 \text{ for } i\ne j \, \text{ and } \, F_{s_k}(A_1^k,\dots , A_m^k ) = m_k, \end{aligned}$$

(4.1)

where $m_k=m_{s_k}$ defined in (2.2).

Let $(A_1,\dots , A_m) \in {\mathcal {A}}_1(\Omega )^m$ be such that $\text {cap}_1(A_i\cap A_j,\Omega )=0$ for $i\ne j$. Since $0<s_k\uparrow 1$, we can assume $0<\varepsilon _0<s_k\uparrow 1$, for some fixed $\varepsilon _0$.

Now, recalling Corollary A.7 and Remark A.8, we know that $(A_1,\dots ,A_m)$ belongs to

$$\begin{aligned} \{(B_1,\dots ,B_m):B_i\in {\mathcal {A}}_{s_k}(\Omega ), \, \text {cap}_{s_k}(B_i\cap B_j,\Omega )=0 \text{ for } i\ne j \}, \end{aligned}$$

for every $k\in {\mathbb {N}}$. This fact and condition $(H_1)$ imply that

$$\begin{aligned} \limsup _{k\rightarrow \infty } F_{s_k}(A_1^k,\dots , A_m^k)\le \lim _{k\rightarrow \infty } F_{s_k}(A_1,\dots , A_m) = F_1(A_1,\dots , A_m). \end{aligned}$$

It follows that

$$\begin{aligned} \limsup _{k\rightarrow \infty } m_k\le m_1. \end{aligned}$$

(4.2)

To see the remaining inequality, let us denote $u_i^k:=u_{A_i^k}^{s_k} \in {\mathcal {K}}_{s_k}$. By Lemma 4.1, there is $u_i\in {\mathcal {K}}_1$ such that, up to a subsequence, $u_i^k \rightarrow u_i$ strongly in $L^2(\Omega )$ and a.e. in $\Omega$.

Denote by $A_i:=\{u_i>0\}\in {\mathcal {A}}_1(\Omega )$ for every $i=1,\dots , m$. We claim that $\text {cap}_1(A_i \cap A_j,\Omega )=0$ for $i\ne j$.

Indeed, let $i\ne j$ be fixed. For each $k\in {\mathbb {N}}$, due to Lemma A.2 and (4.1), we know that

$$\begin{aligned} |\{u_i^k \cdot u_j^k\ne 0 \}|=|A_i^k\cap A_j^k|\le C(n,s_k)\text {cap}_{s_k}(A_i^k \cap A_j^k, \Omega )=0. \end{aligned}$$

Then, $u_i^k \cdot u_j^k=0$ a.e. in ${\mathbb {R}}^n$. Since $u_l^k\rightarrow u_l$ a.e. in $\Omega$ for $l=1,2$, we conclude $u_i \cdot u_j =0$ a.e in $\Omega$, it is still true in ${\mathbb {R}}^n \setminus \Omega$ considering that they belong to $H_0^s(\Omega )$. So, $u_i \cdot u_j=0$ a.e. in ${\mathbb {R}}^n$.

Reminding that we are working with 1-quasi-continuous representative functions in $H_0^1(\Omega )$, the previous identity $u_i \cdot u_j=0$ a.e. in ${\mathbb {R}}^n$ and [20, Lemma 3.3.30] tells that $u_i \cdot u_j =0$ 1-q.e. in ${\mathbb {R}}^n$. That means $\text {cap}_1(A_i\cap A_j, \Omega )=0$.

Consequently, $(A_1,\dots ,A_m)$ is admissible to the problem 2.2 with $s=1$ and we obtain $m_1\le F_1(A_1,\dots ,A_m)$.

Moreover, by Proposition 4.2, there exists $\tilde{A}_i^k \in {\mathcal {A}}_{s_k}(\Omega )$ such that $A_i^k \subset \tilde{A}_i^k$ and $(\tilde{A}_1^k, \dots , \tilde{A}_m^k)$ $\gamma -$converges to $(A_1,\dots , A_m)$.

Finally, from condition $(H_2)$ and the decreasing property of $F_{s_k}$, we conclude that

$$\begin{aligned} m_1&\le F_1(A_1,\dots ,A_m) \le \liminf _{k\rightarrow \infty } F_{s_k}(\tilde{A}_1^{k}, \dots , \tilde{A}_m^{k}) \\&\le \liminf _{k\rightarrow \infty } F_{s_k}(A_1^k, \dots , A_m^k)=\liminf _{k\rightarrow \infty }m_k. \end{aligned}$$

Therefore, from previous conclusion and (4.2) we have the identity (2.4), so that the results follow. $\square$

Appendix A. Some useful properties of s-capacity

The following lemmas address some basic properties of s-capacity and s-quasi-continuous functions. We suppose those results are well known, and we include them for completeness.

Lemma A.1

Let $u,v :{\mathbb {R}}^n \rightarrow {\mathbb {R}}$ be s-quasi-continuous functions. Then, the product $u \cdot v$ is also an s-quasi-continuous function.

Proof

By definition, there exist decreasing sequences $\{A_k\}_{k\in {\mathbb {N}}}$ and $\{B_k\}_{k\in {\mathbb {N}}}$ of open sets such that $\lim _{k\rightarrow \infty }\text {cap}_s(A_k,\Omega )=\lim _{k\rightarrow \infty } \text {cap}_s(B_k,\Omega )=0$ and $u|_{{\mathbb {R}}^n \setminus A_k}$, $v|_{{\mathbb {R}}^n \setminus B_k}$ are continuous.

Consider $C_k:=A_k\cup B_k$. Then, $\{C_k\}_{k\in {\mathbb {N}}}$ is a decreasing sequence of open sets such that $\lim _{k\rightarrow \infty }\text {cap}_s(C_k,\Omega )=0$, since $\text {cap}_s(C_k,\Omega )\le \text {cap}_s(A_k,\Omega )+\text {cap}_s(B_k,\Omega )$ by [28, Proposition 3.6]. Moreover, $(u \cdot v)|_{{\mathbb {R}}^n \setminus C_k}$ is continuous.$\square$

Next lemma gives a relation between the Lebesgue measure and the s-capacity of a subset $A \subset \Omega$. The proof is easy and follows [15, Section 4.7, Theorem 2 VI], where it was shown with the classical capacity measure ($s=1$).

Lemma A.2

For every $A\subset \Omega$, $|A|\le C(\Omega ,s) \text {cap}_s(A,\Omega ),$ where $C(\Omega ,s)$ is the Poincaré’s constant in $H_0^s(\Omega )$.

Proof

For every $\varepsilon >0$, there exists a function $u_{\varepsilon }\in H_0^s(\Omega )$ such that $u_{\varepsilon }\ge 1$ a.e. in a neighborhood of A and

$$\begin{aligned}{}[u_{\varepsilon }]_s^2 \le \text {cap}_s(A,\Omega )+\varepsilon . \end{aligned}$$

On the other hand, by Poincaré’s inequality,

$$\begin{aligned} |A|=\int _{A}1 \, dx \le \int _{{\mathbb {R}}^n} u_{\varepsilon }^2 \, dx \le C(\Omega ,s)[u_{\varepsilon }]_s^2 \le C(\Omega ,s)\left( \text {cap}_s(A,\Omega )+\varepsilon \right) . \end{aligned}$$

Take the limit $\varepsilon \downarrow 0$ to obtain the result. $\square$

For every $A\in {\mathcal {A}}_s(\Omega )$, we will show that $A=\{u_A^s>0\}$ in the sense of $\text {cap}_s(\cdot , \Omega )$. To prove this aim, we need some previous results which are modifications from [11, Lemma 2.1] and [12, Proposition 5.5].

Lemma A.3

Let $A\in {\mathcal {A}}_s(\Omega )$, Then, there exists an increasing sequence $\{v_k\}_{k\in {\mathbb {N}}}\subset H_0^s(\Omega )$ of nonnegative functions, such that $\sup _{k\in {\mathbb {N}}} v_k=1_A$ s-q.e. on $\Omega$.

We omit the proof since it is completely analogous to that of [11, Lemma 2.1].

We prove a density result in $H_0^s(A)$, for $A\in {\mathcal {A}}_s(\Omega )$, which is similar to [12, Proposition 5.5].

Lemma A.4

Let $A\in {\mathcal {A}}_s(\Omega )$. Then, $\{\varphi u_A^s :\varphi \in C_c^{\infty }(\Omega ) \}$ is dense in $H_0^s(A)$.

Proof

In order to prove the lemma, it is sufficient to see that we can approximate any nonnegative function $w\in H_0^s(A)$ with $(-\Delta )^s w\in L^{\infty }(\Omega )$, since $L^{\infty }(\Omega )$ is dense in $H^{-s}(\Omega )$ and $w=w^+ -w^-$. Indeed, for an arbitrary function $w\in H_0^s(\Omega )$, we know that $(-\Delta )^s w=:f \in H^{-s}(\Omega )$.

Denote by $f:=(-\Delta )^s w$. Then,

$$\begin{aligned} (-\Delta )^s w \le \Vert f\Vert _{L^{\infty }(\Omega )} = \Vert f\Vert _{L^{\infty }(\Omega )} (-\Delta )^s u_A^s \quad \text{ in } A. \end{aligned}$$

By comparison, we obtain $0\le w\le c u_A^s$, where $c:=\Vert f\Vert _{L^{\infty }(\Omega )}$.

For every $\varepsilon >0$, consider $(w-c\varepsilon )^+ \in H_0^s(\Omega )$. Thus,

$$\begin{aligned} \{ (w-c\varepsilon )^+>0\} \subset \{ u_A^s >\varepsilon \}. \end{aligned}$$

(A.1)

Notice that $u_A^s \in L^{\infty }(\Omega )$ by [13, Theorem 4.1]. Observe that, using (A.1), $\varepsilon <u_A^s \le \Vert u_A^s\Vert _{L^{\infty }(\Omega )}$ in $\{(w-c\varepsilon )^+>0\}$. Then, the function $\frac{(w-c\varepsilon )^+}{u_A^s}$ belongs to $H_0^s(\Omega )$. So, there exists a sequence $\{\varphi _k^{\varepsilon }\}_{k\in {\mathbb {N}}} \subset C_c^{\infty }(\Omega )$ such that $\varphi _k^{\varepsilon } \rightarrow \frac{(w-c\varepsilon )^+}{u_A^s}$ strongly in $H_0^s(\Omega )$, when $k\rightarrow \infty$. Therefore, $\varphi _k^{\varepsilon } u_A^s \rightarrow (w-c\varepsilon )^+$ strongly in $H_0^s(\Omega )$, when $k\rightarrow \infty$.

On the other hand, $(w-c\varepsilon )^+\rightarrow w$ strongly in $H_0^s(\Omega )$, when $\varepsilon \downarrow 0$.

Consequently, by a diagonal argument, there exist subsequences $\varepsilon _{j}\downarrow 0$ and $\{ \varphi _{k_j}^{\varepsilon _{j}}\}_{j\in {\mathbb {N}}} \subset C_c^{\infty }(\Omega )$ such that $\varphi _{k_j}^{\varepsilon _j} u_A^s \rightarrow w$ strongly in $H_0^s(\Omega )$.$\square$

The following proposition is an essential component to relate domains and functions. It also contributes to the proofs of principal results Theorems 2.6 and 2.8.

Proposition A.5

Let $A \in {\mathcal {A}}_s(\Omega )$. Then, $A=\{u_A^s>0\}$ in sense of $\text {cap}_s(\cdot , \Omega )$. That is, $\text {cap}_s(A \triangle \{u_A^s>0\}, \Omega )=0$.

Proof

It is clear that $u_A^s=0$ s-q.e. on ${\mathbb {R}}^n \setminus A$. So, $\{u_A^s>0\} \subset A$.

To see $A\subset \{u_A^s>0 \}$, we use the previous lemmas.

By Lemma A.3, there exists an increasing sequence $\{v_k\}_{k\in {\mathbb {N}}}\subset H_0^s(\Omega )$ of nonnegative functions, such that $\sup _{k\in {\mathbb {N}}} v_k=1_A$ s-q.e. on $\Omega$.

For every $v_k$, by Lemma A.4, there exists a sequence $\{\varphi _{j}^k\}_{j\in {\mathbb {N}}}\in C_c^{\infty }(\Omega )$ such that $\varphi _{j}^k u_A^s \rightarrow v_k$ strongly in $H_0^s(\Omega )$ and s-q.e., when $j\rightarrow \infty$. Since $\varphi _{j}^k u_A^s=0$ s-q.e. in $\{u_A^s=0\}$, then $v_k=0$ s-q.e. in $\{u_A^s=0\}$. Therefore, $1_A=0$ s-q.e. in $\{u_A^s=0\}$, which implies $A\subset \{u_A^s>0\}$.$\square$

Now, we prove a key estimate used in Sect. 4, which is a simply remark following the proof of [14, Proposition 2.2]. Notice that we are interested in finding a positive constant connecting in some sense $\text {cap}_s(\cdot , \Omega )$ and $\text {cap}_1(\cdot ,\Omega )$. But, we also want that this constant does not depend on s. As our goal in Sect. 4 is related to the limit case $s\uparrow 1$, we can assume $0<\varepsilon _0<s<1$ for some $\varepsilon _0$ and that will be enough to obtain this desired and independent constant.

As we said before, the proof of next lemma follows [14, Proposition 2.2] and, despite the similarity, it is included since we want to analyze how the constant depends on s.

Lemma A.6

Let $\varepsilon _0>0$ and $\varepsilon _0<s <1$. Then, there exits a constant $C>0$ such that for every $u\in H_0^1(\Omega )$

$$\begin{aligned} (1-s)[u]_s^2 \le C \Vert \nabla u\Vert _{L^2(\Omega )}^2. \end{aligned}$$

and $C=C(\Omega ,n,\varepsilon _0)$ does not depend on s.

Proof

Let $u\in C_c^{\infty }(\Omega )$, we split $[u]_s^2$ into two pieces.

For the first part, use the change of variable $z=y-x$ and observe that for $z\in B_1(0)\setminus \{0\}$ and $\varphi (t):=u(x+tz)$ for $t\in [0,1]$ we estimate

$$\begin{aligned} \frac{|u(x+z)-u(x)|}{|z|}=\frac{\left| \int _0^1 \varphi '(t) \, dt \right| }{|z|}= \frac{\left| \int _0^1 \nabla u(x+tz)\cdot z \, dt \right| }{|z|}\le \int _0^1 | \nabla u(x+tz)|\, dt. \end{aligned}$$

Now, use the previous remark and Jensen’s inequality to obtain

$$\begin{aligned} \int _{{\mathbb {R}}^n}\int _{{\mathbb {R}}^n \cap \{|y-x|<1\}} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} \, dx dy&= \int _{{\mathbb {R}}^n}\int _{B_1(0)} \frac{|u(x)-u(z+x)|^2}{|z|^{n+2s}} \, dz dx\\&= \int _{{\mathbb {R}}^n}\int _{B_1(0)} \frac{|u(x)-u(z+x)|^2}{|z|^{2}|z|^{n+2(s-1)}} \, dz dx\\&\le \int _{{\mathbb {R}}^n}\int _{B_1(0)} \left( \int _0^1\frac{|\nabla u(x+tz)|}{|z|^{\frac{n}{2}+s-1}} \, dt\right) ^2 dz dx\\&\le \int _{B_1(0)} \frac{1}{|z|^{n+2(s-1)}} \int _0^1 \Vert \nabla u\Vert _{L^2(\Omega )}^2 dt dz \\&\le \Vert \nabla u\Vert _{L^2(\Omega )}^2\int _{B_1(0)} \frac{1}{|z|^{n+2(s-1)}} dz \\&=\frac{|B_1(0)|}{2(1-s)}\Vert \nabla u\Vert _{L^2(\Omega )}. \end{aligned}$$

For the remaining part, use $|a-b|^2\le 2(a^2+b^2)$ and easily follows

$$\begin{aligned} \int _{{\mathbb {R}}^n}\int _{{\mathbb {R}}^n \cap \{|y-x|\ge 1\}} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} \, dx dy&\le 2\int _{{\mathbb {R}}^n}\int _{{\mathbb {R}}^n \cap \{|y-x|\ge 1\}} \frac{|u(x)|^2+|u(y)|^2}{|x-y|^{n+2s}} \, dx dy\\&\le 4 \int _{{\mathbb {R}}^n}\int _{{\mathbb {R}}^n \cap \{|y-x|\ge 1\}} \frac{|u(x)|^2}{|x-y|^{n+2s}} \, dx dy\\&\le \int _{{\mathbb {R}}^n}|u(x)|^2\left( \int _{ \{|z|\ge 1\}} \frac{1}{|z|^{n+2s}} dz\right) \, dx \\&=\frac{|B_1(0)|}{2s}\Vert u\Vert _{L^2(\Omega )} \\&\le \frac{|B_1(0)|}{2\varepsilon _0}C_1(\Omega ,n) \Vert \nabla u\Vert _{L^2(\Omega )}, \end{aligned}$$

where $C_1(\Omega ,n)$ is the constant of classical Poincaré’s inequality in $H_0^1(\Omega )$.

Then, put together the two estimates to conclude

$$\begin{aligned} (1-s)[u]_s^2&\le (1-s) \left( \frac{C_1(\Omega ,n)}{2\varepsilon _0}+\frac{1}{2(1-s)}\right) |B_1(0)|\Vert \nabla u\Vert _{L^2(\Omega )}\\&\le \left( \frac{C_1(\Omega ,n)}{2\varepsilon _0}+\frac{1}{2}\right) |B_1(0)|\Vert \nabla u\Vert _{L^2(\Omega )}\\&= C(\Omega ,n,\varepsilon _0) \Vert \nabla u\Vert _{L^2(\Omega )}. \end{aligned}$$

$\square$

Automatically, we obtain an estimate relating the s-capacity and the 1-capacity.

Corollary A.7

Let $\varepsilon _0>0$ and $\varepsilon _0<s <1$. Then, there exits a constant $C>0$ such that for every $A\subset \Omega$

$$\begin{aligned} (1-s)\text {cap}_s(A,\Omega ) \le C \text {cap}_1(A,\Omega ), \end{aligned}$$

and $C=C(\Omega ,n,\varepsilon _0)$ does not depend on s.

We deduce other useful remark from Lemma A.6: Every 1-quasi open set is also an s-quasi-open, for $0<s<1$.

Remark A.8

For every $0<s<1$, ${\mathcal {A}}_1(\Omega )\subset {\mathcal {A}}_s(\Omega )$. Moreover, if $0<s<t\le 1$, then ${\mathcal {A}}_t(\Omega ) \subset {\mathcal {A}}_s(\Omega )$.

Proof

Let $A\in {\mathcal {A}}_1(\Omega )$. There exists a decreasing sequence of open sets $\{G_k\}_{k\in {\mathbb {N}}}$ such that $A\cup G_k$ is open and $\text {cap}_1(G_k,\Omega )\rightarrow 0$.

Let $0<s<1$. By Corollary A.7, $\text {cap}_s(G_k,\Omega )\rightarrow 0$. Then, $A\in {\mathcal {A}}_s(\Omega )$.

To prove ${\mathcal {A}}_t(\Omega ) \subset {\mathcal {A}}_s(\Omega )$ for $0<s<t\le 1$, use definitions of capacity and [14, Proposition 2.1].$\square$