Regularity of all minimizers of a class of spectral partition problems

To Sandro Salsa, with admiration and gratitude.

1.   Introduction

Let $\Omega \subset {\mathbb{R}}^N$ be a smooth bounded domain, $m\geq 2$ an integer and $k_1, \ldots, k_m\in {\mathbb{N}}$. Consider the following optimal partition problem: among all $m$-tuples of open disjoint subsets $\omega_1, \dots, \omega_m$ of $\Omega$, belonging to an admissible class, find those that minimize the functional

where $\lambda_i(\omega)$ is the $i$-th eigenvalue of $\omega$ with Dirichlet boundary conditions. Here $F$ and $\varphi_i$ are given functions which satisfy certain monotonicity and coercivity assumptions. The aim of this paper is to show that not only problems of this form have a regular solution, but also that any solution is regular. Examples of functionals that fall in the scope of our results are

and combinations of these functionals.

Optimal partition problems are a particular case of a shape optimization problem that appears quite naturally in engineering, where a cost functional defined on a structure made of several materials is being optimized (each material corresponds to a set of the partition).

The problem of existence and regularity of optimal shapes for spectral costs (meaning cost functionals that depend on the spectrum of an operator set in a specific member of the partition) has been addressed by many authors. They are connected with the study of nodal sets of eigenfunctions of Schrödinger operators ^[3,4,5,17], monotonicity formulas ^{[2,9,14,18,23]} and nonlinear systems of partial differential equations with strong competition terms ^{[9,11,12,13,14,17,20,22,24]}. Moreover, these problems provide examples of monotone functionals which are lower-semicontinuous with respect to the weak $\gamma$-convergence, where existence results of a relaxed formulation (partitions of quasi-open sets) can be achieved by direct methods ^[7,8]. Alternative methods typically involve penalization arguments (see for instance ^{[6,17,18,20,24]}).

The main goal of this paper is to characterize and prove regularity of all possible minimal partitions of problem (1.3) and their eigenfucntions.

1.1.   Open partitions

We contextualize our results by introducing a first natural formulation of the problem. For a given $m \geq 2$, consider the set of open partitions of $\Omega$ in $m$ disjoint subsets, denoted by

Observe that, according to this definition, a partition is not necessarily exhaustive, meaning that possibly $\cup_i \omega_i \subsetneqq \Omega$. To any element $\omega$ of a partition we associate the corresponding eigenvalues of the Laplacian with zero Dirichlet boundary condition $\lambda_1(\omega) \leq \lambda_2(\omega) \leq \dots$, counting multiplicity. It is well-known that these eigenvalues are the critical levels of the Rayleigh quotient

where $H^1_0(\omega)$ is the closure of the subset of $H^1(\Omega)$ functions whose support is contained in $\omega$. A characterization of eigenvalues, which takes naturally into account their multiplicity and is also better suited for our purpose, is given by the Courant-Fisher-Weyl formula, which states that for any $j \geq 1$

where $M$ is any linear subset of $H^1_0(\omega)$ of dimension $j$.

1.2.   Cost functional

We introduce a general class of cost functional for the optimal partition problem. Let $F \in C^1({\mathbb{R}}^m; {\mathbb{R}})$ and, for any $i = 1, \dots, m$, $\varphi_i \in C^1(({\mathbb{R}}^+)^{k_i}; {\mathbb{R}})$, functions that verify the following assumptions.

(H1) Monotonicity: for every $i = 1, \ldots, m$,

(H2) Coercivity: for every $i = 1, \ldots, m$,

(H3) Symmetry, for every $i = 1, \ldots, m$

We consider the following problem: among all partition $(\omega_1, \dots, \omega_m) \in {\mathcal{P}}_m(\Omega)$, find

The goal here is to show that a solution, an optimal partition, exists and also to establish some of its qualitative properties, such as the regularity of the associated eigenfunctions, topological properties of the partitions and the structure of their boundary.

Although this first formulation has a very natural appeal, it comes with an apparent incompatibility between the structure of the set of solutions ${\mathcal{P}}_m(\Omega)$ and the minimization problem. Indeed it does not seem easy to endow the set of the open partitions ${\mathcal{P}}_m(\Omega)$ with a topology that allows any compactness results on sequences of minimizers of the cost functional. There are many ways to circumvent this issue (see for instance ^[6,8,17]), usually by considering a relaxed version of the original problem.

1.3.   Measurable partitions

We adopt here the framework of ^[17], see also ^[15], in that we reformulate our problem in the context of measurable sets. For this reason we extend our notion of partition and consider the set of measurable partitions of $\Omega$ in $m$ almost-disjoint subsets, denoted by

where $|\cdot|$ is the Lebesgue measure. Correspondingly, for any $\omega \subset {\mathbb{R}}^N$ measurable (with non-empty interior) we define the Sobolev-like set

and we introduce the generalized eigenvalues of $\omega$ as

They form a nondecreasing sequence which is associated to an $L^2$–orthonormal sequence of eigenfunctions $\{\phi_j\}_{j\in {\mathbb{N}}}$, which satisfy $-\Delta \phi_j = \widetilde \lambda_j(\omega) \phi_j$ in the weak sense

and belong to $L^\infty(\Omega)$ (see [15,Section 2]).

Remark. The notions of classical eigenvalue $\lambda_k$ and generalized eigenvalue $\widetilde{\lambda}_k$ differ in general, even for Lipschitz sets. Indeed, there are open sets $\Omega \subset {\mathbb{R}}^N$, such that $\lambda_k(\Omega) \neq \widetilde{\lambda}_k(\Omega)$ for some $k$ (in general we have $\widetilde{\lambda}_k(\Omega) \leq \lambda_k(\Omega)$). Taking for instance $\Omega = B_1(0) \setminus \{x_1 = 0\}$, then one easily verifies that $\lambda_1(\Omega) = \lambda_2(\Omega) = \lambda_2(B_1(0))$, while $\widetilde{\lambda}_k(\Omega) = \lambda_k(B_1(0))$ for any $k \in {\mathbb{N}}$. On the other hand, if $\Omega$ has smooth boundary (for instance, $\Omega$ enjoys an exterior cone condition), then the two notions coincide. See ^[15] for a more in depth discussion on this subject.

We can finally introduce a suitable relaxed formulation of the minimization problem: among all partition $(\omega_1, \dots, \omega_m) \in \widetilde{{\mathcal{P}}}_m(\Omega)$, find

We state a general existence theorem for the solutions of this problem

Theorem (^[20]). The optimal partition problem (1.3) coincides with (1.2) and admits an open regular solution $(\omega_{1}, \ldots, \omega_{m})\in {\mathcal{P}}_m(\Omega)$ This partition is that $\widetilde \lambda_j(\omega_i) = \lambda_j(\omega_i)$ for every $i = 1, \ldots, m$, $j = 1, \ldots, k_i$.

Moreover, for all $i = 1, \dots, m$ there exist $k_i$ linearly independent eigenfunctions $u_{i, 1}, \ldots, u_{i, k_i} \in \widetilde{H}^1_0(\omega_i)$ associated to $\widetilde{\lambda}_1(\omega_i), \ldots, \widetilde{\lambda}_{k_i}(\omega_i)$ which are Lispschitz continuous, and $O_i$ coincides with the interior of the support of

Finally, for each $i = 1, \ldots, m$ and $j = 1, \dots, k_i$ there exists $a_{i, j} > 0$ such that given $x_0$ in the regular part between the interface between two adjacent sets $O_p, O_q$ of the partition, the free boundary condition is

For the notion of regular partition, we refer to the next statement. This statement is a combination of [20,Theorem 1.2] and the paragraphs after that, see in particular the relaxed formulation (2.4) therein. It should be noted that the case of functionals depending only with first eigenvalues was treated in ^[1,9,10,14], while ^[24] deals with second eigenvalues.

1.4.   Main results

In this paper we strengthen the previous result, by showing that every solution of (1.3) is equivalent to a regular partition, together with universal results regarding the regularity of eigenfunctions and a free boundary condition. In what follows, $\triangle$ denotes the symmetric difference between two sets.

Theorem 1.1. Let $\omega: = (\omega_1, \ldots, \omega_m)\in \widetilde {\mathcal{P}}_m(\Omega)$ be any minimizer of (1.3). Then there exists a unique open partition $O = (O_1, \ldots, O_m) \in {\mathcal{P}}_m(\Omega)$ such that the following holds.

Equivalence:

● subsets coincide up to negligible sets, $|\omega_i \triangle O_i| = 0$ for all $i = 1, \dots, m$;

● they share the same eigenvalues,

● they share the same eigenfunctions, for all $i = 1, \dots, m$ there exist $k_i$ linearly independent eigenfunctions $\phi_{i, 1}, \ldots, \phi_{i, k_i} \in \widetilde{H}^1_0(\omega_i)$ associated to $\widetilde{\lambda}_1(\omega_i), \ldots, \widetilde{\lambda}_{k_i}(\omega_i)$ and $k_i$ linearly independent eigenfunctions $u_{i, 1}, \ldots, u_{i, k_i} \in H^1_0(O_i)$ associated to $\lambda_1(O_i), \ldots, \lambda_{k_i}(O_i)$ such that, for any $j \in 1, \dots, k_i$, we have

Regularity of the sets: the partition $O$ is regular, in the sense that the free-boundary $\Gamma = \Omega \setminus \bigcup_{i = 1}^{m} O_i$ is a rectifiable set and there exist disjoint sets ${\mathcal{R}}, \Sigma \subset \Gamma$ such that

● $\Gamma = {\mathcal{R}} \cup \Sigma$ has Hausdorff dimension at most $N-1$: ${\mathscr{H}}_\mathit{\text{dim}}(\Gamma)\leq N-1$;

● ${\mathcal{R}}$ is relatively open and $\Sigma$ is relatively close in $\Gamma$;

● ${\mathcal{R}}$ is a collection of hypersurfaces of class $C^{1, \alpha}$ (for some $0 < \alpha < 1$). Moreover, each hypersurface separates locally exactly two different elements of the partition: for every $x_0\in {\mathcal{R}}$, there exists $\rho > 0$ and exactly two indices $i\neq j$ such that $x_0\in \partial O_i \cap \partial O_j$, $B_\rho(x_0)\setminus \Gamma = B_{\varepsilon}(x_0) \cap (O_i\cup O_j)$.

● $\Sigma$ is small in the sense that ${\mathscr{H}}_\mathit{\text{dim}} (\Sigma)\leq N-2$;

● if $N = 2$, the set $\Sigma$ is a locally finite set and ${\mathcal{R}}$ consists of a locally finite collection of curves meeting at singular points.

Spectral gap:

● for each $i = 1, \ldots, m$ it holds

In particular, if $\widetilde E_{i, j}(\omega_i) \subset \widetilde H^1_0(\omega_i)$ denotes the eigenspace associated to $\widetilde \lambda_j(\omega_i)$, then the dimension of the linear space $E_{k_i}: = \mathrm{span}\left(\cup_{j = 1}^{k_i} \widetilde E_{i, j}\right)$ is equal to $k_i$.

Regularity of the eigenfunctions:

● for $i = 1, \ldots, m$, we have

in the sense that each eigenfunction has a continuous representative.

Now, for $i = 1, \ldots, m$, let $\phi_{i, 1}, \ldots, \phi_{i, k_{i}}$ be an $L^2$-orthonormal base of $E_{k_i}$, associated respectively to the eigenvalues $\widetilde \lambda_1(\omega_i)\leq \ldots\leq \widetilde \lambda_{k_i}(\omega_i)$. Then

● for each $i = 1, \ldots, m$, $O_i$ is the interior of the support of

● there exists $a_{i, j} > 0$ such that given $x_0 \in {\mathcal{R}}$ and $O_p, O_q$ the two adjacent sets of the partition at $x_0$, then

The coefficients depend only on the eigenvalues of the optimal partition, through the formula

and $a_{i, m} = a_{i, n}$ if $\widetilde \lambda_m(\omega_i) = \widetilde \lambda_n(\omega_i)$.

The proof of Theorem 1.1 is based on a penalization argument. We exploit a regularized version of the relaxed formulation (1.3), involving eigenfunctions rather than eigenvalues, that is better suited to prove the aforementioned properties of optimal sets. Following ^[20], we consider a singular perturbation and approximate these eigenfunctions by minimal solutions of a nonlinear elliptic system with competition terms. This allows to prove the regularity results concerning eigenfunctions and interfaces. By adding an extra term in the energy functional we are able to select any specific minimizer of which we wish to show regularity.

It should be noted that the previous result in the case of functionals depending on first eigenvalues was proved in ^[17]. The case of higher eigenvalues presents many extra difficulties which are related to the unknown multiplicity of the eigenvalues of an optimal partition and to the fact that some eigenfunctions change sign.

1.5.   Examples

Before presenting the proof of our result, we illustrate a couple of concrete applications for specific choices of cost functionals. As a model case, we consider the first function in (1.1), that is the case of

with $p_i > 0$. Then our theory applies to all minimizer of

which are the shown to be regular in the sense of Theorem 1.1. Moreover, the coefficient in (1.4) are given in this case by

The same results also holds for the (suitably renormalized) limit case $p_i \to 0$, where we find

1.6.   A remark about quasi-open sets

In the theory of optimal partitions with respect to spectral costs we can find another class of partitions, given by quasi-open sets, which is in a sense intermediate between the class of open partitions and the class of measurable partitions. It is defined by

with associated problem

We recall briefly the notions of capacity and of quasi-open sets, taken from [7,Chapter 4]. The (2-)capacity of a set is

A set $A$ is said to be quasi-open if for each ${\varepsilon} > 0$ there is an open set $A_{\varepsilon}$ satisfying ${\rm{cap}}(A \triangle A_{\varepsilon}) < {\varepsilon}$, where $\triangle$ denotes the symmetric difference between sets. There is a close relation between quasi-open sets and Sobolev functions. In fact, each $u\in H^1({\mathbb{R}}^N)$ admits a quasi-continuous representative, this meaning that for each ${\varepsilon} > 0$ there is a continuous function $u_{\varepsilon}$ with ${\rm{cap}}(\{u\neq u_{\varepsilon}\}) < {\varepsilon}$. Now $A$ is a quasi-open set if and only if $A = \{u > 0\}$ for a quasi-continuous function $u$. It follows from the definition that, in the setting of this paper, any open minimal partition is a quasi-open minimal partition, and any quasi-open minimal partition is a measurable minimal partition. Then, thanks to Theorem 1.1, we find that the three formulations are actually equivalent (up to negligible sets).

1.7.   Numerical simulations and open problems

We conclude this introduction providing some numerical simulations. They were obtained implementing the construction in Section 3 (see also ^[20]), via a fixed point iteration and a finite element discretization. Specifically, given an initial guess of the eigenfunctions associated to the optimal partition, iteratively we orthogonalize the eigenfunctions (2.3), reorder them according to their respective eigenvalues (2.4), compute the coefficients (2.5), linearize the Euler-Lagrange equation (2.6) of the penalized functional $E_\beta$ and compute new approximations of the eigenfunctions. We do this, increasing the penalization parameter $\beta$, until we reach a fixed point up to a tolerance in the $H^1$ norm. All the simulations were implemented in FreeFem++ ^[16], a free software available at https://freefem.org/.

In Figure 1, a numerical approximation of the optimal partition of the unit ball associated to the cost functionals

and

The two functionals share, numerically, the same optimal partition. The first functional (1.5) is linear, making the algorithm quite efficient in this case.

On the left of Figure 1 is a representation of the eigenfunctions associated to the first eigenvalues of the partition: They highlight the two sets of the partition, which are symmetric semicircles. On the right the second eigenfunctions of the two sets. Observe the additional nodal lines (in connected sets the second eigenfunctions is sign-changing). In this case the strong symmetry of the two functionals seems to translate in the symmetry of their solutions.

In Figure 2, a numerical approximation for

In this case the functional is no more symmetric and the solution too looses symmetry. Nevertheless, observe that the cost functional is scale-invariant.

On the left of Figure 2 is a representation of the eigenfunctions associated to the first eigenvalues of the partition and the two sets $\omega_1$ (in the center) and $\omega_2$ (the two lobes). On the right the second eigenfunctions of the two sets. Numerically, we observe that the second domain is not connected and that $\lambda_1(\omega_2) = \lambda_2(\omega_2)$. This implies that the first eigenvalue of the second subset of the partition has multiplicity two and one can choose the corresponding eigenfunction to have disjoint supports contained in only one of the two lobes at the time. This suggests that there are minimal partitions made of disconnected sets and where the eigenvalues have multiplicity higher than one (unlike the case of cost functions depending on first eigenvalues only). Any choice of eigenfunctions will still verify (1.4) with the same coefficients. Finally we point out that in this example the equi-partition of angles at singular points seems false (unlike in ^[17]), although at the moment we lack any explicit counterexample of this fact.

In Figure 3, a numerical approximation of the optimal partition of the unit ball associated to

This functional gives a rather good approximation of the cost

which does not fall in the scope of our main result, as it is not strictly monotone with respect to $\lambda_1(\omega_1)$. It can be shown that the optimal partition corresponding to this last function is the two third sector of the circle ($\omega_1$) and a third sector of the circle ($\omega_2$). We obtain a rather similar result for (1.7).

On the left the eigenfunctions associated to the first eigenvalues of the partition and on the right the second eigenfunction of the first subset. We point out a seemingly singular point at the center of the ball. According to Theorem 1.1 all the eigenfunctions in the energy functional are regular, and indeed the first eigenfunction of $\omega_1$ is regular, but it appears that as the exponent in the functional becomes larger and larger (the $l^p$ norm approaches the $l^{\infty}$ norm), the first eigenfunctions loses its regularity. This phenomenon will be the object of an upcoming paper.

2.   The penalization argument: an approximate problem

In order to simplify the presentation, we only detail the proof in the case $m = 2$, $k_1 = k_2 = :k\in {\mathbb{N}}$ and $\varphi_1 = \varphi_2 = :\varphi$. The general case follows by the same argument with some simple modifications. In this particular situation, problem (1.3) becomes

where, we recall,

Following ^[20], this problem has at least one open and regular solution in the sense of Theorem 1.1. Here we show that every solution of this problem is equivalent to an open and regular partition, together with some regularity properties of the associated eigenfunctions and a free boundary condition (1.1).

Keeping this in mind, let $(\omega_1, \omega_2)\in \widetilde {\mathcal{P}}_2(\Omega)$ be a solution of (2.1). We denote by $\{(\widetilde \lambda_i(\omega_1), \phi_i)\}_{i \in {\mathbb{N}}}$ and $\{(\widetilde \lambda_i(\omega_2), \psi_i)\}_{i \in {\mathbb{N}}}$ the sequences of nondecreasing generalized eigenvalues (enumerated with multiplicity) and corresponding orthonormal eigenfunctions of the Laplacian in $\widetilde H^1_0(\omega_1)$ and $\widetilde H^1_0(\omega_2)$, respectively. We point out that, even though the eigenfunctions associated to the generalized eigenvalues belong to some Sobolev-like spaces, they are still $H^1_0(\Omega)$ functions. Thus we have the identities

and similarly for $\{\psi_i\}_{i \in {\mathbb{N}}}$. Here $\delta_{ij}$ denotes the Kronecker symbol, that is $\delta_{ij} = 1$ if $i = j$ and $0$ otherwise.

Remark 2.1. We point out that, a priori, the sets $\mathrm{span}\{\phi_1, \ldots, \phi_k\}$ and $\mathrm{span}\{\psi_1, \ldots, \psi_k\}$ may not contain all the eigenfunctions associated to $\widetilde \lambda_k(\omega_1)$ and $\widetilde \lambda_k(\omega_2)$. However, we shall see later on that this is never the case, thanks to the spectral gap property (cfr. Theorem 1.1).

We denote $\boldsymbol{\phi} = (\phi_1, \ldots, \phi_k)$ and $\boldsymbol{\psi} = (\psi_1, \ldots, \psi_k)$ and we introduce two linear subspaces of $L^2(\Omega)$ generated by $\boldsymbol{\phi}$ and $\boldsymbol{\psi}$, together with their orthogonal projections:

Exploiting the orthogonality of $\boldsymbol{\phi}$ and $\boldsymbol{\psi}$ we find that for every $w\in L^2(\Omega)$ the projections are

where $\langle \cdot, \cdot \rangle_{L^2(\Omega)}$ denotes the usual scalar product in $L^2(\Omega)$.

Our aim is to define an energy functional and an associated minimization problem whose solutions are close to those of (2.1). In order to achieve this, we need to introduce a regularized energy functional with two additional terms. For the first one, inspired by ^[20], we relax the disjointedness constraint of the supports of the eigenfunctions $\boldsymbol{\phi}$ and $\boldsymbol{\psi}$ by introducing a competition term between groups of eigenfunctions; this allows to prove the regularity of both the partition and of the eigenfunctions. For the second one, using the projection operators $P^\perp$ and $Q^\perp$, we introduce a penalization that enables us to select the specific minimizer to which the sequence of approximated minimizers converges. This allows to prove that the singular limits are, up to orthogonal transformation, the original eigenfunctions. We need a couple of technical tools before introducing the approximating functionals.

Given ${\bf{u}}, {\bf{v}}\in H^1_0(\Omega; {\mathbb{R}}^k)$, define the $k \times k$ symmetric and positive definite matrices

Observe that for any orthogonal matrix $O\in {\mathcal{O}}_k({\mathbb{R}})$ we have

In particular $M(O {\bf{u}})$ and $M({\bf{u}})$ have the same spectrum.

We extend the function $\varphi:({\mathbb{R}}^+)^k\to {\mathbb{R}}$ to the set of symmetric and positive definite matrices in the following way: given such a matrix $M$, we let

where $\gamma_1, \dots, \gamma_k$ are the (positive) eigenvalues of $M$ (with an abuse of notation, we identify the function acting on the eigenvalues with the function acting on the matrices). Observe that such function is well defined by the symmetry assumption (H3). By definition, we have

Since the original function (acting on the eigenvalues) is smooth and symmetric, we find that $\varphi$ is also a $C^1$ function in the set of symmetric and positive definite matrices. We denote

the (tangent) derivative, in the set of symmetric matrices, of $\varphi$ at $M$ with respect to the component $(i, j)$. Here $E_{ij}$ is the matrix whose component $(i, j)$ is equal to $1$, while all other components are $0$.

Example 2.2. In some notable cases the extended functions can be computed explicitly. For the map $(s_1, \ldots, s_k)\mapsto \left(\sum_{i = 1}^k (s_i)^p \right)^{1/p}$, we have $\varphi(M): = \left(\sum_{i = 1}^k(\gamma_i)^p\right)^{1/p} = \left(\rm{trace}(M^p)\right)^{1/p}$, which coincides with the $p$-Schatten norm of a symmetric and positive definite matrix $M$. For $(s_1, \ldots, s_k)\mapsto \prod_{i = 1}^k s_i$, we have $\varphi(M): = \prod_{i = 1}^k \gamma_i = \det(M)$. These examples are related to (1.1).

Lemma 2.3. ([20,Lemma 3.6]). For every diagonal matrix $D = {\rm{diag}}(\gamma_1, \ldots, \gamma_k)$, we have

We are now ready to introduce the family of approximating functionals. Fix any exponent $1/2 < q < 2^*/4 = N/[2(N-2)^+]$. For $\beta > 0$ we define the $C^1$ energy functional $E_{\beta}:H^1_0(\Omega, {\mathbb{R}}^k)\times H^1_0(\Omega, {\mathbb{R}}^k)\to {\mathbb{R}}$ as

and the least energy level

where

The functional and the set $\Sigma(L^2)$ are invariant under multiplication by orthogonal matrices

and

One should keep in mind that $E_{\beta}$ and $c_{\beta}$ also depend on the vectors of eigenfunctions $\boldsymbol{\phi}$, $\boldsymbol{\psi}$. However, in order to simplify the notation, we will not point out this dependence explicitly.

Lemma 2.4. For each $\beta > 0$ we have

and $c_\beta$ is finite.

Proof. For any $({\bf{u}}, {\bf{v}}) \in \Sigma(L^2)$, take $O_1, O_2\in \mathcal{O}_k({\mathbb{R}})$ in such a way that $O_1M({\bf{u}})O_1^T$, $O_2 N({\bf{v}})O_2^T$ are diagonal and the elements on the diagonal are ordered nondecreasingly. Let ${\bf{\widetilde u}} = O_1 {\bf{u}}$, ${\bf{\widetilde v}} = O_2 {\bf{v}}$. Exploiting the monotonicity of $F$ and $\varphi$, and the invariance of $\Sigma(L^2)$ and $\varphi$ under orthogonal transformations, we find that

Then, recalling that $\beta > 0$, we conclude

We have established that for any $\beta > 0$, the functional $E_\beta$ is bounded from below in $\Sigma(L^2)$. We now show that the infimum is always attained, making the least energy level $c_\beta$ in (2.2) a critical level for $E_\beta$. For notation convenience, let

Proposition 2.5. For any $\beta > 0$, we have the following:

(a) the value $c_{\beta}$ is a critical level for the functional $E_{\beta}$ and ${G_\beta}$ is not empty. Moreover, for every $({\bf{u}}, {\bf{v}}) = ((u_{1}, \ldots, u_{k}), (v_{1}, \ldots, v_{k}))\in M_\beta$, we have

(b) For any $O_1, O_2 \in {\mathcal{O}}_k({\mathbb{R}})$ orthogonal matrices,

Therefore, if $({\bf{u}}, {\bf{v}}) \in {G_\beta}$ we can further assume that it verifies

In particular, $M({\bf{u}}), N({\bf{v}})$ are orthogonal matrices, and

$(c)$ For $i, j = 1, \dots, k$ there exist Lagrange multipliers $\mu_{ij, \beta}, \nu_{ij, \beta} > 0$, and coefficients

such that the components of $({\bf{u}}, {\bf{v}})$ solve the system

In view of the previous result, whenever we refer to ${G_\beta}$ we assume that its functions verify the additional conditions (2.3) and (2.4).

Proof. The result follows by the critical point theory of functionals in Hilbert spaces. First, some preliminary remarks :

1). $\Sigma(L^2)$ is a $C^1$ submanifold of $H^1_0(\Omega, {\mathbb{R}}^k)$ of codimension $k(k+1)/2$ (see [20,Lemma 3.7]).

2). $E_{\beta} : H^1_0(\Omega) \times H^1_0(\Omega) \to {\mathbb{R}}^+$ is a $C^1$ functional and, for any $\boldsymbol{\xi}, \boldsymbol{\eta} \in H^1_0(\Omega, {\mathbb{R}}^k)$, we have

Let $\beta > 0$. By Lemma 2.4 we have $c_\beta > -\infty$. We take a minimizing sequence ${\bf{u}}_n = (u_{1, n}, \ldots, u_{k, n})$, ${\bf{v}}_n = (v_{1, n}, \ldots, v_{k, n})\in \Sigma(L^2)$, $E_{\beta}({\bf{u}}_n, {\bf{v}}_n)\to c_{\beta}$ as $n\to \infty$. By Ekeland's Variational Principle and by property (1) listed above, we can suppose without loss of generality that $E_{\beta}|_{\Sigma(L^2)}'({\bf{u}}_n, {\bf{v}}_n)\to 0$ in $H^{-1}(\Omega, {\mathbb{R}}^k)$. For each $n \in {\mathbb{N}}$ take $O_{1, n}, O_{2, n} \in \mathcal{O}_k({\mathbb{R}})$ such that $O_{1, n} M({\bf{u}}_n) O_{1, n}^T$ and $O_{2, n} M({\bf{v}}_n) O_{2, n}^T$ are diagonal matrices and let

Then $E_\beta({\bf{\widetilde{u}}}_n, {\bf{\widetilde{v}}}_n) = E_\beta({\bf{u}}_n, {\bf{v}}_n)$, ${\bf{\widetilde{u}}}_n, {\bf{\widetilde{v}}}_n \in \Sigma(L^2)$ and

Therefore

for large $n$. Since ${\bf{\widetilde u}}_n, {\bf{\widetilde v}}_n\in \Sigma(L^2)$ then

Combining this information with (H1)–(H2) we deduce that ${\bf{\widetilde u}}_n, {\bf{\widetilde v}}_n$ are bounded sequences in $H^1_0(\Omega, {\mathbb{R}}^k)$, so that (up to subsequence) ${\bf{\widetilde u}}_n\rightharpoonup {\bf{\widetilde u}}$, ${\bf{\widetilde v}}_n\rightharpoonup {\bf{\widetilde v}}$ weakly in $H^1_0(\Omega, {\mathbb{R}}^k)$, strongly in $L^r(\Omega; {\mathbb{R}}^k)$, for every $1\leq r < 2^*$. We can now conclude exactly as in [20,Theorem 3.8], observing that $\frac{\partial}{\partial E_{ij}} \varphi(M({\bf{\widetilde{u}}}_n)) = \frac{\partial}{\partial E_{ij}} \varphi(N({\bf{\widetilde{v}}}_n)) = 0$ for $i\neq j$ (recall Lemma 2.3), that

for some $\delta > 0$ independent from $n$, and that ${\bf{\widetilde u}}_n, {\bf{\widetilde v}}_n$ satisfy (2.6) up to an $\rm{o}_n(1)$ perturbation in $H^{-1}(\Omega, {\mathbb{R}}^k)$. We can then conclude that actually ${\bf{\widetilde u}}_n, {\bf{\widetilde v}}_n$ converge strongly to ${\bf{\widetilde u}}, {\bf{\widetilde v}}$ in $H^1_0(\Omega, {\mathbb{R}}^k)$, which solve (2.6).

3.   Asymptotic limits: Proof of Theorem 1.1

We study the entirety of ${G_\beta}$, the set of critical points of $E_\beta$ at level $c_\beta$, in order to establish its limit when $\beta \to + \infty$. Our main aim is to show that the functions in ${G_\beta}$ are uniformly Hölder continuous in $\beta$. This allows to prove strong convergence in $H^1$ to $(\boldsymbol{\phi}, \boldsymbol{\psi})$, together with the desired regularity results in Theorem 1.1.

3.1.   Uniform bounds

Recall the definition of $\widetilde c$ from (2.1). We start with some easier bounds of the $L^\infty$ and $H^1$ norms.

Proposition 3.1 (Uniform $L^\infty$ and $H^1$ bounds). We have

There exists $C > 0$ independent of $\beta$ such that for any $({\bf{u}}_{\beta}, {\bf{v}}_{\beta}) \in {G_\beta}$ we have

and

Furthermore,

Proof. Since $\phi_i = 0$ a.e. in $\Omega \setminus \omega_1$, $\psi_i = 0$ a.e. in $\Omega \setminus \omega_2$ and $|\omega_1\cap \omega_2| = 0,$ then $\phi_i \cdot \phi_j = 0$ a.e. in $\Omega$ for every $i, j$, hence

Moreover, $P^\perp \phi_i = Q^\perp \psi_i = 0$, as $\phi_i\in L(\boldsymbol{\phi})$ and $\psi_i\in L(\boldsymbol{\psi})$. Therefore, since $\boldsymbol{\phi}, \boldsymbol{\psi}\in \Sigma(L^2)$,

By the monotonicity assumptions on $F$ and $\varphi$, and since $\beta > 0$, we see that

Combining this with Lemma 2.4 and our assumptions of $F$ and $\varphi$, (H1)–(H2), we conclude that there exists a constant $C > 0$ such that

Since $F$ and $\varphi$ are of class $C^1$, by (2.5) we conclude that $1/C\leq a_{i, \beta}, b_{i, \beta}\leq C$ for some $C > 0$.

The only thing left to prove is the $L^\infty$ uniform estimate. Let $i, l\in \{1, \ldots, k\}$. Testing the equation of $u_{i, \beta}$ in (2.6) by $u_{l, \beta}$ yields

and hence $|\mu_{il, p, \beta}|\leq C$ independently of $\beta > 0$. Recall that $P^\perp u_{i, \beta} = u_{i, \beta}-\sum_{j = 1}^k \langle u_{i, \beta}, \phi_j \rangle_{L^2(\Omega)} \phi_j$. By Kato's inequality, we have

By summing up for $i = 1, \dots, k$ and letting $w_{\beta}: = \sum_{i = 1}^k |u_{i, \beta}|\geq 0$, we have

Since $\{w_{\beta}\}$ is uniformly bounded in $L^2(\Omega)$, a Brezis-Kato type argument allows us to conclude. Indeed, assume that $w_{\beta}\in L^{2+\delta}(\Omega)$ for some $\delta\geq 0$. To simplify, we omit the dependent of $w$ on $\beta$ for the remainder of the proof, and consider $N\geq 3$ (otherwise the proof is simpler). Testing (3.2) by $w^{1+\delta}$, using Sobolev and Hölder inequalities, and denoting the best Sobolev constant of $H^1_0(\Omega)\hookrightarrow L^{2^*}(\Omega)$ by $C_S$ we find

Hence there exists a constant $\kappa > 0$ such that

We wish to iterate this inequality in order to obtain a bound for the $L^\infty$ norm of $w$. To this end, let $\{\delta_n\}_n$ be the sequence of positive real numbers such that $\delta_0 = 0$ and $2+\delta_{n+1} = 2^*(2+\delta_n)/2$. We immediately note that $\delta_n\geq (2^*/2)^{n-1}$, thus

As a consequence

and the proof is concluded, as $w = w_{\beta}$ is uniformly bounded in $L^2(\Omega)$.

We proceed our analysis of the family of solutions ${G_\beta}$, focusing this time on stronger compactness results independent of the separation parameter $\beta > 0$. Our goal is to show that it is possible to take the limit as $\beta \to + \infty$ in the family of minimizers of Proposition 2.5. In particular, we want to apply the well-established framework of ^[19,21,25]. We start by some uniform estimates of the $C^{0, \alpha}$ norms of the solutions. Here we scheme through the proof of this result without entering too much into the details since the result, even though expected to hold, is not present in this from in the literature due to a different form of the competition term (cfr. in particular ^[21]).

Proposition 3.2 (Uniform Hölder bounds). For any given $\alpha\in (0, 1)$ there exists a constant $C_{\alpha} > 0$, which may depend on $\alpha$ but not on $\beta$, such that for any $({\bf{u}}_{\beta}, {\bf{v}}_{\beta}) \in {G_\beta}$

The proof is based on a contradiction argument, to which we dedicate the rest of this subsection. Let us assume that, for some $\alpha < 1$, there exists a sequence of solutions $({\bf{u}}_{n}, {\bf{v}}_{n})$ whose $\alpha$-Hölder quotient is not bounded. Since the function $({\bf{u}}_{\beta}, {\bf{v}}_{\beta})$ are smooth for $\beta$ bounded, it follows that necessarily $\beta_n \to + \infty$ and that there exists a sequence of points $(x_n, y_n) \in \bar \Omega \times \Omega$ such that

Letting $r_n = |x_n - y_n| \to 0$, we introduce a new family of functions, which are rescaled versions of $({\bf{u}}_n, {\bf{v}}_n)$. Namely, for any $i = 1, \dots, k$, we let

for $x \in \Omega_n = \frac{\Omega - x_n}{r_n}$. From the definition, we observe that the functions $({\bf{\bar u}}_{n}, {\bf{\bar v}}_{n})$, although they may not be uniformly bounded in $0$ for instance, they have uniformly bounded Hölder quotient of exponent $\alpha$ and moreover for each $n$ there exists a component in $({\bf{\bar u}}_{n}, {\bf{\bar v}}_{n})$ whose oscillation in $B_1$ is equal to 1, that is

Without loss of generality, we assume that

Finally, a direct computation shows that $({\bf{\bar u}}_{n}, {\bf{\bar v}}_{n})$ solves

where the competition parameter is $M_n = \beta_n L_n^{4q-2} r_n^{2\alpha(2q-1)+2}$, and

uniformly in $\bar \Omega_n$ by Proposition 3.1 and since $\phi_j, \psi_j\in L^\infty(\Omega)$ for every $j$.

We now split the rest of the contradiction argument into several lemmas.

Lemma 3.3. The functions in $({\bf{\bar u}}_{n}, {\bf{\bar v}}_{n})$ are uniformly locally bounded in $C^{0, \alpha}(\Omega_n)$. In particular, both

are bounded uniformly.

We adapt the proof of [25,Lemma 6.10] to our present context, which is based an a contradiction argument. We need an integral estimate on the size of the competition term. First of all we observe that if either $\{d_n\}$ or $\{e_n\}$ is unbounded, then necessarily $\Omega_n \to {\mathbb{R}}^n$ by the uniform estimate on the Hölder quotients of the blow-up sequence and since ${\bf{u}}_n = {\bf{v}}_n = 0$ on $\partial \Omega_n$. In particular, we may assume that for any $x \in {\mathbb{R}}^n$ and $R > 0$, $B_R(x) \subset \Omega_n$ for any $n$ sufficiently large.

Lemma 3.4. Assume that either $d_n \to +\infty$ or $e_n \to +\infty$. For any $R > 0$ there exists $C(R) \geq 0$ such that for any $x \in {\mathbb{R}}^N$ and $n$ large enough

Proof. The proof follows verify closely the proof of [25,Lemma 6.10], thus we provide here and a sketch of it in the case $x = 0$. We consider the system (3.4). Multiplying the equation in $\bar{u}_{j, n}$ by $\bar{u}_{j, n}$, integrating by parts in $B_R(0)$ and summing over $j$, we find

Exploiting the uniform Hölder bounds of the blow-up sequence we have

On the other hand, taking also (3.1) into account, we can bound the same integral term from below as follows.

We can reach an analogous conclusion by taking into account the equations satisfied by $\bar{{\bf{v}}}_n$. The conclusion follows by joining the two estimates together with (3.5).

Proof of Lemma 3.3. To prove the result we argue by contradiction, excluding different possibilities for the sequences $\{d_n\}$ and $\{e_n\}$. Specifically we show that the assumption that the one of these two sequences is unbounded is incompatible with the uniform Hölder bounds of the blow-up sequence.

Case 1. We start by excluding the case in which both sequences $d_n$ and $e_n$ are unbounded. Exploiting the uniform bounds of the $C^{0, \alpha}$-seminorm of $\bar{{\bf{u}}}_n$ and $\bar{{\bf{v}}}_n$ we find from Lemma 3.4 that for some $R > 0$ there exists $\bar n$ such that if $n\geq \bar n$ then

In particular, since $d_n, e_n\to +\infty$, we obtain that in this case $M_n \to 0$. Moreover there exists $\Lambda \in {\mathbb{R}}$ such that

uniformly in any compact set of $\Omega_n$. Indeed for any $K \subset {\mathbb{R}}^n$

We introduce now an auxiliary sequence of functions by letting $w_n : = \bar u_{1, n} - \bar u_{1, n}(0)$. The sequence $\{w_n\}$ is uniformly bounded in $C^{0, \alpha}_{ \mathrm{loc}}$ and, up to striking out a subsequence, there exists $w \in C^{0, \alpha}_ \mathrm{loc}({\mathbb{R}}^n)$ such that $w_n \to w$ locally uniformly (and in $C^{0, \gamma}_ \mathrm{loc}({\mathbb{R}}^n)$ for any $\gamma \in (0, \alpha)$), $w$ is globally Hölder continuous of exponent $\alpha < 1$, $w$ is not constant and it solves the equation (for $a_i: = \lim a_{1, n})$

a contradiction. Indeed $w = h + \Lambda/(2n) |x|^2$ where $h$ is harmonic which grows at most quadratically (since $|h(x)| \leq \Lambda/(2n) |x|^2 + |w(x)|$), thus $h$ is a harmonic polynomial of degree at most $2$, but since $w$ is globally Hölder continuous this implies that $h(x) \sim - \Lambda/(2n) |x|^2$ for $|x|\to +\infty$, which is impossible.

Case 2. We exclude the case in which the sequence $\{d_n\}$ is bounded while $\{e_n\}$ is unbounded. Observe that, in this case, the sequence $\{\bar{{\bf{u}}}_{n}\}$ is uniformly bounded in $C^{0, \alpha}_{ \mathrm{loc}}$ and, up to striking out a subsequence, there exists a vector ${\bf{w}} \in C^{0, \alpha}({\mathbb{R}}^n)$ such that $\bar{{\bf{u}}}_n \to {\bf{w}}$ locally uniformly, ${\bf{w}}$ is globally Hölder continuous of exponent $\alpha$, at least its first component $w_1$ is not constant by (3.3). Since at least $w_1$ is not identically $0$ in $B_1$, we can again exploit Lemma 3.4 in order to conclude that there exist $R > 0$ small and constants $C, C' > 0$ such that

Thus $M_n \to 0$ bounded and there exists a constant $\Lambda \geq 0$ such that

uniformly on compact subsets of ${\mathbb{R}}^n$. We conclude that ${\bf{w}}$ has at least one component (its first one) not constant and it solves

a contradiction by applying [21,Lemma A.3] to $|w_i|$.

Case 3. Similarly, we now exclude the possibility $\{d_n\}$ is unbounded, $\{e_n\}$ is bounded and there exists $x \in {\mathbb{R}}^n$ and $C$ such that $e_n(x) \geq C > 0$. Indeed, as in the previous case we find that there exists $C > 0$ such that

thus $M_n \to 0$ and there exists $\Lambda$

Then, by assumption the sequence $\{\bar{{\bf{v}}}_{n}\}$ is uniformly bounded in $C^{0, \alpha}_{ \mathrm{loc}}$ and, up to striking out a subsequence, there exists a vector ${\bf{z}} \in C^{0, \alpha}({\mathbb{R}}^n)$ such that $\bar{{\bf{v}}}_n \to {\bf{z}}$ locally uniformly, ${\bf{z}}$ is globally Hölder continuous of exponent $\alpha$, at least one component of ${\bf{z}}$ is not zero and it solves

which implies that $\Lambda = 0$ (and ${\bf{z}}$ constant). But then letting $w_n : = \bar u_{1, n} - \bar u_{1, n}(0)$, then $\{w_n\}$ is uniformly bounded in $C^{0, \alpha}_{ \mathrm{loc}}$ and, up to striking out a subsequence, there exists $w \in C^{0, \alpha}_ \mathrm{loc}({\mathbb{R}}^n)$ such that $w_n \to w$ locally uniformly, $w$ is globally Hölder continuous of exponent $\alpha < 1$, $w$ is not constant and it solves

in contradiction with the classical theorem by Liouville on entire harmonic functions.

Case 4. Thus we need to exclude the case $\{d_n\}$ is unbounded but $\{e_n\}$ is bounded and $e_n(x) \to 0$ locally uniformly. Again by Lemma 3.4 we find that for any $x \in \Omega_n$ and $R > 0$ we have

Let $\eta \in C^{\infty}_0({\mathbb{R}}^n)$ be any test function. By multiplying the equation in $\bar{u}_{1, n}$ by $\eta$ and integrating by parts we find

for any $R > 0$ such that ${\mathrm{supp\, }} \eta \subset B_R$. Letting once more $w_n : = \bar u_{1, n} - \bar u_{1, n}(0)$, the sequence $\{w_n\}$ is uniformly bounded in $C^{0, \alpha}_{ \mathrm{loc}}$ and, up to striking out a subsequence, there exists $w \in C^{0, \alpha}_ \mathrm{loc}({\mathbb{R}}^n)$ such that $w_n \to w$ locally uniformly (and in $C^{0, \gamma}_ \mathrm{loc}({\mathbb{R}}^n)$ for any $\gamma \in (0, \alpha)$), $w$ is globally Hölder continuous of exponent $\alpha$, $w$ is not constant and it solves the equation

a contradiction.

As a consequence of the previous result, we have that, up to striking out a subsequence, the sequence $\{ ({\bf{\bar u}}_n, {\bf{\bar v}}_n) \}_{n \in {\mathbb{N}}}$ converges in $C^{0, \gamma}_ \mathrm{loc}$ for any $\gamma < \alpha$ to some limiting entire profile $({\bf{\bar u}}, {\bf{\bar v}}) \in C^{0, \alpha}$. Reasoning as in [19,pp. 293–294] we have the following.

Lemma 3.5. The convergence of (a subsequence of) $({\bf{\bar u}}_n, {\bf{\bar v}}_n)$ to its limit $({\bf{\bar u}}, {\bf{\bar v}})$ is also strong in $H^1_ \mathrm{loc}({\mathbb{R}}^N)$.

In order to conclude, we have to analyze the following three possible case: $M_n \to 0$, $M_n$ bounded and $M_n \to \infty$.

Lemma 3.6. There exists $C > 0$ such that $M_n \geq C$ for all $n$.

Proof. Indeed, assume by contradiction that there exists a subsequence in $({\bf{\bar u}}_n, {\bf{\bar v}}_n)$ for which $M_n \to 0$. Then, from the local uniform convergence of $({\bf{\bar u}}_n, {\bf{\bar v}}_n)$ we obtain that the limit $({\bf{\bar u}}, {\bf{\bar v}})$ is made of entire harmonic functions with bounded $C^{0, \alpha}$ semi-norm. Consequently they all must be constant, in contrast with the limit of the oscillation in $B_1$ of the first component.

Lemma 3.7. It must be that $\lim_n M_n = +\infty$.

Proof. We may reason as before, assuming that $M_n \to 1$. We then end up with limiting functions $({\bf{\bar u}}, {\bf{\bar v}})$ which solve

and the conclusion follows as in [20,Claim 2. pag 18].

Finally, let us address the case $M_n \to \infty$. In this case, in order to find a contradiction, we need to ensure the validity of an Almgren-type monotonicity formula for the limit profiles $({\bf{\bar u}}, {\bf{\bar v}})$. To this end, we let first show the following

Lemma 3.8. For any $x \in {\mathbb{R}}^N$ and almost every $r > 0$, the following identity holds

Proof. The proof follows mainly by a direct computation. For easier notation, let us consider the case $x_0 = 0$. Testing the equation in $({\bf{\bar u}}_{n}, {\bf{\bar v}}_{n})$ by $(x\cdot \nabla {\bf{\bar u}}_{n}, x\cdot \nabla {\bf{\bar v}}_{n})$ and summing over $i = 1, \dots, k$, we obtain integrating by parts

We observe that, due to the strong $H^1$ convergence, the right hand side of the previous expression passes to the limit for almost every radius $r > 0$. On the other hand, replacing the equation in the left hand side, we find

We now go back to the equations in $({\bf{\bar u}}_{n}, {\bf{\bar v}}_{n})$. By Kato's inequality we find that there exists a positive constant $C$, independent of $n$, such that

and similarly for $\bar v_{i, n}$. Let $r > 0$ be any fixed radius, we multiply the previous inequality by a smooth cut-function $\eta \in C_0^\infty(B_{3r})$ such that

Integrating by parts yields the estimate

We obtain that

and thus, by Fubini's theorem, for almost every radius $r > 0$ the right hand side in (3.6) vanishes as $n \to +\infty$. Finally, we observe that thanks to the $H^1$ converge of $({\bf{\bar u}}_n, {\bf{\bar v}}_n)$ and the uniform vanishing of $(\varepsilon_{n}, \delta_n)$ (see Eq 3.5), we have

for every radius $r > 0$. The proof follows recollecting the previous observations.

We are in position to conclude the uniform regularity result.

Proof of Proposition 3.2. As of now, we have obtained that, if there is no uniform Hölder bound, then necessarily $M_n \to \infty$. From this point on, the conclusion follows exactly as in [20,Step B. page 19].

3.2.   Conclusion of the proof of Theorem 1.1

From the previous results we can completely characterize the limit profiles as $\beta\to \infty$.

Proposition 3.9 (Limit as $\beta\to \infty$). Let $({\bf{u}}_{\beta}, {\bf{v}}_{\beta}) \in {G_\beta}$. Then

Moreover, there exist ${\bf{u}} = (u_1, \ldots, u_k), {\bf{v}} = (v_1, \ldots, v_k)\in C^{0, \alpha}(\overline{\Omega}; {\mathbb{R}}^k)\cap H^1_0(\Omega, {\mathbb{R}}^k)$ such that, up to subsequence:

1). ${\bf{u}}_{\beta}\to {\bf{u}}$, ${\bf{v}}_{\beta}\to {\bf{v}}$ as $\beta\to +\infty$, strongly in $H^1_0(\Omega, {\mathbb{R}}^k)$ and in $C^{0, \alpha}(\overline{\Omega}, {\mathbb{R}}^k)$ for every $\alpha\in (0, 1)$.

2). $u_{i}\cdot v_{j} = 0$ in $\Omega$ for every $i, j = 1, \ldots, k$, and

3). ${\bf{u}}, {\bf{v}}\in \Sigma(L^2)$;

4). we have

As a consequence we have

Proof. We only sketch the proof of these results, referring to [19,p. 294] for a complete and detailed proof. Recall the uniform bounds in Propositions 3.1 and 3.2. Since $C^{0, \alpha}(\overline \Omega) \hookrightarrow C^{0, \gamma}(\overline \Omega)$ is a compact embedding whenever $0 < \gamma < \alpha < 1$, we have (up to a subsequence)

weakly in $H^1_0(\Omega, {\mathbb{R}}^k)$ and strongly in $C^{0, \alpha}(\overline \Omega, {\mathbb{R}}^k)\cap L^p(\Omega)$ for every $\alpha \in (0, 1)$, $p\in [1, +\infty]$. By combining this information with Proposition 3.1 we have items (2) and (3). By Kato's inequality and the bounds mentioned before, we have the existence of $C > 0$ independent on $\beta$ such that

and the same holds for the equation of $v_{i, \beta}$. Since $\Omega$ is smooth $\partial_\nu |u_{i, \beta}|, \partial_\nu |v_{i, \beta}|\leq 0$ on $\partial \Omega$ and an integration of the previous differential inequality yields

We can deduce (3.7). Moreover, testing the equation of $u_{i, \beta}$ with $u_{i, \beta}-u_{i}$ and the one of $v_{i, \beta}$ with $v_{i, \beta}-v_{i}$ implies that in (3.8) the $H^1_0$–convergence is actually strong, so that (1) is proved. Finally, (4) is a direct consequence of this strong convergence combined with (2.3)–(2.4)

Proposition 3.10. From the family of functions $({\bf{u}}_{\beta}, {\bf{v}}_{\beta})$ in Proposition 2.5 we consider any converging subsequence, and let $({\bf{u}}, {\bf{v}}): = \lim_{\beta\to \infty} ({\bf{u}}_{\beta}, {\bf{v}}_{\beta})$ be any limit profile, as in the previous lemma. Then:

1). regarding the parameters, we have:

2). the limit profiles satisfy

3). for any $x_0 \in {\mathbb{R}}^N$ and $r \in (0, \mathrm{dist}(x_0, \partial \Omega))$, the following identity holds

Proof. The positivity of the coefficients in (3.9) follows directly from Proposition 3.1. Testing the equation of $u_{i, \beta}$ in (2.6) by $u_{j, \beta}$, we see that

as $\beta\to \infty$ by (3.7), and the same for $\nu_{ij, \beta}$. From this follows (1) and (2). As for (3), it follows exactly as in the proof of [20,Corollary 3.16], taking again into account the strong $H^1_0$–convergence of minimizers (Proposition 3.9-(1)) and the vanishing property of the interaction term (3.7).

In order to reach the conclusion of Theorem 1.1, it is convenient to introduce the following definition. Given a measurable set $\omega\subset {\mathbb{R}}^n$, we define $\widetilde \lambda_k(\omega, \boldsymbol{\phi})$ as the $k$-eigenvalue (counting multiplicities) of the operator $-\Delta +P^\perp$ in $\widetilde H^1_0(\omega)$, which can be characterized as

We define $\widetilde \lambda_k(\omega, \boldsymbol{\psi})$ is an analogous way. Clearly, we have

Conclusion of the proof of Theorem 1.1. Let

and $(O_1, O_2): = (\{|{\bf{u}}| > 0 \}, \{|{\bf{v}}| > 0\})$. We recall that ${\bf{u}}$ and ${\bf{v}}$ are continuous functions, thus $O_1$ and $O_2$ are open subsets of $\Omega$. By Proposition 3.9-(4) and inequality (3.10),

for every $i = 1, \ldots, k$. Therefore, using the monotonicity of $F$ and $\varphi$ together with Propositions 3.1, 3.9 and 3.10,

Therefore all inequalities are in fact equalities, $(O_1, O_2)$ is an (open) optimal partition for $c = \widetilde c$, and (by the strict monotonicity of $F$ and $\varphi$) $\lambda_i(O_1) = \lambda_i(O_1, \boldsymbol{\phi})$, $\lambda_i(O_2) = \lambda_i(O_2, \boldsymbol{\psi})$ for every $i = 1, \ldots, k$.

We now claim that $P^\perp u_i = Q^\perp v_i = 0$. Indeed, for $i = 1$:

so that $P^\perp u_1 = 0$. Moreover,

and

hence $P^\perp u_2 = 0$. By iterating this procedure, we obtain $P^\perp u_i$ = 0 for $i = 1, \dots, k$ and, analogously, $Q^\perp v_i = 0$, which proves our claim.

From this we deduce that

and $\lambda_i(\omega_1) = \lambda_i(O_1)$ for $i = 1, \ldots, k$. Moreover ${\bf{u}}\in L(\boldsymbol{\phi})$, ${\bf{v}}\in L(\boldsymbol{\psi})$, that is,

for $M: = (\langle u_i, \phi_j\rangle_{L^2(\Omega)})_{i, j}, N_{ij}: = (\langle v_i, \psi_i\rangle_{L^2(\Omega)})_{i, j}\in {\mathbb{R}}^{k\times k}$ and, since $({\bf{u}}, {\bf{v}}), (\boldsymbol{\phi}, \boldsymbol{\psi})\in \Sigma(L^2)$, then actually $M, N\in {\mathcal{O}}_k({\mathbb{R}})$, being block diagonal matrices:

where the dimension of each block is at most equal to the dimension of the eigenspace of the associated eigenvalue, and each block is itself an orthogonal matrix.

This has many important consequences:

$1).$ In the local Pohozaev identities of Proposition 3.10-(3) we have $P^\perp u_i = Q^\perp v_i = 0$, which corresponds to the statement in [20,Corollary 3.16]. Therefore we are in the exact framework of Sections 3 and 4 of ^[20], which implies by Theorem 2.2 therein that $u_i, v_i$ are Lipschitz continuous, $(O_1, O_2)$ is a regular partition, and, given $x_0$ in the regular part of the free boundary,

where

Since $\varphi$ is symmetric, then $a_i = a_j$ whenever $\lambda_i(\omega_1) = \lambda_j(\omega_1)$, and the same holds true for the coefficients $b_i$. Combining this remark with the orthogonality of the block matrices in (3.12), we deduce that also

Moreover we find that (3.13) does not depend on the starting configuration $\boldsymbol{\varphi}, \boldsymbol{\psi}$. $2).$ Since $M$ and $N$ are invertible, $\phi = M^{-1} {\bf{u}}$ and $\psi = N^{-1} {\bf{v}}$ a.e. in $\Omega$, and since ${\bf{u}}, {\bf{v}}$ are Lipschitz continuous, then each $\phi_i$ and $\psi_i$ has a Lipschitz continuous representative.

$3).$ For a.e. $x\in \Omega$ we find

Therefore we have $O_j \subseteq \omega_j$ up to a set of Lebesgue measure zero, $\lambda_i(O_j) = \widetilde \lambda_i(O_j) \geq \widetilde \lambda_i(\omega_j)$ for $j = 1, 2$, $i = 1, \ldots, k$. Combining this with the strict monotonicity of $F$ and $\varphi$ and (3.11), we obtain the equality between the eigenvalues. Moreover, the regularity results of $(O_1, O_2)$ allow to conclude that $|O_i\triangle \omega_i | = 0$.

We are left to show the spectral gap property, that is, to prove that $\widetilde \lambda_k(\omega_1) < \widetilde \lambda_{k+1}(\omega_1)$. For this purpose, let $E \subset \widetilde H^1_0(\omega_1)$ be the (generalized) eigenspace associated to the eigenvalue $\widetilde \lambda_k(\omega_1)$ and let $\ell \in {\mathbb{N}}$ be the number of eigenvalue of $\omega_1$ that are strictly less than $\widetilde \lambda_k(\omega_1)$. Our goal is to show that

Assume, in view of a contradiction, that $\widetilde \lambda_k(\omega_1) = \widetilde \lambda_{k+1}(\omega_1)$ or, more generally, that

To start off, we apply the previous reasoning to any vector $\boldsymbol{\phi} = (\phi_1, \dots, \phi_{\ell}, \bar \phi_{\ell+1}, \dots, \bar \phi_{k})$ where $\bar \phi_{\ell+1}, \dots, \bar \phi_{k}$ are $k-\ell$ orthonormal functions in $E$. This shows that all the eigenfunctions in $E$ have a Lipschitz representative and that $E$ is made of standard eigenfunctions. In particular, by (3.13), replacing one eigenfunction at the time, for any $\phi_i \perp \phi_j$ in any orthonormal base of $E$ we deduce

on the regular part of the free boundary. Let now $S\subset \overline{\Omega}$ stand for the support of $E$

We claim that, under (3.14), $S$ has a unique connected component. Assume the opposite and pick two normalized functions $\phi', \phi'' \in E$ with disjoint supports (this is possible since $S$ is disconnected, and $\phi', \phi''$ are orthonormal by construction), and consider other $\mathrm{dim}(E)-2$ functions to complete an orthonormal base of $E$. We immediately find a contradiction with (3.15). Hence, up to a change of sign, letting $w: = \phi_{i}-\phi_{j}$ for any $\phi_i \perp \phi_j$ in any orthonormal base of $E$, we find

But then, by Hopf's lemma, we have $w = 0$ that is $\phi_i = \phi_j$, a contradiction. The same reasoning holds true for $\widetilde \lambda_k(\omega_2)$.

Acknowledgments

H. Tavares was supported by the Portuguese government through FCT – Fundação para a Ciência e a Tecnologia, I.P., under the projects PTDC/MAT-PUR/28686/2017, UID/MAT/04561/2013 and UIDB/MAT/04459/2020. H. Tavares would also like to acknowledge the Faculty of Sciences of the University of Lisbon for granting a semestral sabbatical leave, during which part of this work was developed.

This work was partially supported by the project ANR-18-CE40-0013 SHAPO financed by the French Agence Nationale de la Recherche (ANR).

Conflict of interest

The authors declare no conflict of interest.