In this very interesting paper the authors prove existence of an optimal partition for the multiphase shape optimization problem which consists of minimizing the sum of the first Robin Laplacian eigenvalue of \(k\) mutually disjoint open sets which have a \(\mathcal{H}^{d-1}\)-countably rectifiable boundary and are contained in a given set \(D\) in \(\mathbb{R}^d\). More specifically, given a open bounded set \(D\subset\mathbb{R}^d\) with Lipschitz boundary, one considers the problem of minimizing \[ \inf\left\{\sum_{i=1}^k\lambda_1(\Omega_i,\beta) \;: \;(\Omega_1,\ldots,\Omega_k)\in \mathcal{A}(D)\right\}, \] where \(\mathcal{A}(D)\) is a class of \(k\)-tuples of open domains contained in \(D\) described below, and \(\lambda_1(\Omega,\beta)\) is the first Robin Laplacian eigenvalue, for a fixed parameter \(\beta>0\). When \(\Omega\) is sufficiently smooth, \[ \lambda_1(\Omega,\beta)=\inf_{u\in H^1(\Omega)\setminus \{0\}}\frac{\int_{\Omega}|\nabla u|^2dx+\beta\int_{\partial\Omega}u^2d\mathcal{H}^{d-1}}{\int_{\Omega}u^2dx}. \] The authors prove existence of optimal partitions in the class \[ \begin{aligned} \mathcal{A}(D)=\Big\{&(\Omega_1,\ldots,\Omega_k)\;: \;\Omega_i\subset D, \Omega_i\cap\Omega_j=\emptyset \text{ for } i\neq j, \Omega \text{ is open, } \partial\Omega_i \text{ is }\\ &\mathcal{H}^{d-1}-\text{countably rectifiable with } \mathcal{H}^{d-1}(\partial\Omega_i)<\infty\Big\}. \end{aligned} \] Even though the sets \(\Omega_i\) need not be Lipschitz, the definition of first Robin eigenvalue can be extended in this setting. The main result of this paper proves existence of a solution for the optimal partition problem above for the generalized first Robin eigenvalue, in the class \(\mathcal{A}(D)\).
The main ingredient of the proof is a relaxed formulation of the problem, which is a minimization on functions rather than sets, where the space of special functions of bounded variation, \(SBV(\mathbb{R}^d)\), is crucial. The authors consider \[ \begin{aligned} \mathcal{F}(D)=\Big\{(u_1,\ldots,u_k)& \in (SBV^{1/2}(\mathbb{R}^d))^k \;: \;\text{supp}(u_i)\subset\overline{D}, u_i\geq 0, \\ &u_i\cdot u_j=0 \text{ in } D\Big\}, \end{aligned} \] where \(SBV^{1/2}(\mathbb{R}^d)\) is the space of nonnegative functions \(u\in L^2(\mathbb{R}^d)\) with \(u^2\in SBV(\mathbb{R}^d)\). Denoting with \(J_{u_i}\) the jump set of \(u_i\), the relaxed problem considered is the minimization of \[ \sum_{i=1}^k \frac{\int_{\mathbb{R}^d}|\nabla u_i|^2dx+\beta\int_{J_{u_i}}((u_i^+)^2+(u_i^-)^2)d\mathcal{H}^{d-1}}{\int_{\mathbb{R}^d}u_i^2dx}, \] for \((u_1,\ldots,u_k)\in\mathcal{F}(D)\).
The authors first prove existence of a solution of the relaxed problem, which follows from the compactness and lower semicontinuity in \(SBV^{1/2}(\mathbb{R}^d)\). Next, they are able to prove upper and lower bounds on the supports of \(u_i\), which lead to the proof that \(\mathcal{H}^{d-1}(J_{u_i})<\infty\) and \(u_i\in SBV(\mathbb{R}^d)\). The crucial step in the proof of their main result is the proof that the jump sets \(J_{u_i}\) are essentially closed in \(D\). The key ingredients in this part are a uniform density estimate from below for the supports of \(u_i\) (obtained by applying the Faber-Krahn inequality for the first Robin Laplacian eigenvalue), the local isoperimetric inequality, and the fact that \(u_i\) is an almost-quasi minimizer for the Mumford-Shah functional “well inside” its support. Finally, the authors are able to prove that if \(\Omega_i\) is the connected component of \(\mathbb{R}^d\setminus\overline{J_{u_i}}\) where \(u_i\) does not vanish, then \((\Omega_1,\ldots,\Omega_k)\in \mathcal{A}(D)\) and solve the main problem.