Multiphase Shape Optimization Problems

TL;DR: In this article, the analysis of multiphase shape optimization problems with Dirichlet Laplacian functions is studied, where each cell is itself a subsolution for a single-phase shape optimization problem, from which properties like finite perimeter, inner density, separation by open sets, absence of triple junction points, etc.

Abstract: This paper is devoted to the analysis of multiphase shape optimization problems, which can formally be written as $\min \{{g}((F_1(\Omega_1),\dots ,F_h(\Omega_h))+ m|\bigcup_{i=1}^h\Omega_i| :\ \Omega_i \subset D,\ \Omega_i\cap \Omega_j =\emptyset\},$ where $D\subseteq \mathbb{R}^d$ is a given bounded open set, $|\Omega_i|$ is the Lebesgue measure of $\Omega_i$, and $m$ is a positive constant. For a large class of such functionals, we analyze qualitative properties of the cells and the interaction between them. Each cell is itself a subsolution for a (single-phase) shape optimization problem, from which we deduce properties like finite perimeter, inner density, separation by open sets, absence of triple junction points, etc. As main examples we consider functionals involving the eigenvalues of the Dirichlet Laplacian of each cell, i.e., $F_i=\lambda_{k_i}$.