Multiphase shape optimization problems

TL;DR

This paper analyzes multiphase shape optimization problems involving multiple disjoint regions within a domain, focusing on their qualitative properties and interactions, especially when eigenvalues of the Laplacian are involved.

Contribution

It provides a detailed analysis of the qualitative properties of solutions to multiphase shape optimization problems, including eigenvalue-based functionals, and establishes properties like finite perimeter and separation.

Findings

Cells are subsolutions for single-phase problems.

Cells have finite perimeter and are internally dense.

Cells are separated by open sets with no triple junctions.

Abstract

This paper is devoted to the analysis of multiphase shape optimization problems, which can formally be written as $\min\Big\{{g}(F_1(\Omega_1),\dots,F_h(\Omega_h))+ m\vert\,\bigcup_{i=1}^h\Omega_i\vert :\ \Omega_i\subset D,\ \Omega_i\cap \Omega_j =\emptyset\Big\},$ where $D\subset\mathcal{R}^d$ is a given bounded open set, $\vert\Omega_i\vert$ is the Lebesgue measure of $\Omega_i$ and $m$ is a positive constant. For a large class of such functionals, we analyse qualitative properties of the cells $\Omega_i$ and the interaction between them. Each cell is itself 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. $F_i=\lambda_{k_i}$.