Paper detail

Multiphase shape optimization problems

This paper is devoted to the analysis of multiphase shape optimization problems, which can formally be written as $\min\Big\{{g}(F_1(Ω_1),\dots,F_h(Ω_h))+ m\vert\,\bigcup_{i=1}^hΩ_i\vert :\ Ω_i\subset D,\ Ω_i\cap Ω_j =\emptyset\Big\},$ where $D\subset\mathcal{R}^d$ is a given bounded open set, $\vertΩ_i\vert$ is the Lebesgue measure of $Ω_i$ and $m$ is a positive constant. For a large class of such functionals, we analyse qualitative properties of the cells $Ω_i$ 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. $F_i=λ_{k_i}$.