Calculus of Variations and Geometric Measure Theory

D. Bucur - B. Velichkov

Multiphase shape optimization problems

created by velichkov on 24 Mar 2013
modified on 21 Apr 2018


Published Paper

Inserted: 24 mar 2013
Last Updated: 21 apr 2018

Journal: SIAM J. Control Optim.
Year: 2014


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. $F_i=\lambda_{k_i}$.