Calculus of Variations and Geometric Measure Theory

B. Bogosel - B. Velichkov

A multiphase shape optimization problem for eigenvalues: qualitative study and numerical results

created by velichkov on 03 Dec 2013
modified on 21 Apr 2018


Published Paper

Inserted: 3 dec 2013
Last Updated: 21 apr 2018

Journal: SIAM Numer. Anal.
Year: 2013


We consider the multiphase shape optimization problem

$\min\Big\{\sum_{i=1}^h\big(\lambda_1(\Omega_i)+c \vert
\Omega_i \vert
\big):\ \Omega_i\ \hbox{open},\ \Omega_i\subset D,\ \Omega_i\cap\Omega_j=\emptyset \Big\},$

where $c>0$ is a given constant and $D\subset\mathbb{R}^2$ is a bounded open set with Lipschitz boundary. We give some new results concerning the qualitative properties of the optimal sets and the regularity of the corresponding eigenfunctions. We also provide some numerical results for the optimal configuration.

Keywords: monotonicity formula, shape optimization, eigenvalues, multiphase, optimal partition