A Multiphase Shape Optimization Problem for Eigenvalues: Qualitative Study and Numerical Results

TL;DR

This paper investigates a multiphase shape optimization problem involving eigenvalues, providing new qualitative insights, regularity results, and numerical solutions for optimal partitions within a bounded domain.

Contribution

The paper introduces new qualitative properties and regularity results for optimal sets in a multiphase eigenvalue optimization problem, along with numerical demonstrations.

Findings

Optimal sets exhibit specific qualitative properties.

Eigenfunctions associated with optimal partitions have regularity features.

Numerical results illustrate the structure of optimal partitions.

Abstract

We consider the multiphase shape optimization problem $$\min\Big\{\sum_{i=1}^h\lambda_1(\Omega_i)+\alpha|\Omega_i|:\ \Omega_i\ \hbox{open},\ \Omega_i\subset D,\ \Omega_i\cap\Omega_j=\emptyset\Big\},$$ where $\alpha>0$ is a given constant and $ D\subset\Bbb{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 numerical results for the optimal partitions.