Minimization of the ground state of the mixture of two conducting materials in a small contrast regime

TL;DR

This paper analyzes how to optimally distribute two conducting materials with similar conductivities to minimize the first eigenvalue of an elliptic operator, providing asymptotic expansions and numerical results.

Contribution

It offers a complete asymptotic expansion of the first eigenvalue in the low contrast regime and introduces a relaxation method for optimization.

Findings

Asymptotic expansion of the first eigenvalue for small conductivity contrast

Numerical simulations in 2D and 3D demonstrating the approach

Optimization of material distribution to minimize eigenvalues

Abstract

We consider the problem of distributing two conducting materials with a prescribed volume ratio in a given domain so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions. The gap between the two conductivities is assumed to be small (low contrast regime). For any geometrical configuration of the mixture, we provide a complete asymptotic expansion of the first eigenvalue. We then consider a relaxation approach to minimize the second order approximation with respect to the mixture. We present numerical simulations in dimensions two and three.