Optimal ground state energy of two-phase conductors

TL;DR

This paper investigates the optimal distribution of two conducting materials to minimize the first eigenvalue of a Dirichlet operator, disproving a previous conjecture in dimensions two and higher.

Contribution

It provides a counterexample to the conjecture that the optimal configuration is always a centered ball of the highest conductivity material in dimensions n ≥ 2.

Findings

The conjecture does not hold in all dimensions n ≥ 2.

Counterexamples show non-central distributions can be optimal.

The optimal configuration depends on the dimension and other factors.

Abstract

Consider the problem of distributing two conducting materials in a ball with fixed proportion in order to minimize the first eigenvalue of a Dirichlet operator. It was conjectured that the optimal distribution consists of putting the material with the highest conductivity in a ball around the center. In this paper, we show that the conjecture is not true for all dimensions $n \geq 2$.