Regularity of the minimizers in the composite membrane problem in R^2

TL;DR

This paper proves that the boundary of the optimal subset in the composite membrane problem in the plane is analytic, establishing regularity properties of minimizers in this eigenvalue optimization problem.

Contribution

It demonstrates the analyticity of the boundary of minimizers in the composite membrane problem in R^2, a significant regularity result for this class of eigenvalue problems.

Findings

Boundary of minimizers is analytic.

Minimizers exist with regular boundaries.

Regularity results apply specifically in R^2.

Abstract

We study the regularity of minimizers to the composite membrane problem in the plane (ie given a domain omega and a positive number A, smaller than the measure of omega, minimize the first Dirichlet eigenvalue for the Schrodinger operator with potential equal to a fixed multiple of the characteristic function of a subset D of omega, with measure A). We show that for minimizers, the boundary of D is analytic.