Shapes of drums with lowest base frequency under non-isotropic perimeter constraints

TL;DR

This paper investigates the shapes of drums with minimal fundamental frequency under perimeter constraints defined by general norms, establishing existence, uniqueness, convexity, and geometric features of optimal domains.

Contribution

It introduces a comprehensive analysis of perimeter-constrained eigenvalue minimization for arbitrary norms, including conditions for geometric features and stability of near-minimizers.

Findings

Existence and uniqueness of minimizers up to translation.

Optimal domains are convex but may have facets and corners.

Near-minimizers are close to the optimal shape in Hausdorff distance.

Abstract

We study the minimizers of the sum of the principal Dirichlet eigenvalue of the negative Laplacian and the perimeter with respect to a general norm in the class of Jordan domains in the plane. This is equivalent (modulo scaling) to minimizing the said eigenvalue (or the base frequency of a drum of this shape) subject to a hard constraint on the perimeter. We show that, for all norms, a minimizer exists, is unique up to spatial translations and is convex but not necessarily smooth. We give conditions on the norm that characterize the appearance of facets and corners. We also demonstrate that near minimizers have to be close to the optimal ones in the Hausdorff distance. Our motivation for considering this class of variational problems comes from a study of random walks in random environment interacting through the boundary of their support.