Where best to place a Dirichlet condition in an anisotropic membrane?

TL;DR

This paper investigates optimal placement of Dirichlet conditions on an anisotropic membrane to maximize the first eigenvalue, using advanced variational methods and varifold theory to understand the limit distribution of optimal sets.

Contribution

It introduces a novel approach employing varifolds to analyze the asymptotic behavior of optimal Dirichlet set placements in anisotropic eigenvalue problems.

Findings

Characterization of the limit distribution via $mma$-convergence of functionals.

Use of varifolds to preserve local orientation information.

Identification of optimal set configurations as their length tends to infinity.

Abstract

We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with non constant coefficients, over a fixed domain $\Omega$. Dirichlet conditions are imposed along $\partial \Omega$ and, in addition, along a set $\Sigma$ of prescribed length ($1$-dimensional Hausdorff measure). We look for the best shape and position for the supplementary Dirichlet region $\Sigma$ in order to maximize the first eigenvalue. The limit distribution of the optimal sets, as their prescribed length tends to infinity, is characterized via $\Gamma$-convergence of suitable functionals defined over varifolds: the use of varifolds, as opposed to probability measures, allows one to keep track of the local orientation of the optimal sets (which comply with the anisotropy of the problem), and not just of their limit distribution.