# Optimizing the first Dirichlet eigenvalue of the Laplacian with an obstacle

## TL;DR

This paper investigates how to optimally place a connected obstacle within a domain to maximize the first Dirichlet eigenvalue, analyzing properties of solutions and specific cases like disks and rings.

## Contribution

It introduces a new approach using outer Minkowski content for perimeter, proving key properties and analyzing symmetry of solutions in specific geometries.

## Key findings

- Maximizers exhibit certain qualitative properties.
- Symmetry results are established for specific domains.
- Outer Minkowski content is shown to be a suitable perimeter measure.

## Abstract

Inside a fixed bounded domain $\Omega$ of the plane, we look for the best compact connected set $K$, of given perimeter, in order to maximize the first Dirichlet eigenvalue $\lambda_1(\Omega\setminus K)$. We discuss some of the qualitative properties of the maximizers, moving toward existence, regularity and geometry. Then we study the problem in specific domains: disks, rings, and, more generally, disks with convex holes. In these situations, we prove symmetry and, in some cases non symmetry results, identifying the solution. We choose to work with the outer Minkowski content as the "good" notion of perimeter. Therefore, we are led to prove some new properties for it as its lower semicontinuity with respect to the Hausdorff convergence and the fact that the outer Minkowski content is equal to the Hausdorff lower semicontinuous envelope of the classical perimeter.