# Existence and regularity of Faber Krahn minimizers in a Riemannian manifold

## TL;DR

This paper investigates the existence and regularity of minimizers for the first Dirichlet eigenvalue in Riemannian manifolds, extending classical Euclidean results to more general geometric settings.

## Contribution

It establishes existence results in compact manifolds, provides counterexamples in non-compact cases, and develops regularity theory for the minimizers using free boundary problem techniques.

## Key findings

- Existence of minimizers in compact Riemannian manifolds.
- Counterexamples showing non-existence in some non-compact cases.
- Solutions are smooth up to a residual set of codimension 5 or higher.

## Abstract

In this paper, we study the minimization of $\lambda_{1}(\Omega)$, the first Dirichlet eigenvalue of the Laplace-Beltrami operator, within the class of open sets $\Omega$ of fixed volume in a Riemmanian manifold $(M,g)$. In the Euclidian setting (when $(M,g)=(\mathbb{R}^n,e)$), the well-known Faber-Krahn inequality asserts that the solution of such problem is any ball of suitable volume. Even if similar results are known or may be expected for Riemannian manifolds with symmetries, we cannot expect to find explicit solutions for general manifolds $(M,g)$. In this paper we study existence and regularity properties for this spectral shape optimization problem in a Riemannian setting, in a similar fashion as for the isoperimetric problem. We first give an existence result in the context of compact Riemannian manifolds, and we discuss the case of non-compact manifolds by giving a counter-example to existence. We then focus on the regularity theory for this problem, and using the tools coming from the theory of free boundary problems, we show that solutions are smooth up to a possible residual set of co-dimension 5 or higher.