On the boundary of the attainable set of the Dirichlet spectrum

TL;DR

This paper provides an elementary proof that the boundary of the attainable set of the first two Dirichlet eigenvalues has a horizontal tangent at a specific point related to two disjoint balls, clarifying geometric properties of spectral sets.

Contribution

It offers a new elementary proof of the boundary's geometric property for the Dirichlet spectrum's attainable set at a particular point.

Findings

The boundary of the attainable set has a horizontal tangent at the lowest point.

The proof simplifies understanding of spectral set boundaries.

Clarifies geometric structure of eigenvalue pairs for disjoint domains.

Abstract

Denoting by $\mathcal{E}\subseteq \R^2$ the set of the pairs $(\lambda_1(\Omega),\lambda_2(\Omega))$ for all the open sets $\Omega\subseteq\R^N$ with unit measure, and by $\Theta\subseteq\R^N$ the union of two disjoint balls of half measure, we give an elementary proof of the fact that $\partial\E$ has horizontal tangent at its lowest point $(\lambda_1(\Theta),\lambda_2(\Theta))$.