Polygonal Faber-Krahn inequality: Local minimality via validated computing

TL;DR

This paper proves that regular pentagons and hexagons are local minimizers for the first Dirichlet-Laplace eigenvalue among n-gons with fixed area, using validated numerical methods and explicit estimates.

Contribution

It introduces a rigorous computational approach to establish local minimality of regular pentagons and hexagons for a spectral geometric problem.

Findings

Regular pentagons and hexagons are local minimizers.

Validated finite element methods confirm the minimality.

Explicit a priori estimates support the numerical results.

Abstract

The main result of the paper shows that the regular $n$-gon is a local minimizer for the first Dirichlet-Laplace eigenvalue among $n$-gons having fixed area for $n \in \{5,6\}$. The eigenvalue is seen as a function of the coordinates of the vertices in $\Bbb R^{2n}$. Relying on fine regularity results of the first eigenfunction in a convex polygon, an explicit a priori estimate is given for the eigenvalues of the Hessian matrix associated to the discrete problem, whose coefficients involve the solutions of some Poisson equations with singular right hand sides. The a priori estimates, in conjunction with certified finite element approximations of these singular PDEs imply the local minimality for $n \in \{5,6\}$. All computations, including the finite element computations, are realized using interval arithmetic.