On the Polygonal Faber-Krahn Inequality

TL;DR

This paper proves that the conjecture that regular polygons minimize the first eigenvalue of the Dirichlet-Laplace operator among polygons with fixed area can be reduced to finite numerical computations for polygons with five or more sides, and confirms this for pentagons to octagons.

Contribution

It reduces the proof of the polygonal Faber-Krahn inequality to finite certified numerical computations for polygons with at least five sides, and verifies the conjecture for specific cases.

Findings

Confirmed the conjecture for pentagons to octagons through numerical certification.

Reduced the proof of the inequality to a finite number of computational steps.

Established the local minimality of regular polygons via a single numerical computation.

Abstract

It has been conjectured by P\'{o}lya and Szeg\"{o} seventy years ago that the planar set which minimizes the first eigenvalue of the Dirichlet-Laplace operator among polygons with $n$ sides and fixed area is the regular polygon. Despite its apparent simplicity, this result has only been proved for triangles and quadrilaterals. In this paper we prove that for each $n \ge 5$ the proof of the conjecture can be reduced to a finite number of certified numerical computations. Moreover, the local minimality of the regular polygon can be reduced to a single numerical computation. For $n=5, 6,7, 8$ we perform this computation and certify the numerical approximation by finite elements, up to machine errors.