On the Polygonal Faber-Krahn Inequality

Beniamin Bogosel; Dorin Bucur

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arXiv · 2203.16409

On the Polygonal Faber-Krahn Inequality

Beniamin Bogosel·Dorin Bucur

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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.

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BibTeX RIS

Beniamin Bogosel, Dorin Bucur. 2022-03-30. On the Polygonal Faber-Krahn Inequality. https://arxiv.org/abs/2203.16409

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