The Steklov spectrum of convex polygonal domains I: spectral finiteness

TL;DR

This paper investigates the Steklov eigenvalue problem on convex polygons, revealing spectral finiteness and providing bounds for isospectral domains, advancing understanding of spectral geometry in polygonal shapes.

Contribution

It establishes that almost all convex polygons have finitely many Steklov isospectral non-congruent counterparts and derives explicit bounds for their number.

Findings

Finiteness of Steklov isospectral non-congruent convex polygons.

Explicit upper bounds for the number of such polygons.

Isoperimetric bounds relating eigenvalues to interior angles.

Abstract

We explore the Steklov eigenvalue problem on convex polygons, focusing mainly on the inverse Steklov problem. Our primary finding reveals that, for almost all convex polygonal domains, there exist at most finitely many non-congruent domains with the same Steklov spectrum. Moreover, we obtain explicit upper bounds for the maximum number of mutually Steklov isospectral non-congruent polygonal domains. Along the way, we obtain isoperimetric bounds for the Steklov eigenvalues of a convex polygon in terms of the minimal interior angle of the polygon.