On the optimization of the Robin eigenvalues in some classes of polygons

TL;DR

This paper investigates how the shape of polygons affects the first Robin Laplacian eigenvalues, establishing optimization results for polygons with a bounded number of sides under perimeter or volume constraints, depending on the sign of the Robin parameter.

Contribution

It provides the first shape optimization results for Robin eigenvalues in classes of polygons with bounded sides, considering both maximization and minimization depending on the Robin parameter sign.

Findings

Shape minimizers for positive Robin parameter are identified within polygon classes.

Shape maximizers for negative Robin parameter are characterized.

Results depend on constraints like perimeter or volume, and on the number of sides.

Abstract

Given the eigenvalue problem for the Laplacian with Robin boundary conditions, (with $\beta\in\R\setminus\{0\}$ the Robin parameter), we consider a shape minimization problem for a function of the first eigenvalues if $\beta>0$ and a shape maximization problem if $\beta<0$. Both problems are settled in a suitable class of generalized polygons with an upper bound on the number of sides, under either perimeter or volume constraint.