Extremal domains and P\'{o}lya-type inequalities for the Robin Laplacian on rectangles and unions of rectangles

TL;DR

This paper investigates eigenvalues of the Robin Laplacian on rectangles and unions of rectangles, establishing Pólya-type inequalities with optimal exponents and characterizing extremal domains for the eigenvalues.

Contribution

It determines the optimal exponents for Pólya-type inequalities for the Robin Laplacian on these domains and characterizes extremal domains in the parameter space.

Findings

Eigenvalues satisfy Pólya-type inequalities with specific exponents.

Optimal exponents differ between rectangles and unions of rectangles.

Extremal domains are characterized in the (k,α)-plane.

Abstract

We show that eigenvalues of the Robin Laplacian with a positive boundary parameter $\alpha$ on rectangles and unions of rectangtes satisfy P\'{o}lya-type inequalities, albeit with an exponent smaller than that of the corresponding Weyl asympotics for a fixed domain. We determine the optimal exponents in either case, showing that they are different in the two situations. Our approach to proving these results includes a characterisation of the corresponding extremal domains for the $k$th eigenvalue in regions of the $(k,\alpha)$-plane.