Sharp bounds for higher mixed Steklov-Robin eigenvalues on domains with holes

Abstract

This article is concerned with mixed Steklov--Robin eigenvalues on bounded domains in ${\mathbb{R}}^n,n\ge 2$, with Lipschitz boundary. Specifically, we consider domains with symmetry of order $4$ containing a spherical hole. We obtain isoperimetric inequalities for the $k$-th Steklov-Robin eigenvalues for each $k\in \{2,3,\dots ,n+1\}$. We provide examples to emphasize the fact that the symmetry assumptions, on the family of domains considered, are crucial.