Sharp bounds for higher Steklov–Dirichlet eigenvalues on domains with spherical holes

Abstract

We consider a mixed Steklov–Dirichlet eigenvalue problem on a smooth bounded domain having a spherical hole. In this article, we take Dirichlet condition on the boundary of the spherical hole and Steklov condition on the other boundary component/s. Under certain symmetry assumptions on multiconnected domains in ℝ𝑛 having a spherical hole, we obtain isoperimetric inequalities for the k-th Steklov–Dirichlet eigenvalues for each 𝑘 ∈{2,3,…,𝑛 +1}. We provide examples to emphasise the fact that the symmetry assumptions, on the family of domains considered, are crucial. We also extend Theorem 3.1 of “Gavitone et al. (2023), An isoperimetric inequality for the first Steklov–Dirichlet Laplacian eigenvalue of convex sets with a spherical hole, Pacific Journal of Mathematics, 320(2): 241–259” not only from Euclidean domains to domains in space forms but also from convex domains to star-shaped domains. In particular, we obtain sharp lower and upper bounds for the first Steklov–Dirichlet eigenvalue on the family of all bounded star-shaped domains on the hemisphere as well as on the hyperbolic space.