Reverse Faber-Krahn inequality for the $p$-Laplacian in Hyperbolic space

TL;DR

This paper investigates the optimization of the first eigenvalue of the $p$-Laplacian in hyperbolic space, showing that certain symmetric annular domains maximize the eigenvalue under specific geometric constraints.

Contribution

It establishes a reverse Faber-Krahn inequality for the $p$-Laplacian in hyperbolic space, identifying the maximizing domains among multiply-connected regions.

Findings

Concentric annular regions maximize the first eigenvalue for given volume.

Derived Nagy's inequality for outer parallel sets in hyperbolic space.

Extended shape optimization results to multiply-connected domains.

Abstract

In this paper, we study the shape optimization problem for the first eigenvalue of the $p$-Laplace operator with the mixed Neumann-Dirichlet boundary conditions on multiply-connected domains in hyperbolic space. Precisely, we establish that among all multiply-connected domains of a given volume and prescribed $(n-1)$-th quermassintegral of the convex Dirichlet boundary (inner boundary), the concentric annular region produces the largest first eigenvalue. We also derive Nagy's type inequality for outer parallel sets of a convex domain in the hyperbolic space.