First eigenvalue for p-Laplacian with mixed boundary conditions on manifolds

TL;DR

This paper investigates eigenvalue problems of the p-Laplacian on manifolds with interior holes, establishing inequalities, comparison theorems, and sharp estimates for mixed boundary conditions.

Contribution

It introduces new Faber-Krahn-type inequalities, comparison theorems, and sharp eigenvalue estimates for p-Laplacian problems with mixed boundary conditions on manifolds.

Findings

Proved Faber-Krahn-type inequalities for p-Laplacian eigenvalues.

Established Cheng-type eigenvalue comparison theorems on manifolds.

Provided sharp estimates and upper bounds for eigenvalues with mixed boundary conditions.

Abstract

In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison theorem for eigenvalues with inner Dirichlet and outer Neumann boundary in minimal submanifolds in Euclidean space. Lastly we give a sharp estimate of the eigenvalue (with outer Dirichlet and inner Neumann boundaries) in terms of the Dircihlet eigenvalue, and also we give an upper bound of the eigenvalue with inner Dirichlet and outer Neumann problems by the diameter of the hole inside.