A Reilly type integral formula and its applications

TL;DR

This paper develops a Reilly type integral formula for the $\,\phi$-Laplacian and applies it to derive geometric inequalities, eigenvalue relations, and partial results related to Schur lemmas, advancing the understanding of these mathematical concepts.

Contribution

It introduces a new Reilly type integral formula for the $\,\phi$-Laplacian and applies it to establish inequalities and eigenvalue relationships, extending previous results.

Findings

Derived Heintze-Karcher and Minkowski inequalities

Established eigenvalue relationships for Wentzell boundary conditions

Extended partial results of Li and Xia on Schur lemmas

Abstract

In this paper, we achieve a Reilly type integral formula associated with the $\phi$-Laplacian. As its applications, we obtain Heintze-Karcher and Minkowski type inequalities. Furthermore, almost Schur lemmas are also given. They recover the partial results of Li and Xia in [15]. On the other hand, we also study eigenvalue problem for Wentzell boundary conditions and obtain eigenvalue relationships.