An overdetermined problem related to the p-Laplacian on Riemannian manifolds

TL;DR

This paper investigates an overdetermined boundary value problem for the p-Laplacian on compact Riemannian manifolds with positive Ricci curvature, deriving new inequalities and geometric characterizations.

Contribution

It introduces a novel P-function linked to the first eigenvalue of the p-Laplacian, leading to new integral identities and geometric results.

Findings

Established a Heintze-Karcher type inequality.

Proved a Soap Bubble Theorem.

Developed a new P-function related to eigenvalues.

Abstract

In this paper, we study the overdetermined problem for the p-Laplacian equation on a compact Riemannian manifold with positive Ricci curvature. By introducing a new P-function which is related to the first nonzero eigenvalue for p-Laplacian, we obtain some integral identities. As their applications, the Heintze-Karcher type inequality and the Soap Bubble Theorem have been achieved.