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

Guangyue Huang, Chunlei Luo, Hongru Song

TL;DR

This work extends the classical Serrin–Weinberger framework to the nonlinear $p$-Laplacian on compact Riemannian manifolds with positive Ricci curvature. By introducing a P-function tied to the first nonzero $p$-Laplacian eigenvalue and analyzing its linearization, the authors derive integral identities that yield a Heintze–Karcher-type inequality and a Soap Bubble-type rigidity theorem. The results provide sharp geometric inequalities for the boundary mean curvature and establish rigidity conditions characterizing geodesic balls under overdetermined data. Overall, the paper advances understanding of overdetermined problems in nonlinear elliptic settings on curved spaces and connects spectral, geometric, and boundary-geometry aspects.

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.

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

TL;DR

This work extends the classical Serrin–Weinberger framework to the nonlinear -Laplacian on compact Riemannian manifolds with positive Ricci curvature. By introducing a P-function tied to the first nonzero -Laplacian eigenvalue and analyzing its linearization, the authors derive integral identities that yield a Heintze–Karcher-type inequality and a Soap Bubble-type rigidity theorem. The results provide sharp geometric inequalities for the boundary mean curvature and establish rigidity conditions characterizing geodesic balls under overdetermined data. Overall, the paper advances understanding of overdetermined problems in nonlinear elliptic settings on curved spaces and connects spectral, geometric, and boundary-geometry aspects.

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.
Paper Structure (7 sections, 6 theorems, 57 equations)

This paper contains 7 sections, 6 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Linearized $p$-operator and some lemmas
  3. Proof of main results
  4. Proof of Theorem \ref{['1-th-1']}
  5. Proof of Theorem \ref{['1-th-2']}
  6. Proof of Theorem \ref{['1-th-3']}
  7. Proof of Theorem \ref{['1-th-4']}

Key Result

Theorem 1.1

Let $(M,g)$ be an $n$-dimensional compact Riemannian manifold with boundary $\partial M$ and the Ricci curvature satisfying ${\rm Ric}\geq(n-1)K$, where $K$ is a positive constant. Suppose $u$ is a weak solution to 1-6, and let $\mathcal{L}_{u}$ be the operator defined in 2-1. Then for $p\geq2$, we where $|M|$ denotes the volume of $M$.

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2