Modica type estimates and curvature results for overdetermined p-Laplace problems

2025/06/17 by Yuanyuan Lian, Jing Wu, Lian, Yuanyuan +1
Computer Science · Mathematics · #35B50 #35J92 #35N25 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2506.14579

openalex publication_date 2025/06/17 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

In this paper we prove Modica type estimates for the following overdetermined p-Laplace problem \begincases div (|∇ u|^p-2∇ u)+f(u) =0amp; in Ω, ugt;0 amp;in Ω, u=0 amp;on ∂Ω, ∂_ν u=-κamp;on ∂Ω, \endcases where 12 we also assume that if F(u₀)=0, F(u)=O(|u-u₀|^p) as u→ u₀), then either the mean curvature of ∂ Ω is strictly negative or Ω is a half-space.

Citations

• Symmetry in Serrin-type overdetermined problems
• Rigidity results for the capillary overdetermined problem
• Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space
• The p-Laplacian overdetermined problem on Riemannian manifolds
• Sign-changing solution for an overdetermined elliptic problem on unbounded domain
• Overdetermined elliptic problems in onduloid-type domains with general nonlinearities
• Splitting theorems on complete Riemannian manifolds with nonnegative Ricci curvature
• On the stability for Alexandrov's Soap Bubble theorem
• Solutions to overdetermined elliptic problems in nontrivial exterior\n domains
• A rigidity result for overdetermined elliptic problems in the plane
• On Serrin's overdetermined problem and a conjecture of Berestycki, Caffarelli and Nirenberg
• Structure of one-phase free boundaries in the plane
• Bifurcating extremal domains for the first eigenvalue of the Laplacian
• A Symmetry Result Related to Some Overdetermined Boundary Value Problems
• A symmetry problem in potential theory
• Remark on the preceding paper of Serrin

Related

• The p-Laplacian overdetermined problem on Riemannian manifolds
• Rigidity results for the capillary overdetermined problem
• A partially overdetermined problem for p-Laplace equation in convex cones
• Modica type gradient estimates for an inhomogeneous variant of the normalized p-laplacian evolution
• Concavity and perturbed concavity for p-Laplace equations