Nonlocal ordered mean curvature with integrable kernels
Abstract
In this paper we introduce and study the concept of nonlocal ordered curvature. In the classical (differential) setting, the problem was introduced by Li and Nirenberg in [1, 2] where they conjectured (and proved in some cases) that if a bounded smooth surface has its mean curvature ordered in a particular direction, then the surface must be symmetric with respect to some hyperplane orthogonal to that direction. The conjecture was finally settled by Li et al in 2021 [3]. Here we study the counterpart problem in the nonlocal setting, where the nonlocal mean curvature of a set ?, at any point x on its boundary, is defined as H?J(x)=??cJ(x?y)dy???J(x?y)dy and the kernel function J is radially symmetric, non-increasing, integrable and compactly supported. Using a generalization of Alexandrov’s moving plane method, we prove a similar result in the nonlocal setting.
Department(s)
Mathematics
Document Type
Article
DOI
10.1016/j.na.2025.114028
Keywords
Alexandrov’s moving plane method, Finite horizon, Integrable kernel of interaction, Nonlocal mean curvature, Ordered curvature
Publication Date
5-1-2026
Recommended Citation
Biswas, Animesh; Foss, Mikil D.; and Radu, Petronela, "Nonlocal ordered mean curvature with integrable kernels" (2026). Faculty Scholarship. 8.
https://bearworks.missouristate.edu/articles00/8
Journal Title
Nonlinear Analysis Theory Methods and Applications
