Nonlocal Ordered Mean Curvature with Integrable Kernel

TL;DR

This paper extends the classical concept of ordered mean curvature to a nonlocal setting with integrable kernels, proving symmetry results using a generalized moving plane method.

Contribution

It introduces the nonlocal ordered curvature concept and proves a symmetry result analogous to the classical case using a generalized Alexandrov's moving plane method.

Findings

Established nonlocal mean curvature symmetry under ordered conditions.

Generalized Alexandrov's moving plane method for nonlocal curvature.

Proved nonlocal symmetry result for sets with ordered curvature.

Abstract

In this paper we introduce and study the concept of nonlocal ordered curvature. In the classical (differential) setting, the problem was introduced by Nirenberg and Li, where they conjectured 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 proved by Li et al in 2022. Here we study the counterpart problem in the nonlocal setting, where the nonlocal mean curvature of a set $\Omega$, at any point $x$ on its boundary, is defined as $H_\Omega^J(x) = \int_{\Omega^c} J(x-y) dy - \int_\Omega J(x-y) dy$ 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.