Nonlocal ordered mean curvature with integrable kernels

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 Ω, at any point x on its boundary, is defined as H Ω J ( x ) = ∫ Ω c J ( x − y ) d y − ∫ Ω J ( x − y ) d y 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.