Nonlocal Mean Curvature with Integrable Kernel

Abstract

We study the prescribed constant mean curvature problem in the nonlocal setting where the nonlocal curvature has been defined as $$ H^J_{\Omega}(x):=\int_{\mathbb{R} ^n} J(x-y)(\chi_{\Omega^c}(y)-\chi_{\Omega}(y))dy, $$ where $x \in \mathbb{R}^n$, $\Omega \subset \mathbb{R}^n$, $\chi$ is the characteristic function for a set, $J$ is a radially symmetric, nonegative, nonincreasing convolution kernel. Several papers have studied the case of nonlocal curvature with nonintegrable singularity, a generalization of the classical curvature concept, which requires the regularity of the boundary to be above $C^2$. Nonlocal curvature of this form appears in many different applications, such as image processing, curvature driven motion, deformations. In this work, we focus on the problem of constant nonlocal curvature defined via integrable kernel. Our results offer some extensions to the constant mean curvature problem for nonintegrable kernels, where counterparts to Alexandrov theorem in the nonlocal framework were established independently by two separate groups. Ciraolo, Figalli, Maggi, Novaga, and respectively, Cabr\'e, Fall, Sol\`a-Morales, Weth. Using the nonlocal Alexandrov's theorem we identify surfaces of constant mean nonlocal curvature for different integrable kernels as unions of balls situated at distance $\delta$ apart, where $\delta$ measures the radius of nonlocal interactions.