Multiply-periodic hypersurfaces with constant nonlocal mean curvature

Abstract

We study hypersurfaces with fractional mean curvature in N-dimensional Euclidean space. These hypersurfaces are critical points of the fractional perimeter under a volume constraint. We use local inversion arguments to prove existence of smooth branches of multiply-periodic hypersurfaces bifurcating from suitable parallel hyperplanes.

On an electrostatic problem and a new class of exceptional subdomains of $\mathbb{R}^3$

“…where c α is a constant. The problem of constructing surfaces of the type Z ϕ with constant nonlocal mean curvature has been considered recently by Cabré, Fall and Weth in [3], and Minlend, Niang and Thiam [17]. The difference between (1.4) and…”

“…(1.11) is two-fold. First, the case α = 0 corresponds to a limiting case which is not admitted in [3,17] but more closely related to the 0-fractional perimeter, which has been studied recently in [7]. Moreover, the normal ν + is taken in at x in (1.4), while it evaluated at y in (1.11).…”

“…Moreover, the normal ν + is taken in at x in (1.4), while it evaluated at y in (1.11). Despite these differences which strongly affect the analysis, the construction here follows the general strategy of [3,17] which is based on an application of the Crandall-Rabinowitz bifurcation theorem, see [5,Theorem 1.7]. The main difficulties in this approach are the following.…”