Nonlocal free boundary minimal surfaces

TL;DR

This paper introduces a nonlocal version of free boundary minimal hypersurfaces based on fractional perimeter, revealing new phenomena like boundary-less critical points and volume constraints, with analysis of regularity and stickiness.

Contribution

It defines nonlocal free boundary minimal hypersurfaces, derives their Euler-Lagrange equations, and uncovers novel properties not present in classical cases.

Findings

Existence of critical points without boundary

Strong volume constraints for unbounded hypersurfaces

Regularity and stickiness properties analyzed

Abstract

We introduce the nonlocal analogue of the classical free boundary minimal hypersurfaces in an open domain $\Omega$ of $\mathbb{R}^n$ as the (boundaries of) critical points of the fractional perimeter $\operatorname{Per}_s(\cdot,\,\Omega )$ with respect to inner variations leaving $\Omega$ invariant. We deduce the Euler-Lagrange equations and prove a few surprising features, such as the existence of critical points without boundary and a strong volume constraint in $\Omega$ for unbounded hypersurfaces. Moreover, we investigate stickiness properties and regularity across the boundary.