Nonlocal free boundary minimal surfaces

Abstract

We introduce the nonlocal analogue of the classical free boundary minimal hypersurfaces in an open domain $\hat{I}\text{c◯}$ of ${\mathbb{R}}^n$ as the (boundaries of) critical points of the fractional perimeter ${\mathrm{Per}}_s(\cdot ,\hat{I}\text{c◯})$ with respect to inner variations leaving $\hat{I}\text{c◯}$ 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 $\hat{I}\text{c◯}$ for unbounded hypersurfaces. Moreover, we investigate stickiness properties and regularity across the boundary.

43 pages, 5 figures. All comments are welcome!