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arXiv · 2508.00337

Nonlocal free boundary minimal surfaces

Marco Badran·Serena Dipierro·Enrico Valdinoci
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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.

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Marco Badran, Serena Dipierro, Enrico Valdinoci. 2025-08-01. Nonlocal free boundary minimal surfaces. https://arxiv.org/abs/2508.00337

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