Boundary continuity of nonlocal minimal surfaces in domains with singularities and a problem posed by Borthagaray, Li, and Nochetto

Abstract

Differently from their classical counterpart, nonlocal minimal surfaces are known to present boundary discontinuities, by sticking at the boundary of smooth domains. It has been observed numerically by J. P. Borthagaray, W. Li, and R. H. Nochetto ``that stickiness is larger near the concave portions of the boundary than near the convex ones, and that it is absent in the corners of the square'', leading to the conjecture ``that there is a relation between the amount of stickiness on $\partial\Omega$ and the nonlocal mean curvature of $\partial\Omega$''. In this paper, we give a positive answer to this conjecture, by showing that the nonlocal minimal surfaces are continuous at convex corners of the domain boundary and discontinuous at concave corners. More generally, we show that boundary continuity for nonlocal minimal surfaces holds true at all points in which the domain is not better than $C^{1,s}$, with the singularity pointing outward, while, as pointed out by a concrete example, discontinuities may occur at all point in which the domain possesses an interior touching set of class $C^{1,\alpha}$ with $\alpha>s$.