Continuity of s-minimal functions

TL;DR

This paper investigates the continuity properties of minimizers of a fractional seminorm related to nonlocal perimeter functionals, establishing conditions for interior and boundary continuity and demonstrating their optimality.

Contribution

It provides new results on the boundedness and continuity of minimizers of fractional perimeter problems, including boundary behavior under mild assumptions.

Findings

Minimizers are bounded and continuous inside the domain.

Under mild conditions, minimizers are continuous up to the boundary.

Continuity across the boundary may fail in general.

Abstract

We consider the minimization property of a Gagliardo-Slobodeckij seminorm which can be seen as the fractional counterpart of the classical problem of functions of least gradient and which is related to the minimization of the nonlocal perimeter functional. We discuss continuity properties for this kind of problem. In particular, we show that, under natural structural assumptions, the minimizers are bounded and continuous in the interior of the ambient domain (and, in fact, also continuous up to the boundary under some mild additional hypothesis). We show that these results are also essentially optimal, since in general the minimizer is not necessarily continuous across the boundary.