Asymptotics as $s\searrow 0$ of the nonlocal nonparametric Plateau problem with obstacles

TL;DR

This paper studies a nonlocal obstacle problem for minimal graphs, revealing a stickiness phenomenon where solutions adhere to obstacles when the fractional parameter is small.

Contribution

It introduces a new functional and geometric framework for the problem, proves existence and estimates, and highlights a novel stickiness behavior for small fractional parameters.

Findings

Solutions exhibit stickiness to obstacles at small fractional parameters.

Nonlocal minimal graphs can fail to be continuous across boundaries.

The paper establishes equivalence between two problem settings.

Abstract

In this paper, we introduce a functional and a geometric setting for an obstacle problem for nonlocal minimal graphs. In particular we study existence of solutions, a priori estimates, and we prove the equivalence of the two settings. We then observe a striking stickiness phenomena when the fractional parameter is small and the data at infinity is not too large: the nonlocal minimal graphs adhere entirely to the obstacle and leave the remainder of the domain asymptotically empty. We thus provide a class of examples where continuity of nonlocal minimal graphs across the boundary and across the obstacle may fail.