The obstacle problem for nonlinear integro-differential operators

TL;DR

This paper studies the obstacle problem for nonlinear nonlocal integro-differential operators, establishing existence, uniqueness, and regularity of solutions that inherit properties from the obstacle, with a focus on fractional p-Laplacian models.

Contribution

It proves existence, uniqueness, and regularity results for the obstacle problem involving nonlinear nonlocal operators like the fractional p-Laplacian.

Findings

Solutions exist and are unique.

Solutions inherit regularity from the obstacle.

Solutions are bounded, continuous, and Hölder continuous up to the boundary.

Abstract

We investigate the obstacle problem for a class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional $p$-Laplacian operator with measurable coefficients. Amongst other results, we will prove both the existence and uniqueness of the solutions to the obstacle problem, and that these solutions inherit regularity properties, such as boundedness, continuity and H\"older continuity (up to the boundary), from the obstacle.