An implicit two-phase obstacle-type problem for the $p$-Laplacian with the fractional gradient

TL;DR

This paper investigates an inhomogeneous two-phase obstacle problem involving the fractional p-Laplacian, establishing existence, uniqueness, and convergence results, and exploring dependence on the level-set function for a broader class of problems.

Contribution

It introduces a novel analysis of the fractional p-Laplacian obstacle problem, including convergence to the classical case and implicit level-set dependence.

Findings

Solutions exist and are unique.

Solutions converge to the classical problem as s approaches 1.

Continuous dependence on the level-set function is established.

Abstract

In this work we study an inhomogeneous two-phase obstacle-type problem associated to the $s$-fractional $p$-Laplacian. Besides the existence and uniqueness of solutions, we study the convergence of the solutions when $s\to 1$ to the classical problem. We also study the continuous dependence with respect to the level-set function $v$, yielding an existence result for a two-phase obstacle-type problem with an implicit level-set function.