On the Stability of the $s$-Nonlocal $p$-Obstacle Problem and their Coincidence Sets and Free Boundaries

TL;DR

This paper investigates the stability and convergence of solutions to nonlocal $p$-obstacle problems as the fractional parameter approaches 1, connecting nonlocal and local obstacle problems and analyzing the behavior of their coincidence sets and free boundaries.

Contribution

It establishes the weak stability and convergence of solutions, coincidence sets, and free boundaries from nonlocal to local $p$-obstacle problems as the fractional parameter tends to 1.

Findings

Solutions converge to the local obstacle problem as $s o 1$.

Characteristic functions of coincidence sets converge strongly under nondegeneracy.

Free boundaries converge in Hausdorff distance under certain conditions.

Abstract

We show that the solutions to the nonlocal obstacle problems for the nonlocal $-\Delta_p^s$ operator, when the fractional parameter $s\to\sigma$ for $0<\sigma\leq1$, converge to the solution of the corresponding obstacle problem for $-\Delta_p^\sigma$, being $\sigma=1$ the classical obstacle problem for the local $p$-Laplacian. We discuss the weak stability of the quasi-characteristic functions of coincidence sets of the solution with the obstacle, which is a strong convergence of their characteristic functions when $s\nearrow 1$ under a nondegeneracy condition. This stability can be shown also in terms of the convergence of the free boundaries, as well as of the coincidence sets, in Hausdorff distance when $s\nearrow 1$, under non-degeneracy local assumptions on the external force and a local topological property of the coincidence set of the limit classical obstacle problem for the local $p$-Laplacian, essentially when the limit coincidence set is the closure of its interior.