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

We show that the solutions to the nonlocal obstacle problems for the nonlocal $-{\mathrm{\Delta }}_p^s$ operator, when the fractional parameter $s\to \sigma$ for $0<\sigma \le 1$, converge to the solution of the corresponding obstacle problem for $-{\mathrm{\Delta }}_p^{\sigma }$, 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.