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 -Δₚˢ operator, when the fractional parameter s→σ for 0<σ≤1, converge to the solution of the corresponding obstacle problem for -Δₚ^σ, being σ=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 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 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.