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

TL;DR

This work analyzes the stability of the nonlocal obstacle problem for the fractional $p$-Laplacian as the fractional parameter $s$ tends to a limit $\sigma\le1$, and in particular as $s\\to1$ connects to the classical obstacle problem. It develops a robust variational framework with penalization to obtain uniform Hölder regularity, proves $\\Gamma$-convergence of the energy functionals $\\mathcal{J}_{s,p}$ to $\\mathcal{J}_{1,p}$, and establishes convergence of solutions $u^s$ to $u^\sigma$ in $W^{r,p}_0(\\Omega)$ for every $0\le r<\sigma$, together with weak-$^*$ convergence of the quasi-characteristics $\\vartheta^s$. The authors further prove convergence of the coincidence sets in measure, and under a local nondegeneracy/topology assumption, convergence of the free boundaries in Hausdorff distance, thereby extending local obstacle problem stability results to the nonlocal regime. These results provide a precise bridge between nonlocal and local obstacle problems, clarifying how active/contact regions and free boundaries behave under the nonlocal-to-local transition and establishing conditions for geometric stability of the solution structure.

Abstract

We show that the solutions to the nonlocal obstacle problems for the nonlocal $-Δ_p^s$ operator, when the fractional parameter $s\toσ$ for $0<σ\leq1$, converge to the solution of the corresponding obstacle problem for $-Δ_p^σ$, 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 $s\nearrow 1$ 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.

Theorems & Definitions (48)

• Lemma 2.1: Poincaré inequality, Theorem 6.1 of BonderSalort2019JFAFracOrliczSobolev
• Lemma 2.2
• Remark 2.3
• Proposition 2.4
• proof
• Theorem 2.5
• proof
• Lemma 2.6
• Theorem 2.7
• proof