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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.18106 |
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| _version_ | 1866916733465395200 |
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| author | Lo, Catharine W. K. Rodrigues, José Francisco |
| author_facet | Lo, Catharine W. K. Rodrigues, José Francisco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18106 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Stability of the $s$-Nonlocal $p$-Obstacle Problem and their Coincidence Sets and Free Boundaries Lo, Catharine W. K. Rodrigues, José Francisco Analysis of PDEs 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. |
| title | On the Stability of the $s$-Nonlocal $p$-Obstacle Problem and their Coincidence Sets and Free Boundaries |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.18106 |