Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian
We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, $\min \left\{(-\Delta)^s u, u-\varphi\right\}=0$ in $\mathbb{R}^n$, for general obstacles $\varphi$. Our main result establishes the complete structure and regularity of the singular set. To prove i...
| Authors: | , |
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| Format: | article |
| Status: | Versión aceptada para publicación |
| Publication Date: | 2019 |
| Country: | España |
| Institution: | Universidad de Barcelona |
| Repository: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/194048 |
| Online Access: | https://hdl.handle.net/2445/194048 |
| Access Level: | Open access |
| Keyword: | Operadors diferencials parcials Teoria d'operadors Equacions en derivades parcials Processos estocàstics Partial differential operators Operator theory Partial differential equations Stochastic processes |
| Summary: | We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, $\min \left\{(-\Delta)^s u, u-\varphi\right\}=0$ in $\mathbb{R}^n$, for general obstacles $\varphi$. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in those of Garofalo-Petrosyan to all $s \in(0,1)$. |
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