Generic regularity of free boundaries for the obstacle problem
The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in $\mathbb{R}^n$. By classical results of Caffarelli, the free boundary is $C^{\infty}$ outside a set of singular points. Explicit examples show that the singular set could be in general $(n-1)$-di...
| Authors: | , , |
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| Format: | article |
| Status: | Versión aceptada para publicación |
| Publication Date: | 2020 |
| Country: | España |
| Institution: | Universidad de Barcelona |
| Repository: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/194135 |
| Online Access: | https://hdl.handle.net/2445/194135 |
| Access Level: | Open access |
| Keyword: | Problemes de contorn Equacions en derivades parcials Funcions de variables complexes Distribució (Teoria de la probabilitat) Boundary value problems Partial differential equations Functions of complex variables Distribution (Probability theory) |
| Summary: | The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in $\mathbb{R}^n$. By classical results of Caffarelli, the free boundary is $C^{\infty}$ outside a set of singular points. Explicit examples show that the singular set could be in general $(n-1)$-dimensional - that is, as large as the regular set. Our main result establishes that, generically, the singular set has zero $\mathcal{H}^{n-4}$ measure (in particular, it has codimension 3 inside the free boundary). In particular, for $n \leq 4$, the free boundary is generically a $C^{\infty}$ manifold. This solves a conjecture of Schaeffer (dating back to 1974 ) on the generic regularity of free boundaries in dimensions $n \leq 4$ |
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