Free boundary regularity for almost every solution to the Signorini problem
We investigate the regularity of the free boundary for the Signorini problem in $\mathbb{R}^{n+1}$. It is known that regular points are $(n-1)$-dimensional and $C^{\infty}$. However, even for $C^{\infty}$ obstacles $\varphi$, the set of non-regular (or degenerate) points could be very large-e.g. wit...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2445/190628 |
| Acceso en línea: | https://hdl.handle.net/2445/190628 |
| Access Level: | acceso abierto |
| Palabra clave: | Equacions en derivades parcials Mecànica Sòlids Partial differential equations Mechanics Solids |
| Sumario: | We investigate the regularity of the free boundary for the Signorini problem in $\mathbb{R}^{n+1}$. It is known that regular points are $(n-1)$-dimensional and $C^{\infty}$. However, even for $C^{\infty}$ obstacles $\varphi$, the set of non-regular (or degenerate) points could be very large-e.g. with infinite $\mathcal{H}^{n-1}$ measure. The only two assumptions under which a nice structure result for degenerate points has been established are when $\varphi$ is analytic, and when $\Delta \varphi<0$. However, even in these cases, the set of degenerate points is in general $(n-1)$-dimensional-as large as the set of regular points. In this work, we show for the first time that, 'usually', the set of degenerate points is small. Namely, we prove that, given any $C^{\infty}$ obstacle, for almost every solution the nonregular part of the free boundary is at most $(n-2)$-dimensional. This is the first result in this direction for the Signorini problem. Furthermore, we prove analogous results for the obstacle problem for the fractional Laplacian $(-\Delta)^s$, and for the parabolic Signorini problem. In the parabolic Signorini problem, our main result establishes that the non-regular part of the free boundary is $\left(n-1-\alpha_{\circ}\right)$-dimensional for almost all times $t$, for some $\alpha_{\circ}>0$. Finally, we construct some new examples of free boundaries with degenerate points. |
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