Global regularity for the free boundary in the obstacle problem for the fractional Laplacian

Begoña Barrios; Alessio Figalli; Xavier Ros-Oton

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arXiv · 1506.04684

Global regularity for the free boundary in the obstacle problem for the fractional Laplacian

Begoña Barrios·Alessio Figalli·Xavier Ros-Oton

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Abstract

We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle $\varphi$ satisfies $\Delta \varphi\leq 0$ near the contact region. Our main result establishes that the free boundary consists of a set of regular points, which is known to be a $(n-1)$-dimensional $C^{1,\alpha}$ manifold by the results in \cite{CSS}, and a set of singular points, which we prove to be contained in a union of $k$-dimensional $C^1$-submanifold, $k=0,\ldots,n-1$. Such a complete result on the structure of the free boundary was known only in the case of the classical Laplacian \cite{C-obst1,C-obst2}, and it is new even for the Signorini problem (which corresponds to the particular case of the $\frac12$-fractional Laplacian). A key ingredient behind our results is the validity of a new non-degeneracy condition $\sup_{B_r(x_0)}(u-\varphi)\geq c\,r^2$, valid at all free boundary points $x_0$.

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Begoña Barrios, Alessio Figalli, Xavier Ros-Oton. 2015-06-15. Global regularity for the free boundary in the obstacle problem for the fractional Laplacian. https://arxiv.org/abs/1506.04684

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