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Regularity of the obstacle problem for a fractional power of the Laplace operator. (English) Zbl 1141.49035
Given a function $\varphi$ and $s \in (0,1)$, in this paper is considered the following obstacle problem: 1. $u \ge \varphi$ in $\Bbb R^n$; 2. $(\Delta)^s u \ge 0$ in $\Bbb R^n$; 3. $(\Delta)^s u= 0$ for those $x$ such that $u(x) >\varphi(x)$; 4. $\lim_{\vert x\vert \rightarrow +\infty} u(x)=0$. The author proves that if $\varphi$ is in $C^{1,s}$ then the solution $u$ is in $C^{1,\alpha}$ for all $\alpha <s$. In the case where the contact set $u=\varphi$ is convex, the optimal regularity $u \in C^{1,s}$ is obtained. Moreover, when $\varphi$ is in $C^{1,\beta}$ with $\beta < s$ the solution is in $C^{1,\alpha}$ for all $\alpha <\beta$. Finally some applications of the results are presented. In particular, interesting considerations on a well known Signorini problem are given.

49N60Regularity of solutions in calculus of variations
35B65Smoothness and regularity of solutions of PDE
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