## 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. $$u \geq \varphi$$ in $$\mathbb R^n$$; 2. $$(\Delta)^s u \geq 0$$ in $$\mathbb R^n$$; 3. $$(\Delta)^s u= 0$$ for those $$x$$ such that $$u(x) >\varphi(x)$$; 4. $$\lim_{| x| \rightarrow +\infty} u(x)=0$$.
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.

### MSC:

 49N60 Regularity of solutions in optimal control 35B65 Smoothness and regularity of solutions to PDEs 93B07 Observability

### Keywords:

optimal regularity; Signorini problem
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### References:

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