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The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. (English) Zbl 1285.35020
Summary: We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if $$u$$ is a solution of $$(-\Delta)^su=g$$ in $$\varOmega$$, $$u\equiv 0$$ in $$\mathbb R^n\backslash\varOmega$$, for some $$s\in (0,1)$$ and $$g\in L^\infty(\varOmega)$$, then $$u$$ is $$C^s(\mathbb R^n)$$ and $$u/\delta^s|_\varOmega$$ is $$C^\alpha$$ up to the boundary $$\partial\varOmega$$ for some $$\alpha\in (0,1)$$, where $$\delta(x)=\mathrm{dist}(x,\partial\varOmega)$$. For this, we develop a fractional analog of the Krylov boundary Harnack method.
Moreover, under further regularity assumptions on $$g$$ we obtain higher order Hölder estimates for $$u$$ and $$u/\delta^s$$. Namely, the $$C^\beta$$ norms of $$u$$ and $$u/\delta^s$$ in the sets $$\{x\in \varOmega:\delta(x)\geqslant\rho\}$$ are controlled by $$C\rho^{s-\beta}$$ and $$C\rho^{\alpha-\beta}$$, respectively.
These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian [X. Ros-Oton and J. Serra, C. R., Math., Acad. Sci. Paris 350, No. 9–10, 505–508 (2012; Zbl 1273.35301), “The Pohozaev identity for the fractional Laplacian” (submitted), arxiv:1207.5986]

##### MSC:
 35J40 Boundary value problems for higher-order elliptic equations 35B65 Smoothness and regularity of solutions to PDEs
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