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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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