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The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary
Ros Oton, Xavier; Serra Montolí, Joaquim
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I
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 (-d)su=g in O, u=0 in Rn\O, for some s¿(0, 1) and g¿L8(O), then u is Cs(Rn) and u/ds|O is Ca up to the boundary ¿O for some a¿(0, 1), where d(x)=dist(x, ¿O). 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/ds. Namely, the Cß norms of u and u/ds in the sets {x¿O:d(x)=¿} are controlled by C¿s-ß and C¿a-ß, respectively.These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian (Ros-Oton and Serra, 2012 [19,20]). © 2013 Elsevier Masson SAS.
Boundary element methods
Fractional Laplacian
Boundary regularity
Dirichlet problem
Fractional Laplacian
Attribution-NonCommercial-NoDerivs 3.0 Spain

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