1. The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary.
- Author
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Ros-Oton, Xavier and Serra, Joaquim
- Subjects
- *
DIRICHLET problem , *FRACTIONAL calculus , *LAPLACIAN operator , *BOUNDARY value problems , *ESTIMATES - Abstract
Abstract: 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 in Ω, in , for some and , then u is and is up to the boundary ∂Ω for some , where . 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 . Namely, the norms of u and in the sets are controlled by and , 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]). [Copyright &y& Elsevier]
- Published
- 2014
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