Variational problems with free boundaries for the fractional Laplacian
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In this article, the authors discuss properties of a variational interface problem involving the fractional Laplacian, including optimal regularity, non-degeneracy, and smoothness of the free boundary.Abstract:
We discuss properties (optimal regularity, non-degeneracy, smoothness of the free boundary...) of a variational interface problem involving the fractional Laplacian; Due to the non-locality of the Dirichlet problem, the task is nontrivial. This difficulty is by-passed by an extension formula, discovered by the first author and Silvestre, which reduces the study to that of a co-dimension 2 (degenerate) free boundary.read more
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Hitchhiker's guide to the fractional Sobolev spaces
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The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary
Xavier Ros-Oton,Joaquim Serra +1 more
TL;DR: In this article, the Pohozaev identity up to the boundary of the Dirichlet problem for the fractional Laplacian was shown to hold for the case of ( − Δ ) s u = g in Ω, u ≡ 0 in R n \ Ω, for some s ∈ ( 0, 1 ) and g ∈ L ∞ ( Ω ), then u is C s ( R n ) and u / δ s | Ω is C α up to boundary ∂Ω for some α ∈( 0
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An Extension Problem Related to the Fractional Laplacian
TL;DR: In this article, the square root of the Laplacian (−△) 1/2 operator was obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition.
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