Integration by Parts Formula for Regional Fractional Laplacian

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Nonhomogeneous boundary conditions for the spectral fractional Laplacian (2015, background): “Let us also mention the work of Guan [18], who considered similar issues in the context of the regional fractional Laplacian and generalizations thereof.”
The best constant in a fractional Hardy inequality (2008, methodology): “12)], [24], [25]: Δ D u(x) = Ad,−α lim ε→0+ ∫”
Regularity of spectral fractional Dirichlet and Neumann problems (2014, background): “a singularity at ∂Ω (balancing the singularity of the ﬁrst term). This case appe ars e.g. in Lieb and Yau [35], Chen and Kim [15], Bogdan, Burdzy and Chen [7]. For 1 2 <a<1, it is shown in Guan [23] how P0 represents a Neumann condition (d2−2a∂ nu)|∂Ω = 0. Nonhomogeneous Neumann and Robin prob”
Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations (2015, background): “, [23]), defined, for smooth functions, as (compare with (10))”