The Dirichlet elliptic problem involving regional fractional Laplacian

الأصل · EN

In this paper, we consider the solutions for elliptic equations involving regional fractional Laplacian equation0 =1pt arraylll (-Δ)αΩu=f & in Ω,2mm] (-Δ)α u=g & on ∂ Ω, array equation where Ω is a bounded open domain in Rⁿ (N≥ 2) with C² boundary ∂Ω, α∈(12,1) and the operator (-Δ)αΩ denotes the regional fractional Laplacian. We prove that when g≡0, problem (0) admits a unique weak solution in the cases that f∈ L²(Ω), f∈ L¹(Ω, ρβdx) and f∈ M(Ω,ρβ), here ρ(x)= dist(x,∂Ω), β=2α-1 and M(Ω,ρβ) is a space of all Radon measures ν satisfying ∫Ωρβd|ν|<+∞. Finally, we provide an Integral by Parts Formula for the classical solution of (0) with general boundary data g.