The Dirichlet elliptic problem involving regional fractional Laplacian

Abstract

In this paper, we study the solutions of elliptic equations involving regional fractional Laplacian (E) (−Δ)Ωαu=f in a bounded regular domain Ω in RN(N≥2) with C2 boundary ∂Ω, subject to Dirichlet boundary g on ∂Ω, where α∈(12,1) and the operator (−Δ)Ωα denotes the regional fractional Laplacian. We prove that when g ≡ 0, problem (E) admits a unique weak solution under the hypotheses that f ∈ L2(Ω), f ∈ L1(Ω, ρβdx), and f∈M(Ω,ρβ), where 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 integration by parts formula for the classical solution of (E) with boundary data g.