A new look at the fractional Poisson problem via the logarithmic Laplacian

Abstract

We analyze the s-dependence of solutions us to the family of fractional Poisson problems(−Δ)su=fin Ω,u≡0on RN∖Ω in an open bounded set Ω⊂RN, s∈(0,1). In the case where Ω is of class C2 and f∈Cα(Ω‾) for some α>0, we show that the map (0,1)→L∞(Ω), s↦us is of class C1, and we characterize the derivative ∂sus in terms of the logarithmic Laplacian of f. As a corollary, we derive pointwise monotonicity properties of the solution map s↦us under suitable assumptions on f and Ω. Moreover, we derive explicit bounds for the corresponding Green operator on arbitrary bounded domains which are new even for the case s=1, i.e., for the local Dirichlet problem −Δu=f in Ω, u≡0 on ∂Ω.