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

Abstract

We analyze the s-dependence of solutions u s to the family of fractional Poisson problemsin an open bounded set Ω ⊂ R N , s ∈ (0, 1). In the case where Ω is of class C 2 and f ∈ C α (Ω) for some α > 0, we show that the map (0, 1) → L ∞ (Ω), s → u s is of class C 1 , and we characterize the derivative ∂ s u s in terms of the logarithmic Laplacian of f . As a corollary, we derive pointwise monotonicity properties of the solution map s → u s 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 −∆u = f in Ω, u ≡ 0 on ∂ Ω.

Fractional Laplacians on ellipsoids

“…Nevertheless, we believe that the study of p.p.p. in the fractional regime is relevant, since it offers a novel perspective on the subject using the continuity of the solution mapping, see [24].…”

“…The function u 1 also has applications in fluid mechanics (modelling the pressure gradient of a flow in a viscous fluid), see [25] and the references therein. A solution of (1.1) in general domains for any s > 0 is usually also called torsion function, and its explicit expression is often useful for checking inequalities and to formulate or disproof general conjectures (see, for example, [24,25,29]). Theorem 1.1 relies on the following more general result.…”