An extension problem related to inverse fractional operators

Abstract

It is well known from the work of Caffarelli and Silvestre that the fractional Laplacian $(-Δ_x)^{\fracσ{2}}$ for $σ\in (0,2)$ can be obtained as a Dirichlet-to-Neumann map through an extension problem to the upper half space. In this paper we emphasize that the inverse fractional Laplacian $(-Δ_x)^{-\fracσ{2}}$ has a similar property: it can be obtained as a Neumann-to-Dirichlet map via an extension problem to the upper half space. We also show an explicit formula for the solution of the extension problem. Moreover, we deal with powers of a more general class of second order differential operators defined in open subsets of $\mathbb{R}^N$ using the results of Stinga and Torrea. From this characterization we show possible applications among which we mention the numerical analysis of a wide class of nonlinear and nonlocal equations.