An extension problem related to inverse fractional operators
This provides a new theoretical characterization for inverse fractional operators, which may enable new numerical methods for nonlocal PDEs, but the results are primarily theoretical and incremental.
The paper shows that the inverse fractional Laplacian can be obtained as a Neumann-to-Dirichlet map via an extension problem, analogous to the known characterization of the fractional Laplacian. It provides an explicit formula and extends the result to more general operators, with potential applications in numerical analysis of nonlocal equations.
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.