An extension problem related to inverse fractional operators

TL;DR

This paper demonstrates that the inverse fractional Laplacian can be represented as a Neumann-to-Dirichlet map through an extension problem, extending the known fractional Laplacian characterization and exploring broader differential operators.

Contribution

It introduces an extension problem for the inverse fractional Laplacian as a Neumann-to-Dirichlet map, providing explicit solutions and extending the framework to general second order operators.

Findings

Inverse fractional Laplacian characterized as Neumann-to-Dirichlet map

Explicit formula for the extension problem solution

Applications to numerical analysis of nonlinear nonlocal equations

Abstract

It is well known from the work of Caffarelli and Silvestre that the fractional Laplacian $(-\Delta_x)^{\frac{\sigma}{2}}$ for $\sigma \in (0,2)$ 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 $(-\Delta_x)^{-\frac{\sigma}{2}}$ 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.