An extension problem related to inverse fractional operators

Félix del Teso

Published 2016-03-25Version 1

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 show 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 a new extension problem to the upper half space. We also obtain an explicit formula for the solution of the new extension problem. Moreover, we deal with powers of a more general class of second order differential operators defined in open subsets of $\mathbb{R}^N$. From this characterization we show possible applications.