An extension problem related to inverse fractional operators

Félix del Teso

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arXiv · 1603.07988

An extension problem related to inverse fractional operators

Félix del Teso

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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.

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Félix del Teso. 2016-03-25. An extension problem related to inverse fractional operators. https://arxiv.org/abs/1603.07988

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