An Extension Problem Related to the Fractional Laplacian

2006/08/31 by Luis Caffarelli, Luis Silvestre · 9 citations
Mathematics · #math.AP #msc:26A33

paper · pdf · doi:10.1080/03605300600987306

arxiv created 2007/02/13 · crossref issued 2007/08/08 · crossref published 2007/08/08 · crossref published-print 2007/08/08 · crossref created 2007/08/30 · arxiv updated 2010/03/31 · crossref deposited 2019/05/02 · crossref indexed 2026/08/07

Abstract

The operator square root of the Laplacian (-\lap)^1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.

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