An Extension Problem Related to the Fractional Laplacian

The operator square root of the Laplacian (− ▵) 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 article, we obtain similar characterizations for general fractional powers of t...

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Bibliographic Details
Published in:Communications in partial differential equations 2007-08, Vol.32 (8), p.1245-1260
Main Authors: Caffarelli, Luis, Silvestre, Luis
Format: Article
Language:English
Subjects:
Degenerate elliptic equations
Fractional Laplacian
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Summary:The operator square root of the Laplacian (− ▵) 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 article, 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.
ISSN:0360-5302
1532-4133
DOI:10.1080/03605300600987306