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An explicit solution of the Lipschitz extension problem


Author: Adam M. Oberman
Journal: Proc. Amer. Math. Soc. 136 (2008), 4329-4338
MSC (2000): Primary 46A22, 46T20, 58E30, 65D05
DOI: https://doi.org/10.1090/S0002-9939-08-09457-4
Published electronically: June 3, 2008
MathSciNet review: 2431047
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Abstract: Building Lipschitz extensions of functions is a problem of classical analysis. Extensions are not unique: the classical results of Whitney and McShane provide two explicit examples. In certain cases there exists an optimal extension, which is the solution of an elliptic partial differential equation, the infinity Laplace equation. In this work, we find an explicit formula for a sub-optimal extension, which is an improvement over the Whitney and McShane extensions: it can improve the local Lipschitz constant. The formula is found by solving a convex optimization problem for the minimizing extensions at each point. This work extends a previous solution for domains consisting of a finite number of points, which has been used to build convergent numerical schemes for the infinity Laplace equation, and in Image Inpainting applications.


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Additional Information

Adam M. Oberman
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: aoberman@sfu.ca

DOI: https://doi.org/10.1090/S0002-9939-08-09457-4
Keywords: Lipschitz extension, absolute minimizers, infinity Laplacian
Received by editor(s): October 22, 2007
Published electronically: June 3, 2008
Communicated by: Walter Craig
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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