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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
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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  • [ACJ04] Gunnar Aronsson, Michael G. Crandall, and Petri Juutinen.
    A tour of the theory of absolutely minimizing functions.
    Bull. Amer. Math. Soc. (N.S.), 41(4):439-505 (electronic), 2004. MR 2083637 (2005k:35159)
  • [Aro67] Gunnar Aronsson.
    Extension of functions satisfying Lipschitz conditions.
    Ark. Mat., 6:551-561, 1967. MR 0217665 (36:754)
  • [Ber03] Dimitri P. Bertsekas.
    Convex analysis and optimization.
    Athena Scientific, Belmont, MA, 2003.
    With Angelia Nedić and Asuman E. Ozdaglar. MR 2184037 (2006j:90001)
  • [CMS98] Vicent Caselles, Jean-Michel Morel, and Catalina Sbert.
    An axiomatic approach to image interpolation.
    IEEE Trans. Image Process., 7(3):376-386, 1998. MR 1669524 (2000d:94001)
  • [Juu02] Petri Juutinen.
    Absolutely minimizing Lipschitz extensions on a metric space.
    Ann. Acad. Sci. Fenn. Math., 27(1):57-67, 2002. MR 1884349 (2002m:54020)
  • [Kir34] M. D. Kirszbraun.
    Uber die zusammenziehenden und Lipschitzchen Transformationen.
    Fund. Math., 22:77-108, 1934.
  • [LG04] E. Le Gruyer.
    On absolutely minimizing Lipschitz extensions. math/0403158v1, 2004.
  • [LGA96] E. Le Gruyer and J. C. Archer.
    Stability and convergence of extension schemes to continuous functions in general metric spaces.
    SIAM J. Math. Anal., 27(1):274-285, 1996. MR 1373157 (96k:41002)
  • [LGA98] E. Le Gruyer and J. C. Archer.
    Harmonious extensions.
    SIAM J. Math. Anal., 29(1):279-292 (electronic), 1998. MR 1617186 (99d:54008)
  • [LN05] James R. Lee and Assaf Naor.
    Extending Lipschitz functions via random metric partitions.
    Invent. Math., 160(1):59-95, 2005. MR 2129708 (2006c:54013)
  • [LS97] U. Lang and V. Schroeder.
    Kirszbraun's theorem and metric spaces of bounded curvature.
    Geom. Funct. Anal., 7(3):535-560, 1997. MR 1466337 (98d:53062)
  • [McS34] Edward James McShane.
    Extension of range of functions.
    Bull. Amer. Math. Soc., 40:837-842, 1934. MR 1562984
  • [MN06] Manor Mendel and Assaf Naor.
    Some applications of Ball's extension theorem.
    Proc. Amer. Math. Soc., 134(9):2577-2584 (electronic), 2006. MR 2213735 (2007a:46014)
  • [MST06] Facundo Memoli, Guillermo Sapiro, and Paul Thompson.
    Brain and surface warping via minimizing Lipschitz extensions.
    IMA Preprint Series 2092, January 2006.
  • [Obe05] Adam M. Oberman.
    A convergent difference scheme for the infinity Laplacian: Construction of absolutely minimizing Lipschitz extensions.
    Math. Comp., 74(251):1217-1230 (electronic), 2005. MR 2137000 (2006h:65165)
  • [PSSW06] Yuval Peres, O. Schramm, S. Sheffield, and D. Wilson.
    Tug-of-war and the infinity Laplacian
    (preprint). cache/math/pdf/0605/0605002v1.pdf, 2006.
  • [Whi34] Hassler Whitney.
    Analytic extensions of differentiable functions defined in closed sets.
    Trans. Amer. Math. Soc., 36(1):63-89, 1934. MR 1501735

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Additional Information

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

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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