An explicit solution of the Lipschitz extension problem

Oberman, Adam

doi:10.1090/S0002-9939-08-09457-4

An explicit solution of the Lipschitz extension problem
HTML articles powered by AMS MathViewer

by Adam M. Oberman

Proc. Amer. Math. Soc. 136 (2008), 4329-4338

DOI: https://doi.org/10.1090/S0002-9939-08-09457-4

Published electronically: June 3, 2008

PDF | Request permission

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.

References

Gunnar Aronsson, Michael G. Crandall, and Petri Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 4, 439–505. MR 2083637, DOI 10.1090/S0273-0979-04-01035-3
Gunnar Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551–561 (1967). MR 217665, DOI 10.1007/BF02591928
Dimitri P. Bertsekas, Angelia Nedić, and Asuman E. Ozdaglar, Convex analysis and optimization, Athena Scientific, Belmont, MA, 2003. MR 2184037
Vicent Caselles, Jean-Michel Morel, and Catalina Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process. 7 (1998), no. 3, 376–386. MR 1669524, DOI 10.1109/83.661188
Petri Juutinen, Absolutely minimizing Lipschitz extensions on a metric space, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 1, 57–67. MR 1884349
M. D. Kirszbraun. Uber die zusammenziehenden und Lipschitzchen Transformationen. Fund. Math., 22:77–108, 1934.
E. Le Gruyer. On absolutely minimizing Lipschitz extensions. http://arxiv.org/abs/ math/0403158v1, 2004.
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 (1996), no. 1, 274–285. MR 1373157, DOI 10.1137/S003614109325565X
E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Anal. 29 (1998), no. 1, 279–292. MR 1617186, DOI 10.1137/S0036141095294067
James R. Lee and Assaf Naor, Extending Lipschitz functions via random metric partitions, Invent. Math. 160 (2005), no. 1, 59–95. MR 2129708, DOI 10.1007/s00222-004-0400-5
U. Lang and V. Schroeder, Kirszbraun’s theorem and metric spaces of bounded curvature, Geom. Funct. Anal. 7 (1997), no. 3, 535–560. MR 1466337, DOI 10.1007/s000390050018
E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837–842. MR 1562984, DOI 10.1090/S0002-9904-1934-05978-0
Manor Mendel and Assaf Naor, Some applications of Ball’s extension theorem, Proc. Amer. Math. Soc. 134 (2006), no. 9, 2577–2584. MR 2213735, DOI 10.1090/S0002-9939-06-08298-0
Facundo Memoli, Guillermo Sapiro, and Paul Thompson. Brain and surface warping via minimizing Lipschitz extensions. IMA Preprint Series 2092, January 2006.
Adam M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Math. Comp. 74 (2005), no. 251, 1217–1230. MR 2137000, DOI 10.1090/S0025-5718-04-01688-6
Yuval Peres, O. Schramm, S. Sheffield, and D. Wilson. Tug-of-war and the infinity Laplacian (preprint). http://arxiv.org/PS cache/math/pdf/0605/0605002v1.pdf, 2006.
Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89. MR 1501735, DOI 10.1090/S0002-9947-1934-1501735-3