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A tour of the theory of absolutely minimizing functions

Authors: Gunnar Aronsson, Michael G. Crandall and Petri Juutinen
Journal: Bull. Amer. Math. Soc. 41 (2004), 439-505
MSC (2000): Primary 35J70, 49K20, 35B50
Published electronically: August 2, 2004
MathSciNet review: 2083637
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Abstract: These notes are intended to be a rather complete and self-contained exposition of the theory of absolutely minimizing Lipschitz extensions, presented in detail and in a form accessible to readers without any prior knowledge of the subject. In particular, we improve known results regarding existence via arguments that are simpler than those that can be found in the literature. We present a proof of the main known uniqueness result which is largely self-contained and does not rely on the theory of viscosity solutions. A unifying idea in our approach is the use of cone functions. This elementary geometric device renders the theory versatile and transparent. A number of tools and issues routinely encountered in the theory of elliptic partial differential equations are illustrated here in an especially clean manner, free from burdensome technicalities - indeed, usually free from partial differential equations themselves. These include a priori continuity estimates, the Harnack inequality, Perron's method for proving existence results, uniqueness and regularity questions, and some basic tools of viscosity solution theory. We believe that our presentation provides a unified summary of the existing theory as well as new results of interest to experts and researchers and, at the same time, a source which can be used for introducing students to some significant analytical tools.

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Additional Information

Gunnar Aronsson
Affiliation: Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden

Michael G. Crandall
Affiliation: Department of Mathematics, University of California, Santa Barbara, Santa Barbara, California 93106

Petri Juutinen
Affiliation: Department of Mathematics and Statistics, P.O. Box 35, FIN-40014 University of Jyväskylä, Finland

Received by editor(s): July 18, 2003
Received by editor(s) in revised form: May 24, 2004
Published electronically: August 2, 2004
Additional Notes: “Absolutely minimizing" has other meanings besides the one herein. We might more properly say “absolutely minimizing in the Lipschitz sense" instead, but prefer to abbreviate.
The third author is supported by the Academy of Finland, project #80566.
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.