A tour of the theory of absolutely minimizing functions

Aronsson, Gunnar; Crandall, Michael; Juutinen, Petri

doi:10.1090/S0273-0979-04-01035-3

A tour of the theory of absolutely minimizing functions
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by Gunnar Aronsson, Michael G. Crandall and Petri Juutinen PDF

Bull. Amer. Math. Soc. 41 (2004), 439-505 Request permission

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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