The infinity Laplacian, Aronsson's equation and their generalizations

Authors:
E. N. Barron, L. C. Evans and R. Jensen

Journal:
Trans. Amer. Math. Soc. **360** (2008), 77-101

MSC (2000):
Primary 35C99, 35J60; Secondary 49L20, 49L25

Published electronically:
July 25, 2007

MathSciNet review:
2341994

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Abstract | References | Similar Articles | Additional Information

Abstract: The infinity Laplace equation arose originally as a sort of Euler-Lagrange equation governing the absolute minimizer for the variational problem of minimizing the functional ess-sup The more general functional ess-sup leads similarly to the so-called Aronsson equation

In this paper we show that these PDE operators and various interesting generalizations also appear in several other contexts seemingly quite unrelated to variational problems, including two-person game theory with random order of play, rapid switching of states in control problems, etc. The resulting equations can be parabolic and inhomogeneous, equation types precluded in conventional variational problems.

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Additional Information

**E. N. Barron**

Affiliation:
Department of Mathematics and Statistics, Loyola University of Chicago, Chicago, Illinois 60626

Email:
ebarron@luc.edu

**L. C. Evans**

Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, California 94720

Email:
evans@math.berkeley.edu

**R. Jensen**

Affiliation:
Department of Mathematics and Statistics, Loyola University of Chicago, Chicago, Illinois 60626

Email:
rjensen@luc.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-07-04338-3

Keywords:
Infinity Laplacian,
Aronsson equations,
tug of war games,
random evolutions

Received by editor(s):
July 15, 2005

Published electronically:
July 25, 2007

Additional Notes:
The first and third authors were supported in part by NSF Grant DMS-0200169

The second author was supported in part by NSF Grant DMS-0500452.

Article copyright:
© Copyright 2007
American Mathematical Society