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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
DOI: https://doi.org/10.1090/S0002-9947-07-04338-3
Published electronically: July 25, 2007
MathSciNet review: 2341994
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Abstract: The infinity Laplace equation $ \Delta_\infty u=0$ arose originally as a sort of Euler-Lagrange equation governing the absolute minimizer for the $ L^{\infty}$ variational problem of minimizing the functional ess-sup$ _{U}\vert Du\vert.$ The more general functional ess-sup$ _{U}F(x,u,Du)$ leads similarly to the so-called Aronsson equation $ A_F[u]=0.$

In this paper we show that these PDE operators and various interesting generalizations also appear in several other contexts seemingly quite unrelated to $ L^{\infty}$ 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 $ L^{\infty}$ 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: https://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

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