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), 77101
MSC (2000):
Primary 35C99, 35J60; Secondary 49L20, 49L25
Published electronically:
July 25, 2007
MathSciNet review:
2341994
Fulltext PDF Free Access
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Abstract: The infinity Laplace equation arose originally as a sort of EulerLagrange equation governing the absolute minimizer for the variational problem of minimizing the functional esssup The more general functional esssup leads similarly to the socalled 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 twoperson 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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 [A2]
 G. Aronsson, Minimization problems for the functional . II, Arkiv für Mate. 6 (1966), 409431. MR 0203541 (34:3391)
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 [B]
 E. N. Barron, Viscosity solutions and analysis in , in Nonlinear Analysis, Differential Equations and Control, Dordrecht 1999, 160. MR 1695005 (2001c:49045)
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 [BP]
 H. Brezis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Functional Analysis 9 (1972), 6374. MR 0293452 (45:2529)
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 R.V. Kohn and S. Serfaty, A deterministic controlbased approach to motion by mean curvature, preprint, 2004.
 [LG]
 E. Le Gruyer, On absolutely minimizing Lipschitz extensions and the PDE , preprint, 2004.
 [LGA]
 E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Analysis, 29 (1998), 279292. MR 1617186 (99d:54008)
 [O]
 A. Oberman, Convergent difference schemes for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, preprint.
 [PSSW]
 Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug of war and the infinity Laplacian, preprint.
 [P]
 M. Portilheiro, Weak solutions for equations defined by accretive operators I, Proceedings Royal Soc Edinburgh 133A (2003), 11931207. MR 2018332 (2004j:47130)
 [S]
 S. Sheffield, ``Tug of war and the infinity Laplacian'', lecture presented at UC Berkeley, 2004.
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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/S0002994707043383
PII:
S 00029947(07)043383
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 DMS0200169
The second author was supported in part by NSF Grant DMS0500452.
Article copyright:
© Copyright 2007
American Mathematical Society
