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 | 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.

**[A1]**G. Aronsson, Minimization problems for the functional , Arkiv für Mate. 6 (1965), 33-53. MR**0196551 (33:4738)****[A2]**G. Aronsson, Minimization problems for the functional . II, Arkiv für Mate. 6 (1966), 409-431. MR**0203541 (34:3391)****[A3]**G. Aronsson, Extension of functions satisfying Lipschitz conditions, Arkiv für Mate. 6 (1967), 551-561. MR**0217665 (36:754)****[A4]**G. Aronsson, On the partial differential equation , Arkiv für Mate. 7 (1968), 395-425. MR**0237962 (38:6239)****[A5]**G. Aronsson, Minimization problems for the functional . III, Arkiv für Mate. 7 (1969), 509-512. MR**0240690 (39:2035)****[A-C-J]**G. Aronsson, M. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. (N.S.), American Mathematical Society. Bulletin. New Series, 41 (2004), 439-505. MR**2083637 (2005k:35159)****[B]**E. N. Barron, Viscosity solutions and analysis in , in*Nonlinear Analysis, Differential Equations and Control*, Dordrecht 1999, 1-60. MR**1695005 (2001c:49045)****[B-J-W]**E. N. Barron, R. Jensen and C. Wang, The Euler equation and absolute minimizers of functionals, Arch. Rational Mech. Analysis 157 (2001), 255-283. MR**1831173 (2002m:49006)****[B-P]**H. Brezis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Functional Analysis 9 (1972), 63-74. MR**0293452 (45:2529)****[C-G-G]**Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Journal of Differential Geometry 33 (1991), 749-786. MR**1100211 (93a:35093)****[C-I-L]**M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin American Math. Soc., 27 (1992), 1-67. MR**1118699 (92j:35050)****[E1]**L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Royal Society of Edinburgh 111A (1989), 359-375. MR**1007533 (91c:35017)****[E2]**L. C. Evans, Regularity for fully nonlinear elliptic PDE and mean curvature motion (survey paper), in*Viscosity Solutions and Applications*, edited by I. Capuzzo-Dolcetta and P.-L. Lions, Lecture Notes in Mathematics 1660, Springer, 1997. MR**1462701 (98f:35046)****[E-S]**L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, Journal of Differential Geometry 33 (1991), 635-681. MR**1100206 (92h:35097)****[F]**M. Friedlin,*Functional Integration and Partial Differential Equations*, Princeton University Press, Volume 109, Annals of Mathematics, 1985. MR**0833742 (87g:60066)****[J]**R. Jensen, Uniqueness of Lipschitz extensions minimizing the sup-norm of the gradient, Arch. Rational Mech. Analysis 123 (1993), 51-74. MR**1218686 (94g:35063)****[K-S]**R.V. Kohn and S. Serfaty, A deterministic control-based approach to motion by mean curvature, preprint, 2004.**[LG]**E. Le Gruyer, On absolutely minimizing Lipschitz extensions and the PDE , preprint, 2004.**[LG-A]**E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Analysis, 29 (1998), 279-292. MR**1617186 (99d:54008)****[O]**A. Oberman, Convergent difference schemes for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, preprint.**[P-S-S-W]**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), 1193-1207. 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:
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