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.

**[A1]**Gunnar Aronsson,*Minimization problems for the functional 𝑠𝑢𝑝ₓ𝐹(𝑥,𝑓(𝑥),𝑓′(𝑥))*, Ark. Mat.**6**(1965), 33–53 (1965). MR**0196551****[A2]**Gunnar Aronsson,*Minimization problems for the functional 𝑠𝑢𝑝ₓ𝐹(𝑥,𝑓(𝑥),𝑓′(𝑥)). II*, Ark. Mat.**6**(1966), 409–431 (1966). MR**0203541****[A3]**Gunnar Aronsson,*Extension of functions satisfying Lipschitz conditions*, Ark. Mat.**6**(1967), 551–561 (1967). MR**0217665****[A4]**Gunnar Aronsson,*On the partial differential equation 𝑢ₓ²𝑢ₓₓ+2𝑢ₓ𝑢_{𝑦}𝑢_{𝑥𝑦}+𝑢_{𝑦}²𝑢_{𝑦𝑦}=0*, Ark. Mat.**7**(1968), 395–425 (1968). MR**0237962****[A5]**Gunnar Aronsson,*Minimization problems for the functional 𝑠𝑢𝑝ₓ𝐹(𝑥,𝑓(𝑥),𝑓′(𝑥)). III*, Ark. Mat.**7**(1969), 509–512. MR**0240690****[A-C-J]**Gunnar Aronsson, Michael G. Crandall, and Petri Juutinen,*A tour of the theory of absolutely minimizing functions*, Bull. Amer. Math. Soc. (N.S.)**41**(2004), no. 4, 439–505. MR**2083637**, 10.1090/S0273-0979-04-01035-3**[B]**E. N. Barron,*Viscosity solutions and analysis in 𝐿^{∞}*, Nonlinear analysis, differential equations and control (Montreal, QC, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 528, Kluwer Acad. Publ., Dordrecht, 1999, pp. 1–60. MR**1695005****[B-J-W]**E. N. Barron, R. R. Jensen, and C. Y. Wang,*The Euler equation and absolute minimizers of 𝐿^{∞} functionals*, Arch. Ration. Mech. Anal.**157**(2001), no. 4, 255–283. MR**1831173**, 10.1007/PL00004239**[B-P]**H. Brézis and A. Pazy,*Convergence and approximation of semigroups of nonlinear operators in Banach spaces*, J. Functional Analysis**9**(1972), 63–74. MR**0293452****[C-G-G]**Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto,*Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations*, J. Differential Geom.**33**(1991), no. 3, 749–786. MR**1100211****[C-I-L]**Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions,*User’s guide to viscosity solutions of second order partial differential equations*, Bull. Amer. Math. Soc. (N.S.)**27**(1992), no. 1, 1–67. MR**1118699**, 10.1090/S0273-0979-1992-00266-5**[E1]**Lawrence C. Evans,*The perturbed test function method for viscosity solutions of nonlinear PDE*, Proc. Roy. Soc. Edinburgh Sect. A**111**(1989), no. 3-4, 359–375. MR**1007533**, 10.1017/S0308210500018631**[E2]**Lawrence C. Evans,*Regularity for fully nonlinear elliptic equations and motion by mean curvature*, Viscosity solutions and applications (Montecatini Terme, 1995) Lecture Notes in Math., vol. 1660, Springer, Berlin, 1997, pp. 98–133. MR**1462701**, 10.1007/BFb0094296**[E-S]**L. C. Evans and J. Spruck,*Motion of level sets by mean curvature. I*, J. Differential Geom.**33**(1991), no. 3, 635–681. MR**1100206****[F]**Mark Freidlin,*Functional integration and partial differential equations*, Annals of Mathematics Studies, vol. 109, Princeton University Press, Princeton, NJ, 1985. MR**833742****[J]**Robert Jensen,*Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient*, Arch. Rational Mech. Anal.**123**(1993), no. 1, 51–74. MR**1218686**, 10.1007/BF00386368**[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. Anal.**29**(1998), no. 1, 279–292 (electronic). MR**1617186**, 10.1137/S0036141095294067**[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]**Manuel Portilheiro,*Weak solutions for equations defined by accretive operators. I*, Proc. Roy. Soc. Edinburgh Sect. A**133**(2003), no. 5, 1193–1207. MR**2018332**, 10.1017/S0308210500002870**[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