## The infinity Laplacian, Aronsson’s equation and their generalizations

HTML articles powered by AMS MathViewer

- by E. N. Barron, L. C. Evans and R. Jensen PDF
- Trans. Amer. Math. Soc.
**360**(2008), 77-101 Request permission

## 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 $\text {ess-sup}_{U}|Du|.$ The more general functional $\text {ess-sup}_{U}F(x,u,Du)$ 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.## References

- Gunnar Aronsson,
*Minimization problems for the functional $\textrm {sup}_{x}\,F(x,\,f(x),\,f^{\prime } (x))$*, Ark. Mat.**6**(1965), 33–53 (1965). MR**196551**, DOI 10.1007/BF02591326 - Gunnar Aronsson,
*Minimization problems for the functional $\textrm {sup}_{x}\, F(x, f(x),f^\prime (x))$. II*, Ark. Mat.**6**(1966), 409–431 (1966). MR**203541**, DOI 10.1007/BF02590964 - Gunnar Aronsson,
*Extension of functions satisfying Lipschitz conditions*, Ark. Mat.**6**(1967), 551–561 (1967). MR**217665**, DOI 10.1007/BF02591928 - Gunnar Aronsson,
*On the partial differential equation $u_{x}{}^{2}\!u_{xx} +2u_{x}u_{y}u_{xy}+u_{y}{}^{2}\!u_{yy}=0$*, Ark. Mat.**7**(1968), 395–425 (1968). MR**237962**, DOI 10.1007/BF02590989 - Gunnar Aronsson,
*Minimization problems for the functional $\textrm {sup}_{x}\,F(x,\,f(x),\,f^{\prime } \,(x))$. III*, Ark. Mat.**7**(1969), 509–512. MR**240690**, DOI 10.1007/BF02590888 - 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**, DOI 10.1090/S0273-0979-04-01035-3 - E. N. Barron,
*Viscosity solutions and analysis in $L^\infty$*, 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** - E. N. Barron, R. R. Jensen, and C. Y. Wang,
*The Euler equation and absolute minimizers of $L^\infty$ functionals*, Arch. Ration. Mech. Anal.**157**(2001), no. 4, 255–283. MR**1831173**, DOI 10.1007/PL00004239 - 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**, DOI 10.1016/0022-1236(72)90014-6 - 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** - 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**, DOI 10.1090/S0273-0979-1992-00266-5 - 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**, DOI 10.1017/S0308210500018631 - 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**, DOI 10.1007/BFb0094296 - 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** - Mark Freidlin,
*Functional integration and partial differential equations*, Annals of Mathematics Studies, vol. 109, Princeton University Press, Princeton, NJ, 1985. MR**833742**, DOI 10.1515/9781400881598 - 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**, DOI 10.1007/BF00386368 - R.V. Kohn and S. Serfaty, A deterministic control-based approach to motion by mean curvature, preprint, 2004.
- E. Le Gruyer, On absolutely minimizing Lipschitz extensions and the PDE $\Delta _\infty (u)=0$, preprint, 2004.
- E. Le Gruyer and J. C. Archer,
*Harmonious extensions*, SIAM J. Math. Anal.**29**(1998), no. 1, 279–292. MR**1617186**, DOI 10.1137/S0036141095294067 - A. Oberman, Convergent difference schemes for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, preprint.
- Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug of war and the infinity Laplacian, preprint.
- 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**, DOI 10.1017/S0308210500002870 - S. Sheffield, “Tug of war and the infinity Laplacian”, lecture presented at UC Berkeley, 2004.

## Additional Information

**E. N. Barron**- Affiliation: Department of Mathematics and Statistics, Loyola University of Chicago, Chicago, Illinois 60626
- MR Author ID: 31685
- 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
- MR Author ID: 205502
- Email: rjensen@luc.edu
- 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. - © Copyright 2007 American Mathematical Society
- 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
- MathSciNet review: 2341994