The infinity Laplacian, Aronsson’s equation and their generalizations

Barron, E.; Evans, L.; Jensen, R.

doi:10.1090/S0002-9947-07-04338-3

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.