A finite difference approach to the infinity Laplace equation and tug-of-war games

Armstrong, Scott; Smart, Charles

doi:10.1090/S0002-9947-2011-05289-X

A finite difference approach to the infinity Laplace equation and tug-of-war games
HTML articles powered by AMS MathViewer

by Scott N. Armstrong and Charles K. Smart PDF

Trans. Amer. Math. Soc. 364 (2012), 595-636 Request permission

Abstract:

We present a modified version of the two-player “tug-of-war” game introduced by Peres, Schramm, Sheffield, and Wilson (2009). This new tug-of-war game is identical to the original except near the boundary of the domain $\partial \Omega$, but its associated value functions are more regular. The dynamic programming principle implies that the value functions satisfy a certain finite difference equation. By studying this difference equation directly and adapting techniques from viscosity solution theory, we prove a number of new results.

We show that the finite difference equation has unique maximal and minimal solutions, which are identified as the value functions for the two tug-of-war players. We demonstrate uniqueness, and hence the existence of a value for the game, in the case that the running payoff function is nonnegative. We also show that uniqueness holds in certain cases for sign-changing running payoff functions which are sufficiently small. In the limit $\varepsilon \to 0$, we obtain the convergence of the value functions to a viscosity solution of the normalized infinity Laplace equation.

We also obtain several new results for the normalized infinity Laplace equation $-\Delta _\infty u = f$. In particular, we demonstrate the existence of solutions to the Dirichlet problem for any bounded continuous $f$, and continuous boundary data, as well as the uniqueness of solutions to this problem in the generic case. We present a new elementary proof of uniqueness in the case that $f>0$, $f< 0$, or $f\equiv 0$. The stability of the solutions with respect to $f$ is also studied, and an explicit continuous dependence estimate from $f\equiv 0$ is obtained.

References

Scott N. Armstrong and Charles K. Smart, An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions, Calc. Var. Partial Differential Equations 37 (2010), no. 3-4, 381–384. MR 2592977, DOI 10.1007/s00526-009-0267-9
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
G. Barles and Jérôme Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. Partial Differential Equations 26 (2001), no. 11-12, 2323–2337. MR 1876420, DOI 10.1081/PDE-100107824
E. N. Barron, L. C. Evans, and R. Jensen, The infinity Laplacian, Aronsson’s equation and their generalizations, Trans. Amer. Math. Soc. 360 (2008), no. 1, 77–101. MR 2341994, DOI 10.1090/S0002-9947-07-04338-3
Fernando Charro, Jesus García Azorero, and Julio D. Rossi, A mixed problem for the infinity Laplacian via tug-of-war games, Calc. Var. Partial Differential Equations 34 (2009), no. 3, 307–320. MR 2471139, DOI 10.1007/s00526-008-0185-2
M. G. Crandall, L. C. Evans, and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations 13 (2001), no. 2, 123–139. MR 1861094, DOI 10.1007/s005260000065
Michael G. Crandall, A visit with the $\infty$-Laplace equation, Calculus of variations and nonlinear partial differential equations, Lecture Notes in Math., vol. 1927, Springer, Berlin, 2008, pp. 75–122. MR 2408259, DOI 10.1007/978-3-540-75914-0_{3}
Michael G. Crandall, Gunnar Gunnarsson, and Peiyong Wang, Uniqueness of $\infty$-harmonic functions and the eikonal equation, Comm. Partial Differential Equations 32 (2007), no. 10-12, 1587–1615. MR 2372480, DOI 10.1080/03605300601088807
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
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
E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta _\infty (u)=0$, NoDEA Nonlinear Differential Equations Appl. 14 (2007), no. 1-2, 29–55. MR 2346452, DOI 10.1007/s00030-006-4030-z
Guozhen Lu and Peiyong Wang, Infinity Laplace equation with non-trivial right-hand side, preprint.
Guozhen Lu and Peiyong Wang, A PDE perspective of the normalized infinity Laplacian, Comm. Partial Differential Equations 33 (2008), no. 10-12, 1788–1817. MR 2475319, DOI 10.1080/03605300802289253
Adam M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Math. Comp. 74 (2005), no. 251, 1217–1230. MR 2137000, DOI 10.1090/S0025-5718-04-01688-6
Yuval Peres, Gábor Pete, and Stephanie Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones, preprint.
Yuval Peres, Oded Schramm, Scott Sheffield, and David B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 167–210. MR 2449057, DOI 10.1090/S0894-0347-08-00606-1
Yifeng Yu, Uniqueness of values of Aronsson operators and running costs in “tug-of-war” games, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 4, 1299–1308. MR 2542726, DOI 10.1016/j.anihpc.2008.11.001