A finite difference approach to the infinity Laplace equation and tug-of-war games
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- by Scott N. Armstrong and Charles K. Smart PDF
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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
Additional Information
- Scott N. Armstrong
- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
- Email: sarm@math.berkeley.edu
- Charles K. Smart
- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
- MR Author ID: 893148
- Email: smart@math.berkeley.edu
- Received by editor(s): July 8, 2009
- Published electronically: September 14, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 595-636
- MSC (2000): Primary 35J70, 91A15
- DOI: https://doi.org/10.1090/S0002-9947-2011-05289-X
- MathSciNet review: 2846345