# American Institute of Mathematical Sciences

September  2013, 12(5): 1959-1983. doi: 10.3934/cpaa.2013.12.1959

## Tug-of-war games and the infinity Laplacian with spatial dependence

 1 Instituto de Matemática Aplicada del Litoral (IMAL), CONICET-UNL, Departamento de Matemática, Facultad de Ingeniería Química, UNL, Güemes 3450, S3000GLN Santa Fe, Argentina 2 Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, 1428 – Buenos Aires

Received  January 2012 Revised  October 2012 Published  January 2013

In this paper we look for PDEs that arise as limits of values of tug-of-war games when the possible movements of the game are taken in a family of sets that are not necessarily Euclidean balls. In this way we find existence of viscosity solutions to the Dirichlet problem for an equation of the form $- \langle D^2 v\cdot J_x(D v) ; J_x(Dv)\rangle (x) =0$, that is, an infinity Laplacian with spatial dependence. Here $J_x (Dv(x))$ is a vector that depends on the spatial location and the gradient of the solution.
Citation: Ivana Gómez, Julio D. Rossi. Tug-of-war games and the infinity Laplacian with spatial dependence. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1959-1983. doi: 10.3934/cpaa.2013.12.1959
##### References:
 [1] Tonći Antunović, Yuval Peres, Scott Sheffield and Stephanie Somersille, Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition,, Comm. Partial Differential Equations, 37 (2012), 1839.  doi: 10.1080/03605302.2011.642450.  Google Scholar [2] Scott N. Armstrong and Charles K. Smart, A finite difference approach to the infinity Laplace equation and tug-of-war games,, Trans. Amer. Math. Soc., 364 (2012), 595.  doi: 10.1090/S0002-9947-2011-05289-X.  Google Scholar [3] 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), 381.  doi: 10.1007/s00526-009-0267-9.  Google Scholar [4] Scott N. Armstrong, Charles K. Smart and Stephanie J. Somersille, An infinity Laplace equation with gradient term and mixed boundary conditions,, Proc. Amer. Math. Soc., 139 (2011), 1763.  doi: 10.1090/S0002-9939-2010-10666-4.  Google Scholar [5] 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), 439.  doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar [6] 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), 2323.  doi: 10.1081/PDE-100107824.  Google Scholar [7] E. N. Barron, L. C. Evans and R. Jensen, The infinity Laplacian, Aronsson's equation and their generalizations,, Trans. Amer. Math. Soc., 360 (2008), 77.  doi: 10.1090/S0002-9947-07-04338-3.  Google Scholar [8] Marino Belloni and Bernd Kawohl, The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as $p\to\infty$,, ESAIM Control Optim. Calc. Var., 10 (2004), 28.  doi: 10.1051/cocv:2003035.  Google Scholar [9] M. Belloni, B. Kawohl and P. Juutinen, The $p$-Laplace eigenvalue problem as $p\to\infty$ in a Finsler metric,, J. Eur. Math. Soc. (JEMS), 8 (2006), 123.  doi: 10.4171/JEMS/40.  Google Scholar [10] T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as $p\to\infty$ of $\Delta_pu_p=f$ and related extremal problems. , Rend. Sem. Mat. Univ. Politec. Torino, (1991), 15.   Google Scholar [11] Thierry Champion and Luigi De Pascale, Principles of comparison with distance functions for absolute minimizers,, J. Convex. Anal., 14 (2007), 515.   Google Scholar [12] 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), 307.  doi: 10.1007/s00526-008-0185-2.  Google Scholar [13] Fernando Charro and Ireneo Peral, Limit branch of solutions as $p\to\infty$ for a family of sub-diffusive problems related to the $p$-Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1965.  doi: 10.1080/03605300701454792.  Google Scholar [14] Michael G. Crandall, Gunnar Gunnarsson and Peiyong Wang, Uniqueness of $\infty$-harmonic functions and the eikonal equation,, Comm. Partial Differential Equations, 32 (2007), 1587.  doi: 10.1080/03605300601088807.  Google Scholar [15] 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), 1.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar [16] Lawrence C. Evans and Ovidiu Savin, $C^{1,\alpha}$ regularity for infinity harmonic functions in two dimensions,, Calc. Var. Partial Differential Equations, 32 (2008), 325.  doi: 10.1007/s00526-007-0143-4.  Google Scholar [17] Lawrence C. Evans and Charles K. Smart, Everywhere differentiability of infinity harmonic functions,, Calc. Var. Partial Differential Equations, 42 (2011), 289.  doi: 10.1007/s00526-010-0388-1.  Google Scholar [18] E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta_\infty(u)=0$, , NoDEA Nonlinear Differential Equations Appl., 14 (2007), 29.  doi: 10.1007/s00030-006-4030-z.  Google Scholar [19] E. Le Gruyer and J. C. Archer, Harmonious extensions,, SIAM J. Math. Anal., 29 (1998), 279.  doi: 10.1137/S0036141095294067.  Google Scholar [20] Toshihiro Ishibashi and Shigeaki Koike, On fully nonlinear PDEs derived from variational problems of $L^p$ norms,, SIAM J. Math. Anal., 33 (2001), 545.  doi: 10.1137/S0036141000380000.  Google Scholar [21] Robert Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient,, Arch. Rational Mech. Anal., 123 (1993), 51.  doi: 10.1007/BF00386368.  Google Scholar [22] Petri Juutinen and Peter Lindqvist, On the higher eigenvalues for the $\infty$-eigenvalue problem,, Calc. Var. Partial Differential Equations, 23 (2005), 169.  doi: 10.1007/s00526-004-0295-4.  Google Scholar [23] Petri Juutinen, Peter Lindqvist and Juan J. Manfredi, The $\infty$-eigenvalue problem, , Arch. Rational Mech. Anal., 148 (1999), 89.  doi: 10.1007/s002050050157.  Google Scholar [24] Robert V. Kohn and Sylvia Serfaty, A deterministic-control-based approach to motion by curvature,, Comm. Pure Appl. Math., 59 (2006), 344.  doi: 10.1002/cpa.20101.  Google Scholar [25] Ashok P. Maitra and William D. Sudderth, "Discrete Gambling and Stochastic Games,", Applications of Mathematics (New York) 32, (1996).   Google Scholar [26] Juan J. Manfredi, Mikko Parviainen and Julio D. Rossi, Dynamic programming principle for tug-of-war games with noise,, ESAIM Control Optim. Calc. Var., 18 (2012), 81.  doi: 10.1051/cocv/2010046.  Google Scholar [27] Juan J. Manfredi, Mikko Parviainen and Julio D. Rossi, On the definition and properties of $p$-harmonious functions,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 215.  doi: 10.2422/2036-2145.201005_003.  Google Scholar [28] Juan J. Manfredi, Mikko Parviainen and Julio D. Rossi, An asymptotic mean value characterization for $p$-harmonic functions,, Proc. Amer. Math. Soc., 138 (2010), 881.  doi: 10.1090/S0002-9939-09-10183-1.  Google Scholar [29] Juan J. Manfredi, Mikko Parviainen and Julio D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games,, SIAM J. Math. Anal., 42 (2010), 2058.  doi: 10.1137/100782073.  Google Scholar [30] Adam M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions,, Math. Comp., 74 (2005), 1217.  doi: 10.1090/S0025-5718-04-01688-6.  Google Scholar [31] Yuval Peres, Gábor Pete and Stephanie Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones,, Calc. Var. Partial Differential Equations, 38 (2010), 541.  doi: 10.1007/s00526-009-0298-2.  Google Scholar [32] Yuval Peres, Oded Schramm, Scott Sheffield and David B. Wilson, Tug-of-war and the infinity Laplacian,, J. Amer. Math. Soc., 22 (2009), 167.  doi: 10.1090/S0894-0347-08-00606-1.  Google Scholar [33] Yuval Peres and Scott Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian,, Duke Math. J., 145 (2008), 91.  doi: 10.1215/00127094-2008-048.  Google Scholar [34] Julio D. Rossi and Mariel Saez, Optimal regularity for the pseudo infinity Laplacian,, ESAIM Control Optim. Calc. Var., 13 (2007), 294.  doi: 10.1051/cocv:2007018.  Google Scholar [35] Ovidiu Savin, $C^1$ regularity for infinity harmonic functions in two dimensions,, Arch. Ration. Mech. Anal., 176 (2005), 351.  doi: 10.1007/s00205-005-0355-8.  Google Scholar [36] , "Sthocastic Games & Applications," Proceedings of the Nato Advanced Study Institute held in Stony Brook, NY, July 7-17, 1999, Abraham Neyman and Sylvain Sorin (eds.),, NATO Science Series C: Mathematical and Physical Sciences, (2003).   Google Scholar

show all references

##### References:
 [1] Tonći Antunović, Yuval Peres, Scott Sheffield and Stephanie Somersille, Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition,, Comm. Partial Differential Equations, 37 (2012), 1839.  doi: 10.1080/03605302.2011.642450.  Google Scholar [2] Scott N. Armstrong and Charles K. Smart, A finite difference approach to the infinity Laplace equation and tug-of-war games,, Trans. Amer. Math. Soc., 364 (2012), 595.  doi: 10.1090/S0002-9947-2011-05289-X.  Google Scholar [3] 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), 381.  doi: 10.1007/s00526-009-0267-9.  Google Scholar [4] Scott N. Armstrong, Charles K. Smart and Stephanie J. Somersille, An infinity Laplace equation with gradient term and mixed boundary conditions,, Proc. Amer. Math. Soc., 139 (2011), 1763.  doi: 10.1090/S0002-9939-2010-10666-4.  Google Scholar [5] 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), 439.  doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar [6] 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), 2323.  doi: 10.1081/PDE-100107824.  Google Scholar [7] E. N. Barron, L. C. Evans and R. Jensen, The infinity Laplacian, Aronsson's equation and their generalizations,, Trans. Amer. Math. Soc., 360 (2008), 77.  doi: 10.1090/S0002-9947-07-04338-3.  Google Scholar [8] Marino Belloni and Bernd Kawohl, The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as $p\to\infty$,, ESAIM Control Optim. Calc. Var., 10 (2004), 28.  doi: 10.1051/cocv:2003035.  Google Scholar [9] M. Belloni, B. Kawohl and P. Juutinen, The $p$-Laplace eigenvalue problem as $p\to\infty$ in a Finsler metric,, J. Eur. Math. Soc. (JEMS), 8 (2006), 123.  doi: 10.4171/JEMS/40.  Google Scholar [10] T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as $p\to\infty$ of $\Delta_pu_p=f$ and related extremal problems. , Rend. Sem. Mat. Univ. Politec. Torino, (1991), 15.   Google Scholar [11] Thierry Champion and Luigi De Pascale, Principles of comparison with distance functions for absolute minimizers,, J. Convex. Anal., 14 (2007), 515.   Google Scholar [12] 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), 307.  doi: 10.1007/s00526-008-0185-2.  Google Scholar [13] Fernando Charro and Ireneo Peral, Limit branch of solutions as $p\to\infty$ for a family of sub-diffusive problems related to the $p$-Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1965.  doi: 10.1080/03605300701454792.  Google Scholar [14] Michael G. Crandall, Gunnar Gunnarsson and Peiyong Wang, Uniqueness of $\infty$-harmonic functions and the eikonal equation,, Comm. Partial Differential Equations, 32 (2007), 1587.  doi: 10.1080/03605300601088807.  Google Scholar [15] 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), 1.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar [16] Lawrence C. Evans and Ovidiu Savin, $C^{1,\alpha}$ regularity for infinity harmonic functions in two dimensions,, Calc. Var. Partial Differential Equations, 32 (2008), 325.  doi: 10.1007/s00526-007-0143-4.  Google Scholar [17] Lawrence C. Evans and Charles K. Smart, Everywhere differentiability of infinity harmonic functions,, Calc. Var. Partial Differential Equations, 42 (2011), 289.  doi: 10.1007/s00526-010-0388-1.  Google Scholar [18] E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta_\infty(u)=0$, , NoDEA Nonlinear Differential Equations Appl., 14 (2007), 29.  doi: 10.1007/s00030-006-4030-z.  Google Scholar [19] E. Le Gruyer and J. C. Archer, Harmonious extensions,, SIAM J. Math. Anal., 29 (1998), 279.  doi: 10.1137/S0036141095294067.  Google Scholar [20] Toshihiro Ishibashi and Shigeaki Koike, On fully nonlinear PDEs derived from variational problems of $L^p$ norms,, SIAM J. Math. Anal., 33 (2001), 545.  doi: 10.1137/S0036141000380000.  Google Scholar [21] Robert Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient,, Arch. Rational Mech. Anal., 123 (1993), 51.  doi: 10.1007/BF00386368.  Google Scholar [22] Petri Juutinen and Peter Lindqvist, On the higher eigenvalues for the $\infty$-eigenvalue problem,, Calc. Var. Partial Differential Equations, 23 (2005), 169.  doi: 10.1007/s00526-004-0295-4.  Google Scholar [23] Petri Juutinen, Peter Lindqvist and Juan J. Manfredi, The $\infty$-eigenvalue problem, , Arch. Rational Mech. Anal., 148 (1999), 89.  doi: 10.1007/s002050050157.  Google Scholar [24] Robert V. Kohn and Sylvia Serfaty, A deterministic-control-based approach to motion by curvature,, Comm. Pure Appl. Math., 59 (2006), 344.  doi: 10.1002/cpa.20101.  Google Scholar [25] Ashok P. Maitra and William D. Sudderth, "Discrete Gambling and Stochastic Games,", Applications of Mathematics (New York) 32, (1996).   Google Scholar [26] Juan J. Manfredi, Mikko Parviainen and Julio D. Rossi, Dynamic programming principle for tug-of-war games with noise,, ESAIM Control Optim. Calc. Var., 18 (2012), 81.  doi: 10.1051/cocv/2010046.  Google Scholar [27] Juan J. Manfredi, Mikko Parviainen and Julio D. Rossi, On the definition and properties of $p$-harmonious functions,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 215.  doi: 10.2422/2036-2145.201005_003.  Google Scholar [28] Juan J. Manfredi, Mikko Parviainen and Julio D. Rossi, An asymptotic mean value characterization for $p$-harmonic functions,, Proc. Amer. Math. Soc., 138 (2010), 881.  doi: 10.1090/S0002-9939-09-10183-1.  Google Scholar [29] Juan J. Manfredi, Mikko Parviainen and Julio D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games,, SIAM J. Math. Anal., 42 (2010), 2058.  doi: 10.1137/100782073.  Google Scholar [30] Adam M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions,, Math. Comp., 74 (2005), 1217.  doi: 10.1090/S0025-5718-04-01688-6.  Google Scholar [31] Yuval Peres, Gábor Pete and Stephanie Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones,, Calc. Var. Partial Differential Equations, 38 (2010), 541.  doi: 10.1007/s00526-009-0298-2.  Google Scholar [32] Yuval Peres, Oded Schramm, Scott Sheffield and David B. Wilson, Tug-of-war and the infinity Laplacian,, J. Amer. Math. Soc., 22 (2009), 167.  doi: 10.1090/S0894-0347-08-00606-1.  Google Scholar [33] Yuval Peres and Scott Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian,, Duke Math. J., 145 (2008), 91.  doi: 10.1215/00127094-2008-048.  Google Scholar [34] Julio D. Rossi and Mariel Saez, Optimal regularity for the pseudo infinity Laplacian,, ESAIM Control Optim. Calc. Var., 13 (2007), 294.  doi: 10.1051/cocv:2007018.  Google Scholar [35] Ovidiu Savin, $C^1$ regularity for infinity harmonic functions in two dimensions,, Arch. Ration. Mech. Anal., 176 (2005), 351.  doi: 10.1007/s00205-005-0355-8.  Google Scholar [36] , "Sthocastic Games & Applications," Proceedings of the Nato Advanced Study Institute held in Stony Brook, NY, July 7-17, 1999, Abraham Neyman and Sylvain Sorin (eds.),, NATO Science Series C: Mathematical and Physical Sciences, (2003).   Google Scholar
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