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

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

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

Ivana Gómez ^1, and Julio D. Rossi ^2,

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

Abstract
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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.

Keywords: viscosity solutions, Tug-of-war games, infinity Laplacian..

Mathematics Subject Classification: Primary: 35B50, 35J25, 35J70, 49N70, 91A15, 91A2.

Citation: Ivana Gómez, Julio D. Rossi. Tug-of-war games and the infinity Laplacian with spatial dependence. Communications on Pure and 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), 1839-1869. doi: 10.1080/03605302.2011.642450.
[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), 595-636. doi: 10.1090/S0002-9947-2011-05289-X.
[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), 381-384. doi: 10.1007/s00526-009-0267-9.
[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), 1763-1776. doi: 10.1090/S0002-9939-2010-10666-4.
[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), 439-505. doi: 10.1090/S0273-0979-04-01035-3.
[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), 2323-2337. doi: 10.1081/PDE-100107824.
[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), 77-101. doi: 10.1090/S0002-9947-07-04338-3.
[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), 28-52 (electronic). doi: 10.1051/cocv:2003035.
[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), 123-138. doi: 10.4171/JEMS/40.
[10]	T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as $p\to\infty$ of $\Delta_pu_p=f$ and related extremal problems. Some topics in nonlinear PDEs (Turin, 1989), Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (1991), 15-68.
[11]	Thierry Champion and Luigi De Pascale, Principles of comparison with distance functions for absolute minimizers, J. Convex. Anal., 14 (2007), 515-541.
[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), 307-320. doi: 10.1007/s00526-008-0185-2.
[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), 1965-1981. doi: 10.1080/03605300701454792.
[14]	Michael G. Crandall, Gunnar Gunnarsson and Peiyong Wang, Uniqueness of $\infty$-harmonic functions and the eikonal equation, Comm. Partial Differential Equations, 32 (2007), 1587-1615. doi: 10.1080/03605300601088807.
[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), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[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), 325-347. doi: 10.1007/s00526-007-0143-4.
[17]	Lawrence C. Evans and Charles K. Smart, Everywhere differentiability of infinity harmonic functions, Calc. Var. Partial Differential Equations, 42 (2011), 289-299. doi: 10.1007/s00526-010-0388-1.
[18]	E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta_\infty(u)=0$, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 29-55. doi: 10.1007/s00030-006-4030-z.
[19]	E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Anal., 29 (1998), 279-292. doi: 10.1137/S0036141095294067.
[20]	Toshihiro Ishibashi and Shigeaki Koike, On fully nonlinear PDEs derived from variational problems of $L^p$ norms, SIAM J. Math. Anal., 33 (2001), 545-569. doi: 10.1137/S0036141000380000.
[21]	Robert Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74. doi: 10.1007/BF00386368.
[22]	Petri Juutinen and Peter Lindqvist, On the higher eigenvalues for the $\infty$-eigenvalue problem, Calc. Var. Partial Differential Equations, 23 (2005), 169-192. doi: 10.1007/s00526-004-0295-4.
[23]	Petri Juutinen, Peter Lindqvist and Juan J. Manfredi, The $\infty$-eigenvalue problem, Arch. Rational Mech. Anal., 148 (1999), 89-105. doi: 10.1007/s002050050157.
[24]	Robert V. Kohn and Sylvia Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math., 59 (2006), 344-407. doi: 10.1002/cpa.20101.
[25]	Ashok P. Maitra and William D. Sudderth, "Discrete Gambling and Stochastic Games," Applications of Mathematics (New York) 32, Springer-Verlag, 1996.
[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), 81-90. doi: 10.1051/cocv/2010046.
[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), 215-241. doi: 10.2422/2036-2145.201005_003.
[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), 881-889. doi: 10.1090/S0002-9939-09-10183-1.
[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), 2058-2081. doi: 10.1137/100782073.
[30]	Adam M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Math. Comp., 74 (2005), 1217-1230. doi: 10.1090/S0025-5718-04-01688-6.
[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), 541-564. doi: 10.1007/s00526-009-0298-2.
[32]	Yuval Peres, Oded Schramm, Scott Sheffield and David B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210. doi: 10.1090/S0894-0347-08-00606-1.
[33]	Yuval Peres and Scott Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian, Duke Math. J., 145 (2008), 91-120. doi: 10.1215/00127094-2008-048.
[34]	Julio D. Rossi and Mariel Saez, Optimal regularity for the pseudo infinity Laplacian, ESAIM Control Optim. Calc. Var., 13 (2007), 294-304. doi: 10.1051/cocv:2007018.
[35]	Ovidiu Savin, $C^1$ regularity for infinity harmonic functions in two dimensions, Arch. Ration. Mech. Anal., 176 (2005), 351-361. doi: 10.1007/s00205-005-0355-8.
[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).

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), 1839-1869. doi: 10.1080/03605302.2011.642450.
[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), 595-636. doi: 10.1090/S0002-9947-2011-05289-X.
[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), 381-384. doi: 10.1007/s00526-009-0267-9.
[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), 1763-1776. doi: 10.1090/S0002-9939-2010-10666-4.
[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), 439-505. doi: 10.1090/S0273-0979-04-01035-3.
[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), 2323-2337. doi: 10.1081/PDE-100107824.
[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), 77-101. doi: 10.1090/S0002-9947-07-04338-3.
[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), 28-52 (electronic). doi: 10.1051/cocv:2003035.
[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), 123-138. doi: 10.4171/JEMS/40.
[10]	T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as $p\to\infty$ of $\Delta_pu_p=f$ and related extremal problems. Some topics in nonlinear PDEs (Turin, 1989), Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (1991), 15-68.
[11]	Thierry Champion and Luigi De Pascale, Principles of comparison with distance functions for absolute minimizers, J. Convex. Anal., 14 (2007), 515-541.
[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), 307-320. doi: 10.1007/s00526-008-0185-2.
[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), 1965-1981. doi: 10.1080/03605300701454792.
[14]	Michael G. Crandall, Gunnar Gunnarsson and Peiyong Wang, Uniqueness of $\infty$-harmonic functions and the eikonal equation, Comm. Partial Differential Equations, 32 (2007), 1587-1615. doi: 10.1080/03605300601088807.
[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), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[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), 325-347. doi: 10.1007/s00526-007-0143-4.
[17]	Lawrence C. Evans and Charles K. Smart, Everywhere differentiability of infinity harmonic functions, Calc. Var. Partial Differential Equations, 42 (2011), 289-299. doi: 10.1007/s00526-010-0388-1.
[18]	E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta_\infty(u)=0$, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 29-55. doi: 10.1007/s00030-006-4030-z.
[19]	E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Anal., 29 (1998), 279-292. doi: 10.1137/S0036141095294067.
[20]	Toshihiro Ishibashi and Shigeaki Koike, On fully nonlinear PDEs derived from variational problems of $L^p$ norms, SIAM J. Math. Anal., 33 (2001), 545-569. doi: 10.1137/S0036141000380000.
[21]	Robert Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74. doi: 10.1007/BF00386368.
[22]	Petri Juutinen and Peter Lindqvist, On the higher eigenvalues for the $\infty$-eigenvalue problem, Calc. Var. Partial Differential Equations, 23 (2005), 169-192. doi: 10.1007/s00526-004-0295-4.
[23]	Petri Juutinen, Peter Lindqvist and Juan J. Manfredi, The $\infty$-eigenvalue problem, Arch. Rational Mech. Anal., 148 (1999), 89-105. doi: 10.1007/s002050050157.
[24]	Robert V. Kohn and Sylvia Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math., 59 (2006), 344-407. doi: 10.1002/cpa.20101.
[25]	Ashok P. Maitra and William D. Sudderth, "Discrete Gambling and Stochastic Games," Applications of Mathematics (New York) 32, Springer-Verlag, 1996.
[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), 81-90. doi: 10.1051/cocv/2010046.
[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), 215-241. doi: 10.2422/2036-2145.201005_003.
[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), 881-889. doi: 10.1090/S0002-9939-09-10183-1.
[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), 2058-2081. doi: 10.1137/100782073.
[30]	Adam M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Math. Comp., 74 (2005), 1217-1230. doi: 10.1090/S0025-5718-04-01688-6.
[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), 541-564. doi: 10.1007/s00526-009-0298-2.
[32]	Yuval Peres, Oded Schramm, Scott Sheffield and David B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210. doi: 10.1090/S0894-0347-08-00606-1.
[33]	Yuval Peres and Scott Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian, Duke Math. J., 145 (2008), 91-120. doi: 10.1215/00127094-2008-048.
[34]	Julio D. Rossi and Mariel Saez, Optimal regularity for the pseudo infinity Laplacian, ESAIM Control Optim. Calc. Var., 13 (2007), 294-304. doi: 10.1051/cocv:2007018.
[35]	Ovidiu Savin, $C^1$ regularity for infinity harmonic functions in two dimensions, Arch. Ration. Mech. Anal., 176 (2005), 351-361. doi: 10.1007/s00205-005-0355-8.
[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).

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