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), 1839-1869. 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), 595-636. 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), 381-384. 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), 1763-1776. 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), 439-505. 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), 2323-2337. 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), 77-101. 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), 28-52 (electronic). 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), 123-138. 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. Some topics in nonlinear PDEs (Turin, 1989), Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (1991), 15-68.  Google Scholar

[11]

Thierry Champion and Luigi De Pascale, Principles of comparison with distance functions for absolute minimizers, J. Convex. Anal., 14 (2007), 515-541.  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), 307-320. 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), 1965-1981. 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), 1587-1615. 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), 1-67. 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), 325-347. 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), 289-299. 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), 29-55. 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), 279-292. 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), 545-569. 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), 51-74. 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), 169-192. 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), 89-105. 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), 344-407. 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, Springer-Verlag, 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), 81-90. 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), 215-241. 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), 881-889. 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), 2058-2081. 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), 1217-1230. 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), 541-564. 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), 167-210. 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), 91-120. 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), 294-304. 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), 351-361. 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), 1839-1869. 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), 595-636. 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), 381-384. 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), 1763-1776. 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), 439-505. 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), 2323-2337. 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), 77-101. 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), 28-52 (electronic). 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), 123-138. 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. Some topics in nonlinear PDEs (Turin, 1989), Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (1991), 15-68.  Google Scholar

[11]

Thierry Champion and Luigi De Pascale, Principles of comparison with distance functions for absolute minimizers, J. Convex. Anal., 14 (2007), 515-541.  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), 307-320. 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), 1965-1981. 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), 1587-1615. 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), 1-67. 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), 325-347. 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), 289-299. 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), 29-55. 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), 279-292. 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), 545-569. 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), 51-74. 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), 169-192. 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), 89-105. 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), 344-407. 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, Springer-Verlag, 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), 81-90. 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), 215-241. 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), 881-889. 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), 2058-2081. 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), 1217-1230. 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), 541-564. 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), 167-210. 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), 91-120. 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), 294-304. 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), 351-361. 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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