# 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 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).
 [1] Ángel Arroyo, Joonas Heino, Mikko Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-laplacian. Communications on Pure and Applied Analysis, 2017, 16 (3) : 915-944. doi: 10.3934/cpaa.2017044 [2] Juan J. Manfredi, Julio D. Rossi, Stephanie J. Somersille. An obstacle problem for Tug-of-War games. Communications on Pure and Applied Analysis, 2015, 14 (1) : 217-228. doi: 10.3934/cpaa.2015.14.217 [3] Gang Li, Fen Gu, Feida Jiang. Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1449-1462. doi: 10.3934/cpaa.2020071 [4] Bernd Kawohl, Friedemann Schuricht. First eigenfunctions of the 1-Laplacian are viscosity solutions. Communications on Pure and Applied Analysis, 2015, 14 (1) : 329-339. doi: 10.3934/cpaa.2015.14.329 [5] Mostafa Ghelichi, A. M. Goltabar, H. R. Tavakoli, A. Karamodin. Neuro-fuzzy active control optimized by Tug of war optimization method for seismically excited benchmark highway bridge. Numerical Algebra, Control and Optimization, 2021, 11 (3) : 333-351. doi: 10.3934/naco.2020029 [6] Yinbin Deng, Yi Li, Wei Shuai. Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683 [7] Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1837-1855. doi: 10.3934/dcdss.2021007 [8] Jan Burczak, P. Kaplický. Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2401-2445. doi: 10.3934/cpaa.2016042 [9] Fang Liu. An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2395-2421. doi: 10.3934/cpaa.2018114 [10] Goro Akagi, Kazumasa Suzuki. On a certain degenerate parabolic equation associated with the infinity-laplacian. Conference Publications, 2007, 2007 (Special) : 18-27. doi: 10.3934/proc.2007.2007.18 [11] Oliver Juarez-Romero, William Olvera-Lopez, Francisco Sanchez-Sanchez. A simple family of solutions for forest games. Journal of Dynamics and Games, 2017, 4 (2) : 87-96. doi: 10.3934/jdg.2017006 [12] Iryna Pankratova, Andrey Piatnitski. On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 935-970. doi: 10.3934/dcdsb.2009.11.935 [13] Laura Olian Fannio. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 251-264. doi: 10.3934/dcds.1997.3.251 [14] Shu-Yu Hsu. Non-existence and behaviour at infinity of solutions of some elliptic equations. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769 [15] Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489 [16] Guofa Li, Yisheng Huang. Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3971-3989. doi: 10.3934/dcdsb.2021214 [17] Shige Peng, Mingyu Xu. Constrained BSDEs, viscosity solutions of variational inequalities and their applications. Mathematical Control and Related Fields, 2013, 3 (2) : 233-244. doi: 10.3934/mcrf.2013.3.233 [18] Graziano Crasta, Benedetto Piccoli. Viscosity solutions and uniqueness for systems of inhomogeneous balance laws. Discrete and Continuous Dynamical Systems, 1997, 3 (4) : 477-502. doi: 10.3934/dcds.1997.3.477 [19] Inwon C. Kim, Helen K. Lei. Degenerate diffusion with a drift potential: A viscosity solutions approach. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 767-786. doi: 10.3934/dcds.2010.27.767 [20] Boris P. Andreianov, Giuseppe Maria Coclite, Carlotta Donadello. Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5913-5942. doi: 10.3934/dcds.2017257

2020 Impact Factor: 1.916

## Metrics

• HTML views (0)
• Cited by (3)

• on AIMS