# American Institute of Mathematical Sciences

January  2015, 14(1): 217-228. doi: 10.3934/cpaa.2015.14.217

## An obstacle problem for Tug-of-War games

 1 Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States 2 Departamento de Análisis Matemático, Universidad de Alicante, Ap 99, 03080, Alicante, Spain 3 Department of Mathematics, Dartmouth College, Hanover, NH 03755, United States

Received  March 2014 Revised  April 2014 Published  September 2014

We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tug-of-war.
Citation: 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
##### References:
 [1] T. Antunović, Y. Peres and S. Sheffield and S. Somersille, Tug-of-War and infinity Laplace equation with vanishing Neumann boundary conditions, Communications in Partial Differential Equations, 37 (2012), 1839-1869. doi: 10.1080/03605302.2011.642450. [2] S. N. Armstrong, C. K. Smart and S. 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. [3] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3. [4] T. Bhattacharya, E. Di Benedetto and J. Manfredi, Limits as $p \to \infty$ of $\Delta_p u_p = f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, (1991), 15-68. [5] C. Bjorland, L. Caffarelli and A. Figalli, Non-local tug-of-war and the infinity fractional Laplacian, Comm. Pure. Appl. Math., 65 (2012), 337-380. doi: 10.1002/cpa.21379. [6] V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process, 7 (1998), 376-386. doi: 10.1109/83.661188. [7] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [8] A. P. Maitra and W. D. Sudderth, Discrete Gambling and Stochastic Games, Applications of Mathematics 32, Springer-Verlag, 1996. doi: 10.1007/978-1-4612-4002-0. [9] J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization of $p$-harmonic functions, Proc. Amer. Math. Soc., 138 (2010), 881-889. doi: 10.1090/S0002-9939-09-10183-1. [10] J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise, Control Optim. Calc. Var. COCV, 18 (2012), 81-90. doi: 10.1051/cocv/2010046. [11] J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of $p$-harmonious functions, Annali Scuola Normale Sup. Pisa, Clase di Scienze, XI (2012), 215-241. [12] J. J. Manfredi, M. Parviainen and J. 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. [13] Y. Peres, G. Pete and S. 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. [14] Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210. doi: 10.1090/S0894-0347-08-00606-1. [15] Y. Peres and S. 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. [16] J. D. Rossi, E. V. Teixeira and J. M. Urbano, Optimal regularity at the free boundary for the infinity obstacle problem, Preprint.

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##### References:
 [1] T. Antunović, Y. Peres and S. Sheffield and S. Somersille, Tug-of-War and infinity Laplace equation with vanishing Neumann boundary conditions, Communications in Partial Differential Equations, 37 (2012), 1839-1869. doi: 10.1080/03605302.2011.642450. [2] S. N. Armstrong, C. K. Smart and S. 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. [3] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3. [4] T. Bhattacharya, E. Di Benedetto and J. Manfredi, Limits as $p \to \infty$ of $\Delta_p u_p = f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, (1991), 15-68. [5] C. Bjorland, L. Caffarelli and A. Figalli, Non-local tug-of-war and the infinity fractional Laplacian, Comm. Pure. Appl. Math., 65 (2012), 337-380. doi: 10.1002/cpa.21379. [6] V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process, 7 (1998), 376-386. doi: 10.1109/83.661188. [7] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [8] A. P. Maitra and W. D. Sudderth, Discrete Gambling and Stochastic Games, Applications of Mathematics 32, Springer-Verlag, 1996. doi: 10.1007/978-1-4612-4002-0. [9] J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization of $p$-harmonic functions, Proc. Amer. Math. Soc., 138 (2010), 881-889. doi: 10.1090/S0002-9939-09-10183-1. [10] J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise, Control Optim. Calc. Var. COCV, 18 (2012), 81-90. doi: 10.1051/cocv/2010046. [11] J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of $p$-harmonious functions, Annali Scuola Normale Sup. Pisa, Clase di Scienze, XI (2012), 215-241. [12] J. J. Manfredi, M. Parviainen and J. 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. [13] Y. Peres, G. Pete and S. 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. [14] Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210. doi: 10.1090/S0894-0347-08-00606-1. [15] Y. Peres and S. 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. [16] J. D. Rossi, E. V. Teixeira and J. M. Urbano, Optimal regularity at the free boundary for the infinity obstacle problem, Preprint.
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