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 & 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,, \emph{Communications in Partial Differential Equations}, 37 (2012), 1839. doi: 10.1080/03605302.2011.642450. Google Scholar [2] S. N. Armstrong, C. K. Smart and S. J. Somersille, An infinity Laplace equation with gradient term and mixed boundary conditions,, \emph{Proc. Amer. Math. Soc.}, 139 (2011), 1763. doi: 10.1090/S0002-9939-2010-10666-4. Google Scholar [3] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions,, \emph{Bull. Amer. Math. Soc.}, 41 (2004), 439. doi: 10.1090/S0273-0979-04-01035-3. Google Scholar [4] T. Bhattacharya, E. Di Benedetto and J. Manfredi, Limits as $p \to \infty$ of $\Delta_p u_p = f$ and related extremal problems,, \emph{Rend. Sem. Mat. Univ. Politec. Torino}, (1991), 15. Google Scholar [5] C. Bjorland, L. Caffarelli and A. Figalli, Non-local tug-of-war and the infinity fractional Laplacian,, \emph{Comm. Pure. Appl. Math.}, 65 (2012), 337. doi: 10.1002/cpa.21379. Google Scholar [6] V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation,, \emph{IEEE Trans. Image Process}, 7 (1998), 376. doi: 10.1109/83.661188. Google Scholar [7] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations,, \emph{Bull. Amer. Math. Soc.}, 27 (1992), 1. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar [8] A. P. Maitra and W. D. Sudderth, Discrete Gambling and Stochastic Games,, Applications of Mathematics 32, (1996). doi: 10.1007/978-1-4612-4002-0. Google Scholar [9] J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization of $p$-harmonic functions,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 881. doi: 10.1090/S0002-9939-09-10183-1. Google Scholar [10] J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise,, \emph{Control Optim. Calc. Var. COCV}, 18 (2012), 81. doi: 10.1051/cocv/2010046. Google Scholar [11] J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of $p$-harmonious functions,, \emph{Annali Scuola Normale Sup. Pisa, XI (2012), 215. Google Scholar [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,, \emph{SIAM J. Math. Anal.}, 42 (2010), 2058. doi: 10.1137/100782073. Google Scholar [13] Y. Peres, G. Pete and S. Somersille, Biased Tug-of-War, the biased infinity Laplacian and comparison with exponential cones,, \emph{Calc. Var. Partial Differential Equations}, 38 (2010), 541. doi: 10.1007/s00526-009-0298-2. Google Scholar [14] Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian,, \emph{J. Amer. Math. Soc.}, 22 (2009), 167. doi: 10.1090/S0894-0347-08-00606-1. Google Scholar [15] Y. Peres and S. Sheffield, Tug-of-war with noise: a game theoretic view of the $p$-Laplacian,, \emph{Duke Math. J.}, 145 (2008), 91. doi: 10.1215/00127094-2008-048. Google Scholar [16] J. D. Rossi, E. V. Teixeira and J. M. Urbano, Optimal regularity at the free boundary for the infinity obstacle problem,, Preprint., (). Google Scholar

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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,, \emph{Communications in Partial Differential Equations}, 37 (2012), 1839. doi: 10.1080/03605302.2011.642450. Google Scholar [2] S. N. Armstrong, C. K. Smart and S. J. Somersille, An infinity Laplace equation with gradient term and mixed boundary conditions,, \emph{Proc. Amer. Math. Soc.}, 139 (2011), 1763. doi: 10.1090/S0002-9939-2010-10666-4. Google Scholar [3] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions,, \emph{Bull. Amer. Math. Soc.}, 41 (2004), 439. doi: 10.1090/S0273-0979-04-01035-3. Google Scholar [4] T. Bhattacharya, E. Di Benedetto and J. Manfredi, Limits as $p \to \infty$ of $\Delta_p u_p = f$ and related extremal problems,, \emph{Rend. Sem. Mat. Univ. Politec. Torino}, (1991), 15. Google Scholar [5] C. Bjorland, L. Caffarelli and A. Figalli, Non-local tug-of-war and the infinity fractional Laplacian,, \emph{Comm. Pure. Appl. Math.}, 65 (2012), 337. doi: 10.1002/cpa.21379. Google Scholar [6] V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation,, \emph{IEEE Trans. Image Process}, 7 (1998), 376. doi: 10.1109/83.661188. Google Scholar [7] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations,, \emph{Bull. Amer. Math. Soc.}, 27 (1992), 1. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar [8] A. P. Maitra and W. D. Sudderth, Discrete Gambling and Stochastic Games,, Applications of Mathematics 32, (1996). doi: 10.1007/978-1-4612-4002-0. Google Scholar [9] J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization of $p$-harmonic functions,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 881. doi: 10.1090/S0002-9939-09-10183-1. Google Scholar [10] J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise,, \emph{Control Optim. Calc. Var. COCV}, 18 (2012), 81. doi: 10.1051/cocv/2010046. Google Scholar [11] J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of $p$-harmonious functions,, \emph{Annali Scuola Normale Sup. Pisa, XI (2012), 215. Google Scholar [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,, \emph{SIAM J. Math. Anal.}, 42 (2010), 2058. doi: 10.1137/100782073. Google Scholar [13] Y. Peres, G. Pete and S. Somersille, Biased Tug-of-War, the biased infinity Laplacian and comparison with exponential cones,, \emph{Calc. Var. Partial Differential Equations}, 38 (2010), 541. doi: 10.1007/s00526-009-0298-2. Google Scholar [14] Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian,, \emph{J. Amer. Math. Soc.}, 22 (2009), 167. doi: 10.1090/S0894-0347-08-00606-1. Google Scholar [15] Y. Peres and S. Sheffield, Tug-of-war with noise: a game theoretic view of the $p$-Laplacian,, \emph{Duke Math. J.}, 145 (2008), 91. doi: 10.1215/00127094-2008-048. Google Scholar [16] J. D. Rossi, E. V. Teixeira and J. M. Urbano, Optimal regularity at the free boundary for the infinity obstacle problem,, Preprint., (). Google Scholar
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