Article Contents
Article Contents

# General existence of solutions to dynamic programming equations

• We provide an alternative approach to the existence of solutions to dynamic programming equations arising in the discrete game-theoretic interpretations for various nonlinear partial differential equations including the infinity Laplacian, mean curvature flow and Hamilton-Jacobi type. Our general result is similar to Perron's method but adapted to the discrete situation.
Mathematics Subject Classification: 35A35, 49C20, 91A05, 91A15.

 Citation:

•  [1] T. Antunović, Y. Peres, S. Sheffield and S. 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] S. N. Armstrong and C. 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. [3] S. N. Armstrong and C. 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. [4] 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. [5] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1997,doi: 10.1007/978-0-8176-4755-1. [6] G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283. [7] F. Charro, J. García Azorero and J. 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. [8] 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. (N.S.), 27 (1992), 1-67.doi: 10.1090/S0273-0979-1992-00266-5. [9] Y. Giga and Q. Liu, A billiard-based game interpretation of the Neumann problem for the curve shortening equation, Adv. Differential Equations, 14 (2009), 201-240. [10] H. Ishii, Perron's method for Hamilton-Jacobi equations, Duke Math. J., 55 (1987), 369-384.doi: 10.1215/S0012-7094-87-05521-9. [11] H. Ishii, A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of eikonal type, Proc. Amer. Math. Soc., 100 (1987), 247-251.doi: 10.2307/2045953. [12] R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math., 59 (2006), 344-407.doi: 10.1002/cpa.20101. [13] R. V. Kohn and S. Serfaty, A deterministic-control-based approach to fully nonlinear parabolic and elliptic equations, Comm. Pure Appl. Math., 63 (2010), 1298-1350.doi: 10.1002/cpa.20336. [14] S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, vol. 13 of MSJ Memoirs, Mathematical Society of Japan, Tokyo, 2004. [15] E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Anal., 29 (1998), 279-292 (electronic).doi: 10.1137/S0036141095294067. [16] Q. Liu, Waiting time effect for motion by positive second derivatives and applications, Nonlinear Differential Equations Appl., 21 (2014), 589-620.doi: 10.1007/s00030-013-0259-5. [17] Q. Liu, Fattening and comparison principle for level-set equations of mean curvature type, SIAM J. Control Optim., 49 (2011), 2518-2541.doi: 10.1137/100814330. [18] Q. Liu and A. Schikorra, A game-tree approach to discrete infinity Laplacian with running costs, preprint. [19] H. Luiro, M. Parviainen and E. Saksman, On the existence and uniqueness of p-harmonious functions, Differential and Integral Equations, 27 (2014), 201-216. [20] 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. [21] J. J. Manfredi, M. Parviainen and J. 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. [22] J. J. Manfredi, J. D. Rossi and S. Somersille, An obstacle problem for tug-of-war games, preprint. [23] 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. [24] Y. Peres, O. Schramm, S. Sheffield and D. 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. [25] 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. [26] M. B. Rudd and H. A. Van Dyke, Median values, 1-harmonic functions, and functions of least gradient, Commun. Pure Appl. Anal., 12 (2013), 711-719.doi: 10.3934/cpaa.2013.12.711.

• on this site

/