Abstract
The purpose of this article is to obtain mean value characterizations of solutions to some nonlinear PDEs. To motivate the results we review some recent results concerning Tug-of-War games and their relation with PDEs. In particular, we will show that solutions to certain PDEs can be obtained as limits of values of Tug-of-War games when the parameter that controls the length of the possible movements goes to zero. Since the equations under study are nonlinear and not in divergence form we will make extensive use of the concept of viscosity solutions.
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References
G. Aronsson. Extensions of functions satisfying Lipschitz conditions. Ark. Mat. 6 (1967), 551–561.
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), no. 3–4, 381–384.
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.
G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order terms. Comm. Partial Diff. Eq., 26 (2001), 2323–2337.
G. Barles, and P.E. Souganidis,, Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4 (1991), 271–283.
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.
T. Bhattacharya, E. Di Benedetto and J. Manfredi, Limits as p → ∞ of Δp u p = ƒ and related extremal problems. Rend. Sem. Mat. Univ. Politec. Torino, (1991), 15–68.
L. Caffarelli and X. Cabre. Fully Nonlinear Elliptic Equations. Colloquium Publications 43, American Mathematical Society, Providence, RI, 1995.
F. Charro, J. Garcia Azorero and J. D. Rossi. A mixed problem for the infinity laplacian via Tug-of-War games. Calc. Var. Partial Differential Equations, 34(3) (2009), 307–320.
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.
W. H. Fleming, and P.E. Souganidis On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38 (1989), 293–314.
E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE Δ∞ (u) = 0. NoDEA Nonlinear Differential Equations Appl. 14 (2007), 29–55.
E. Le Gruyer and J. C. Archer, Harmonious extensions. SIAM J. Math. Anal. 29 (1998), 279–292.
R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Rational Mech. Anal. 123 (1993), 51–74.
P. Juutinen; Principal eigenvalue of a badly degenerate operator and applications. J. Differential Equations, 236, (2007), 532–550.
P. Juutinen, Absolutely minimizing Lipschitz extension on a metric space. Ann. Acad. Sci. Fenn.. Math. 27 (2002), 57–67.
P. Juutinen, Minimization problems for Lipschitz functions via viscosity solutions, Univ. Jyvaskyla, (1996), 1–39.
P. Juutinen and P. Lindqvist, On the higher eigenvalues for the ∞-eigenvalue problem, Calc. Var. Partial Differential Equations, 23(2) (2005), 169–192.
P. Juutinen, P. Lindqvist and J.J. Manfredi, The ∞-eigenvalue problem, Arch. Rational Mech. Anal., 148, (1999), 89–105.
P. Juutinen, P. Lindqvist and J.J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation. SIAM J. Math. Anal. 33 (2001), 699–717.
R.V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math. 59(3) (2006), 344–407.
A. P. Maitra, W. D. Sudderth, Discrete Gambling and Stochastic Games. Applications of Mathematics 32, Springer-Verlag (1996).
A. P. Maitra, and W. D. Sudderth, Borel stochastic games with lim sup payoff. Ann. Probab., 21(2):861–885, 1996.
J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization of p-harmonic functions. Proc. Amer. Math. Soc., 138, 881–889, (2010).
J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise. To appear in ESAIM. Control, Optim. Calc. Var. COCV.
J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of p-harmonious functions. To appear in Ann. Scuola Normale Sup. Pisa, Clase di Scienze.
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(5), 2058–2081, (2010).
A. Neymann and S. Sorin (eds.), Sthocastic games & applications, pp. 27–36, NATO Science Series (2003).
A. M. Oberman, A convergent difference scheme for the infinity-laplacian: construction of absolutely minimizing Lipschitz extensions, Math. Comp. 74 (2005), 1217–1230.
Y. Peres, G. Pete and S. Somersielle, Biased Tug-of-War, the biased infinity Laplacian and comparison with exponential cones. Calc. Var. Partial Differential Equations 38 (2010), no. 3–4, 541–564.
Y. Peres, O. Schramm, S. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), 167–210.
Y. Peres, S. Sheffield, Tug-of-war with noise: a game theoretic view of the p-Laplacian. Duke Math. J. 145(1) (2008), 91–120.
Varadhan, S. R. S., Probability theory, Courant Lecture Notes in Mathematics, 7
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Rossi, J.D. Asymptotic mean value properties for the P-Laplacian. SeMA 56, 35–62 (2011). https://doi.org/10.1007/BF03322596
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DOI: https://doi.org/10.1007/BF03322596