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Asymptotically optimal strategies in repeated games with incomplete information and vanishing weights
Games for Pucci's maximal operators
| 1. | Departamento de Matemática, FCEyN UBA, Ciudad Universitaria, Pab 1 (1428), Buenos Aires, Argentina |
| 2. | Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA |
In this paper we introduce a game whose value functions converge (as a parameter that measures the size of the steps goes to zero) uniformly to solutions to the second order Pucci maximal operators.
References:
| [1] |
I. Birindelli, G. Galise Giulio and F. Leoni,
Lioville theorems for a family of very degenerate elliptic non lineal operators, Nonlinear Anal., 161 (2017), 198-211.
doi: 10.1016/j.na.2017.06.002. |
| [2] |
I. Birindelli, G. Galise and I. Ishii,
A family of degenerate elliptic operators: Maximum principle and its consequences, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 417-441.
doi: 10.1016/j.anihpc.2017.05.003. |
| [3] |
I. Capuzzo Dolcetta, G. Leoni and L. Vitolo,
On the inequality $F(x, D^2u) \geq f(u) +g(u)|Du|^q$, Math. Ann., 365 (2016), 423-448.
doi: 10.1007/s00208-015-1280-2. |
| [4] |
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. |
| [5] |
F. R. Harvey and H. B. Jr. Lawson,
Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math., 62 (2009), 396-443.
doi: 10.1002/cpa.20265. |
| [6] |
F. R. Harvey and H. B. Jr. Lawson,
$p-$convexity, $p-$plurisubharmonicity and the Levi problem, Indiana Univ. Math. J., 62 (2013), 149-169.
doi: 10.1512/iumj.2013.62.4886. |
| [7] |
P. Lindqvist and J. J. Manfredi,
On the mean value property for the $p-$Laplace equation in the plane, Proc. Amer. Math. Soc., 144 (2016), 143-149.
doi: 10.1090/proc/12675. |
| [8] |
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. |
| [9] |
J. J. Manfredi, M. Parviainen and J. D. Rossi,
Dynamic programming principle for tug-of-war games with noise, ESAIM, Control, Opt. Calc. Var., 18 (2012), 81-90.
doi: 10.1051/cocv/2010046. |
| [10] |
J. J. Manfredi, M. Parviainen and J. D. Rossi,
On the definition and properties of p-harmonious functions, Ann. Scuola Nor. Sup. Pisa, 11 (2012), 215-241.
|
| [11] |
J. P. Sha,
Handlebodies and p-convexity, J. Differential Geometry, 25 (1987), 353-361.
doi: 10.4310/jdg/1214440980. |
| [12] |
H. Wu,
Manifolds of partially positive curvature, Indiana Univ. Math. J., 36 (1987), 525-548.
doi: 10.1512/iumj.1987.36.36029. |
| [13] |
Q. Liu and A. Schikorra,
General existence of solutions to dynamic programming principle, Commun. Pure Appl. Anal., 14 (2015), 167-184.
doi: 10.3934/cpaa.2015.14.167. |
show all references
References:
| [1] |
I. Birindelli, G. Galise Giulio and F. Leoni,
Lioville theorems for a family of very degenerate elliptic non lineal operators, Nonlinear Anal., 161 (2017), 198-211.
doi: 10.1016/j.na.2017.06.002. |
| [2] |
I. Birindelli, G. Galise and I. Ishii,
A family of degenerate elliptic operators: Maximum principle and its consequences, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 417-441.
doi: 10.1016/j.anihpc.2017.05.003. |
| [3] |
I. Capuzzo Dolcetta, G. Leoni and L. Vitolo,
On the inequality $F(x, D^2u) \geq f(u) +g(u)|Du|^q$, Math. Ann., 365 (2016), 423-448.
doi: 10.1007/s00208-015-1280-2. |
| [4] |
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. |
| [5] |
F. R. Harvey and H. B. Jr. Lawson,
Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math., 62 (2009), 396-443.
doi: 10.1002/cpa.20265. |
| [6] |
F. R. Harvey and H. B. Jr. Lawson,
$p-$convexity, $p-$plurisubharmonicity and the Levi problem, Indiana Univ. Math. J., 62 (2013), 149-169.
doi: 10.1512/iumj.2013.62.4886. |
| [7] |
P. Lindqvist and J. J. Manfredi,
On the mean value property for the $p-$Laplace equation in the plane, Proc. Amer. Math. Soc., 144 (2016), 143-149.
doi: 10.1090/proc/12675. |
| [8] |
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. |
| [9] |
J. J. Manfredi, M. Parviainen and J. D. Rossi,
Dynamic programming principle for tug-of-war games with noise, ESAIM, Control, Opt. Calc. Var., 18 (2012), 81-90.
doi: 10.1051/cocv/2010046. |
| [10] |
J. J. Manfredi, M. Parviainen and J. D. Rossi,
On the definition and properties of p-harmonious functions, Ann. Scuola Nor. Sup. Pisa, 11 (2012), 215-241.
|
| [11] |
J. P. Sha,
Handlebodies and p-convexity, J. Differential Geometry, 25 (1987), 353-361.
doi: 10.4310/jdg/1214440980. |
| [12] |
H. Wu,
Manifolds of partially positive curvature, Indiana Univ. Math. J., 36 (1987), 525-548.
doi: 10.1512/iumj.1987.36.36029. |
| [13] |
Q. Liu and A. Schikorra,
General existence of solutions to dynamic programming principle, Commun. Pure Appl. Anal., 14 (2015), 167-184.
doi: 10.3934/cpaa.2015.14.167. |
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