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October  2019, 6(4): 277-289. doi: 10.3934/jdg.2019019

Games for Pucci's maximal operators

Pablo Blanc 1, , Juan J. Manfredi 2, and  Julio D. Rossi 1,,

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

* Corresponding author: Julio D. Rossi

Received  November 2018 Revised  June 2019 Published  October 2019

Fund Project: Partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), by UBACyT grant 20020160100155BA (Argentina) and by MINECO MTM2015-70227-P (Spain).

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.

Keywords: Pucci's maximal operators, dynamic programming principle, games, viscosity solutions.
Mathematics Subject Classification: Primary: 35B50, 49N70; Secondary: 91A15, 91A24.
Citation: Pablo Blanc, Juan J. Manfredi, Julio D. Rossi. Games for Pucci's maximal operators. Journal of Dynamics & Games, 2019, 6 (4) : 277-289. doi: 10.3934/jdg.2019019
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

J. P. Sha, Handlebodies and p-convexity, J. Differential Geometry, 25 (1987), 353-361.  doi: 10.4310/jdg/1214440980.  Google Scholar

[12]

H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J., 36 (1987), 525-548.  doi: 10.1512/iumj.1987.36.36029.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

J. P. Sha, Handlebodies and p-convexity, J. Differential Geometry, 25 (1987), 353-361.  doi: 10.4310/jdg/1214440980.  Google Scholar

[12]

H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J., 36 (1987), 525-548.  doi: 10.1512/iumj.1987.36.36029.  Google Scholar

[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.  Google Scholar

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