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doi: 10.3934/dcds.2022034
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A game theoretical approximation for a parabolic/elliptic system with different operators

Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellon I, Ciudad Universitaria (1428), Buenos Aires, Argentina

*Corresponding author: Julio D. Rossi

To Juan Luis Vazquez in his 75th anniversary with our best wishes.

Received  June 2021 Revised  February 2022 Early access March 2022

Fund Project: partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), PICT-2018-03183 (Argentina) and UBACyT grant 20020160100155BA (Argentina).

In this paper we find viscosity solutions to a coupled system composed by two equations, the first one is parabolic and driven by the infinity Laplacian while the second one is elliptic and involves the usual Laplacian. We prove that there is a two-player zero-sum game played in two different boards with different rules in each board (in the first one we play a Tug-of-War game taking the number of plays into consideration and in the second board we move at random) whose value functions converge uniformly to a viscosity solution to the PDE system.

Citation: Alfredo Miranda, Julio D. Rossi. A game theoretical approximation for a parabolic/elliptic system with different operators. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022034
References:
[1]

T. AntunovicY. PeresS. 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]

A. Arroyo and J. G. Llorente, On the asymptotic mean value property for planar p-harmonic functions, Proc. Amer. Math. Soc., 144 (2016), 3859-3868.  doi: 10.1090/proc/13026.

[4]

G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. Partial Differential Equations, 26 (2001), 2323-2337.  doi: 10.1081/PDE-100107824.

[5]

P. BlancF. CharroJ. D. Rossi and J. J. Manfredi, A nonlinear Mean Value Property for the Monge-Ampère operator, J. Convex Analysis JOCA, 28 (2021), 353-386. 

[6]

P. BlancC. Esteve and J. D. Rossi, The evolution problem associated with eigenvalues of the Hessian, J, London Math. Soc., 102 (2020), 1293-1317.  doi: 10.1112/jlms.12363.

[7]

P. Blanc and J. D. Rossi, Games for eigenvalues of the Hessian and concave/convex envelopes, J. Math. Pures et Appliquees, 127 (2019), 192-215.  doi: 10.1016/j.matpur.2018.08.007.

[8]

P. Blanc and J. D. Rossi, Game Theory and Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications. Vol. 31. 2019. doi: 10.1515/9783110621792.

[9]

F. CharroJ. Garcia 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.

[10]

M. G. Crandall, A visit with the $\infty$-Laplace equation, Calculus of Variations and Nonlinear Partial Differential Equations, Lecture Notes in Math., Springer, Berlin, 1927 (2008), 75-122.  doi: 10.1007/978-3-540-75914-0_3.

[11]

M. G. CrandallH. 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.

[12]

F. del Teso and E. Lindgreen, A mean value formula for the variational $p$-Laplacian, NoDEA Nonlinear Differential Equations Appl., 28 (2021), Paper No. 27, 33 pp, arXiv: 2003.07084v2. doi: 10.1007/s00030-021-00688-6.

[13]

J. L. Doob, What is a martingale?, Amer. Math. Monthly, 78 (1971), 451-463.  doi: 10.1080/00029890.1971.11992788.

[14]

P. A. Ferrari, Teoricas de Probabilidad y Estadistica Matematica, Universidad de Buenos Aires, 2017.

[15]

M. Ishiwata, R. Magnanini and H. Wadade, A natural approach to the asymptotic mean value property for the $p$-Laplacian, Calc. Var. Partial Differential Equations, 56 (2017), Art. 97, 22 pp. doi: 10.1007/s00526-017-1188-7.

[16]

R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74.  doi: 10.1007/BF00386368.

[17]

M. Kac, Random walk and the theory of Brownian motion, Amer. Math. Monthly, 54 (1947), 369-391.  doi: 10.1080/00029890.1947.11990189.

[18]

B. KawohlJ. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 97 (2012), 173-188.  doi: 10.1016/j.matpur.2011.07.001.

[19]

M. Lewicka, A Course on Tug-of-War Games with Random Noise. Introduction and Basic Constructions, Universitext. Springer, Cham, 2020. doi: 10.1007/978-3-030-46209-3.

[20]

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.

[21]

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.

[22]

H. LuiroM. Parviainen and E. Saksman, Harnack's inequality for p-harmonic functions via stochastic games, Comm. Partial Differential Equations, 38 (2013), 1985-2003.  doi: 10.1080/03605302.2013.814068.

[23]

J. J. ManfrediM. 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.

[24]

J. J. ManfrediM. 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.

[25]

J. J. ManfrediM. Parviainen and J. D. Rossi, On the definition and properties of p-harmonious functions, Ann. Scuola Nor. Sup. Pisa Cl. Sci., 11 (2012), 215-241. 

[26]

J. J. ManfrediM. 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.

[27]

A. Miranda and J. D. Rossi, A game theoretical approach for a nonlinear system driven by elliptic operators, SN Partial Diff. Eq. Appl., 1 (2020), Art. 14, 41 pp. doi: 10.1007/s42985-020-00014-2.

[28]

H. Mitake and H. V. Tran, Weakly coupled systems of the infinity Laplace equations, Trans. Amer. Math. Soc., 369 (2017), 1773-1795.  doi: 10.1090/tran6694.

[29]

Y. PeresO. SchrammS. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210.  doi: 10.1090/S0894-0347-08-00606-1.

[30]

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.

[31]

J. D. Rossi, Tug-of-war games and PDEs, Proc. Royal Soc. Edim. A, 141 (2011), 319-369.  doi: 10.1017/S0308210510000041.

[32] D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511813658.

show all references

References:
[1]

T. AntunovicY. PeresS. 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]

A. Arroyo and J. G. Llorente, On the asymptotic mean value property for planar p-harmonic functions, Proc. Amer. Math. Soc., 144 (2016), 3859-3868.  doi: 10.1090/proc/13026.

[4]

G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. Partial Differential Equations, 26 (2001), 2323-2337.  doi: 10.1081/PDE-100107824.

[5]

P. BlancF. CharroJ. D. Rossi and J. J. Manfredi, A nonlinear Mean Value Property for the Monge-Ampère operator, J. Convex Analysis JOCA, 28 (2021), 353-386. 

[6]

P. BlancC. Esteve and J. D. Rossi, The evolution problem associated with eigenvalues of the Hessian, J, London Math. Soc., 102 (2020), 1293-1317.  doi: 10.1112/jlms.12363.

[7]

P. Blanc and J. D. Rossi, Games for eigenvalues of the Hessian and concave/convex envelopes, J. Math. Pures et Appliquees, 127 (2019), 192-215.  doi: 10.1016/j.matpur.2018.08.007.

[8]

P. Blanc and J. D. Rossi, Game Theory and Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications. Vol. 31. 2019. doi: 10.1515/9783110621792.

[9]

F. CharroJ. Garcia 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.

[10]

M. G. Crandall, A visit with the $\infty$-Laplace equation, Calculus of Variations and Nonlinear Partial Differential Equations, Lecture Notes in Math., Springer, Berlin, 1927 (2008), 75-122.  doi: 10.1007/978-3-540-75914-0_3.

[11]

M. G. CrandallH. 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.

[12]

F. del Teso and E. Lindgreen, A mean value formula for the variational $p$-Laplacian, NoDEA Nonlinear Differential Equations Appl., 28 (2021), Paper No. 27, 33 pp, arXiv: 2003.07084v2. doi: 10.1007/s00030-021-00688-6.

[13]

J. L. Doob, What is a martingale?, Amer. Math. Monthly, 78 (1971), 451-463.  doi: 10.1080/00029890.1971.11992788.

[14]

P. A. Ferrari, Teoricas de Probabilidad y Estadistica Matematica, Universidad de Buenos Aires, 2017.

[15]

M. Ishiwata, R. Magnanini and H. Wadade, A natural approach to the asymptotic mean value property for the $p$-Laplacian, Calc. Var. Partial Differential Equations, 56 (2017), Art. 97, 22 pp. doi: 10.1007/s00526-017-1188-7.

[16]

R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74.  doi: 10.1007/BF00386368.

[17]

M. Kac, Random walk and the theory of Brownian motion, Amer. Math. Monthly, 54 (1947), 369-391.  doi: 10.1080/00029890.1947.11990189.

[18]

B. KawohlJ. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 97 (2012), 173-188.  doi: 10.1016/j.matpur.2011.07.001.

[19]

M. Lewicka, A Course on Tug-of-War Games with Random Noise. Introduction and Basic Constructions, Universitext. Springer, Cham, 2020. doi: 10.1007/978-3-030-46209-3.

[20]

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.

[21]

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.

[22]

H. LuiroM. Parviainen and E. Saksman, Harnack's inequality for p-harmonic functions via stochastic games, Comm. Partial Differential Equations, 38 (2013), 1985-2003.  doi: 10.1080/03605302.2013.814068.

[23]

J. J. ManfrediM. 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.

[24]

J. J. ManfrediM. 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.

[25]

J. J. ManfrediM. Parviainen and J. D. Rossi, On the definition and properties of p-harmonious functions, Ann. Scuola Nor. Sup. Pisa Cl. Sci., 11 (2012), 215-241. 

[26]

J. J. ManfrediM. 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.

[27]

A. Miranda and J. D. Rossi, A game theoretical approach for a nonlinear system driven by elliptic operators, SN Partial Diff. Eq. Appl., 1 (2020), Art. 14, 41 pp. doi: 10.1007/s42985-020-00014-2.

[28]

H. Mitake and H. V. Tran, Weakly coupled systems of the infinity Laplace equations, Trans. Amer. Math. Soc., 369 (2017), 1773-1795.  doi: 10.1090/tran6694.

[29]

Y. PeresO. SchrammS. Sheffield and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22 (2009), 167-210.  doi: 10.1090/S0894-0347-08-00606-1.

[30]

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.

[31]

J. D. Rossi, Tug-of-war games and PDEs, Proc. Royal Soc. Edim. A, 141 (2011), 319-369.  doi: 10.1017/S0308210510000041.

[32] D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511813658.
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