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May  2020, 19(5): 2617-2640. doi: 10.3934/cpaa.2020114

Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

Received  December 2018 Revised  October 2019 Published  March 2020

Fund Project: This work was supported by NRF-2015R1A4A1041675

We prove in this article that functions satisfying a dynamic programming principle have a local interior Lipschitz type regularity. This DPP is partly motivated by the connection to the normalized parabolic $ p $-Laplace operator.

Citation: Jeongmin Han. Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2617-2640. doi: 10.3934/cpaa.2020114
References:
[1]

Á. ArroyoJ. Heino and M. Parviainen, Tug-of-war games with varying probabilities and the normalized $p(x)$-Laplacian, Commun. Pure Appl. Anal., 16 (2017), 915-944.  doi: 10.3934/cpaa.2017044.  Google Scholar

[2]

Á. Arroyo, H. Luiro, M. Parviainen and E. Ruosteenoja, Asymptotic Lipschitz regularity for tug-of-war games with varing probabilities, preprint, arXiv: 1806.10838. Google Scholar

[3]

C. BjorlandL. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Commun. Pure Appl. Math., 65 (2012), 337-380.  doi: 10.1002/cpa.21379.  Google Scholar

[4]

F. CharroJ. García Azorero and J. D. Rossi, A mixed problem for the infinity Laplacian via tug-of-war games, Calc. Var. Partial Differ. Equ., 34 (2009), 307-320.  doi: 10.1007/s00526-008-0185-2.  Google Scholar

[5]

L. C. Evans, The 1-Laplacian, the $\infty$-Laplacian and differential games, In Perspectives in Nonlinear Partial Differential Equations, Contemporary Mathematics, Vol. 446, American Mathematical Society, Providence, RI, (2007), 245–254. doi: 10.1090/conm/446/08634.  Google Scholar

[6]

F. FerrariQ. Liu and J. J. Manfredi, On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties, Discrete Contin. Dyn. Syst., 34 (2014), 2779-2793.  doi: 10.3934/dcds.2014.34.2779.  Google Scholar

[7]

T. Jin and L. Silvestre, Hölder gradient estimates for parabolic homogeneous $p$-Laplacian equations, J. Math. Pures Appl., 108 (2017), 63-87.  doi: 10.1016/j.matpur.2016.10.010.  Google Scholar

[8]

B. KawohlJ. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 94 (2012), 173-188.  doi: 10.1016/j.matpur.2011.07.001.  Google Scholar

[9]

R. V. Kohn and S. Serfaty, Second-order PDE's and deterministic games, In ICIAM 07–-6th International Congress on Industrial and Applied Mathematics, Eur. Math. Soc., Zürich, (2009), 239–249. doi: 10.4171/056-1/12.  Google Scholar

[10]

E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta_\infty(u) = 0$, NoDea-Nonlinear Differ. Equ. Appl., (2007), 29–55. doi: 10.1007/s00030-006-4030-z.  Google Scholar

[11]

E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Anal., 29 (1998), 279-292.  doi: 10.1137/S0036141095294067.  Google Scholar

[12]

H. Luiro and M. Parviainen, Regularity for nonlinear stochastic games, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 35 (2018), 1435-1456.  doi: 10.1016/j.anihpc.2017.11.009.  Google Scholar

[13]

H. LuiroM. Parviainen and E. Saksman, Harnack's inequality for $p$-harmonic functions via stochastic games, Commun. Partial Differ. Equ., 38 (2013), 1985-2003.  doi: 10.1080/03605302.2013.814068.  Google Scholar

[14]

H. LuiroM. Parviainen and E. Saksman, On the existence and uniqueness of $p$-harmonious functions, Differ. Integral Equ., 27 (2014), 201-216.   Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

J. J. ManfrediM. Parviainen and J. D. Rossi, On the definition and properties of $p$-harmonious functions, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 11 (2012), 215-241.   Google Scholar

[18]

M. Parviainen and E. Ruosteenoja, Local regularity for time-dependent tug-of-war games with varying probabilities, J. Differ. Equ., 261 (2016), 1357-1398.  doi: 10.1016/j.jde.2016.04.001.  Google Scholar

[19]

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.  Google Scholar

[20]

Y. PeresO. SchrammS. 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.  Google Scholar

[21]

E. Ruosteenoja, Local regularity results for value functions of tug-of-war with noise and running payoff, Adv. Calc. Var., 9 (2016), 1-17.  doi: 10.1515/acv-2014-0021.  Google Scholar

show all references

References:
[1]

Á. ArroyoJ. Heino and M. Parviainen, Tug-of-war games with varying probabilities and the normalized $p(x)$-Laplacian, Commun. Pure Appl. Anal., 16 (2017), 915-944.  doi: 10.3934/cpaa.2017044.  Google Scholar

[2]

Á. Arroyo, H. Luiro, M. Parviainen and E. Ruosteenoja, Asymptotic Lipschitz regularity for tug-of-war games with varing probabilities, preprint, arXiv: 1806.10838. Google Scholar

[3]

C. BjorlandL. Caffarelli and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Commun. Pure Appl. Math., 65 (2012), 337-380.  doi: 10.1002/cpa.21379.  Google Scholar

[4]

F. CharroJ. García Azorero and J. D. Rossi, A mixed problem for the infinity Laplacian via tug-of-war games, Calc. Var. Partial Differ. Equ., 34 (2009), 307-320.  doi: 10.1007/s00526-008-0185-2.  Google Scholar

[5]

L. C. Evans, The 1-Laplacian, the $\infty$-Laplacian and differential games, In Perspectives in Nonlinear Partial Differential Equations, Contemporary Mathematics, Vol. 446, American Mathematical Society, Providence, RI, (2007), 245–254. doi: 10.1090/conm/446/08634.  Google Scholar

[6]

F. FerrariQ. Liu and J. J. Manfredi, On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties, Discrete Contin. Dyn. Syst., 34 (2014), 2779-2793.  doi: 10.3934/dcds.2014.34.2779.  Google Scholar

[7]

T. Jin and L. Silvestre, Hölder gradient estimates for parabolic homogeneous $p$-Laplacian equations, J. Math. Pures Appl., 108 (2017), 63-87.  doi: 10.1016/j.matpur.2016.10.010.  Google Scholar

[8]

B. KawohlJ. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 94 (2012), 173-188.  doi: 10.1016/j.matpur.2011.07.001.  Google Scholar

[9]

R. V. Kohn and S. Serfaty, Second-order PDE's and deterministic games, In ICIAM 07–-6th International Congress on Industrial and Applied Mathematics, Eur. Math. Soc., Zürich, (2009), 239–249. doi: 10.4171/056-1/12.  Google Scholar

[10]

E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta_\infty(u) = 0$, NoDea-Nonlinear Differ. Equ. Appl., (2007), 29–55. doi: 10.1007/s00030-006-4030-z.  Google Scholar

[11]

E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Anal., 29 (1998), 279-292.  doi: 10.1137/S0036141095294067.  Google Scholar

[12]

H. Luiro and M. Parviainen, Regularity for nonlinear stochastic games, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 35 (2018), 1435-1456.  doi: 10.1016/j.anihpc.2017.11.009.  Google Scholar

[13]

H. LuiroM. Parviainen and E. Saksman, Harnack's inequality for $p$-harmonic functions via stochastic games, Commun. Partial Differ. Equ., 38 (2013), 1985-2003.  doi: 10.1080/03605302.2013.814068.  Google Scholar

[14]

H. LuiroM. Parviainen and E. Saksman, On the existence and uniqueness of $p$-harmonious functions, Differ. Integral Equ., 27 (2014), 201-216.   Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

J. J. ManfrediM. Parviainen and J. D. Rossi, On the definition and properties of $p$-harmonious functions, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 11 (2012), 215-241.   Google Scholar

[18]

M. Parviainen and E. Ruosteenoja, Local regularity for time-dependent tug-of-war games with varying probabilities, J. Differ. Equ., 261 (2016), 1357-1398.  doi: 10.1016/j.jde.2016.04.001.  Google Scholar

[19]

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.  Google Scholar

[20]

Y. PeresO. SchrammS. 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.  Google Scholar

[21]

E. Ruosteenoja, Local regularity results for value functions of tug-of-war with noise and running payoff, Adv. Calc. Var., 9 (2016), 1-17.  doi: 10.1515/acv-2014-0021.  Google Scholar

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