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Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea |
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.
References:
| [1] |
Á. Arroyo, J. 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. |
| [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. |
| [3] |
C. Bjorland, L. 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. |
| [4] |
F. Charro, J. 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. |
| [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. |
| [6] |
F. Ferrari, Q. 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. |
| [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. |
| [8] |
B. Kawohl, J. 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. |
| [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. |
| [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. |
| [11] |
E. Le Gruyer and J. C. Archer,
Harmonious extensions, SIAM J. Math. Anal., 29 (1998), 279-292.
doi: 10.1137/S0036141095294067. |
| [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. |
| [13] |
H. Luiro, M. 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. |
| [14] |
H. Luiro, M. Parviainen and E. Saksman,
On the existence and uniqueness of $p$-harmonious functions, Differ. Integral Equ., 27 (2014), 201-216.
|
| [15] |
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 (2010), 2058-2081.
doi: 10.1137/100782073. |
| [16] |
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. |
| [17] |
J. J. Manfredi, M. 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.
|
| [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. |
| [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. |
| [20] |
Y. Peres, O. Schramm, S. 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. |
| [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. |
show all references
References:
| [1] |
Á. Arroyo, J. 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. |
| [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. |
| [3] |
C. Bjorland, L. 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. |
| [4] |
F. Charro, J. 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. |
| [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. |
| [6] |
F. Ferrari, Q. 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. |
| [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. |
| [8] |
B. Kawohl, J. 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. |
| [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. |
| [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. |
| [11] |
E. Le Gruyer and J. C. Archer,
Harmonious extensions, SIAM J. Math. Anal., 29 (1998), 279-292.
doi: 10.1137/S0036141095294067. |
| [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. |
| [13] |
H. Luiro, M. 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. |
| [14] |
H. Luiro, M. Parviainen and E. Saksman,
On the existence and uniqueness of $p$-harmonious functions, Differ. Integral Equ., 27 (2014), 201-216.
|
| [15] |
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 (2010), 2058-2081.
doi: 10.1137/100782073. |
| [16] |
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. |
| [17] |
J. J. Manfredi, M. 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.
|
| [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. |
| [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. |
| [20] |
Y. Peres, O. Schramm, S. 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. |
| [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. |
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