American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution
  • CPAA Home
  • This Issue
  • Next Article
    Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation
May  2020, 19(5): 2617-2640. doi: 10.3934/cpaa.2020114

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

Jeongmin Han

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.

Keywords: Parabolic $ p $-Laplacian, Lipschitz continuity, tug-of-war.
Mathematics Subject Classification: Primary: 35K20; Secondary: 91A15.
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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  Google Scholar

[14]

H. Luiro, M. 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. 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.  Google Scholar

[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.  Google Scholar

[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.   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. 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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  Google Scholar

[14]

H. Luiro, M. 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. 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.  Google Scholar

[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.  Google Scholar

[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.   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. 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.  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

[1]

Ángel Arroyo, Joonas Heino, Mikko Parviainen. Tug-of-war games with varying probabilities and the normalized p(x)-laplacian. Communications on Pure & Applied Analysis, 2017, 16 (3) : 915-944. doi: 10.3934/cpaa.2017044
[2]

Ivana Gómez, Julio D. Rossi. Tug-of-war games and the infinity Laplacian with spatial dependence. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1959-1983. doi: 10.3934/cpaa.2013.12.1959
[3]

Juan J. Manfredi, Julio D. Rossi, Stephanie J. Somersille. An obstacle problem for Tug-of-War games. Communications on Pure & Applied Analysis, 2015, 14 (1) : 217-228. doi: 10.3934/cpaa.2015.14.217
[4]

Mostafa Ghelichi, A. M. Goltabar, H. R. Tavakoli, A. Karamodin. Neuro-fuzzy active control optimized by Tug of war optimization method for seismically excited benchmark highway bridge. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 333-351. doi: 10.3934/naco.2020029
[5]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005
[6]

Pelin G. Geredeli, Azer Khanmamedov. Long-time dynamics of the parabolic $p$-Laplacian equation. Communications on Pure & Applied Analysis, 2013, 12 (2) : 735-754. doi: 10.3934/cpaa.2013.12.735
[7]

Shun Uchida. Solvability of doubly nonlinear parabolic equation with p-laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021033
[8]

Vitali Liskevich, Igor I. Skrypnik, Zeev Sobol. Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1731-1744. doi: 10.3934/cpaa.2013.12.1731
[9]

Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130
[10]

Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058
[11]

Junyong Eom, Ryuichi Sato. Large time behavior of ODE type solutions to parabolic $ p $-Laplacian type equations. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4373-4386. doi: 10.3934/cpaa.2020199
[12]

Yutaka Tsuzuki. Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains. Evolution Equations & Control Theory, 2014, 3 (1) : 191-206. doi: 10.3934/eect.2014.3.191
[13]

Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22
[14]

Michela Eleuteri, Paolo Marcellini, Elvira Mascolo. Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 251-265. doi: 10.3934/dcdss.2019018
[15]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268
[16]

Pavel Jirásek. On Compactness Conditions for the $p$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 715-726. doi: 10.3934/cpaa.2016.15.715
[17]

Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371
[18]

Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040
[19]

Maria do Rosário de Pinho, Ilya Shvartsman. Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505
[20]

Aram L. Karakhanyan. Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 261-277. doi: 10.3934/dcds.2016.36.261

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (182)
  • HTML views (79)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy