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Convergence of dynamic programming principles for the p-Laplacian

Félix del Teso, Juan J. Manfredi ORCID logo and Mikko Parviainen

Abstract

We provide a unified strategy to show that solutions of dynamic programming principles associated to the p-Laplacian converge to the solution of the corresponding Dirichlet problem. Our approach includes all previously known cases for continuous and discrete dynamic programming principles, provides new results, and gives a convergence proof free of probability arguments.


Communicated by Pierre Cardaliaguet


Funding source: Norges Forskningsråd

Award Identifier / Grant number: 250070

Funding source: Academy of Finland

Award Identifier / Grant number: 298641

Funding statement: The first author is supported by the Toppforsk (research excellence) project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway, and by the grant PGC2018-094522-B-I00 from the MICINN of the Spanish Government. The third author is supported by the Academy of Finland project no. 298641.

Acknowledgements

We are very thankful to E. R. Jakobsen for discussions on generalized viscosity solutions and for the reference [10]. The first author wants to thank the University of Pittsburgh for hosting him during a research stay when part of this research was conducted. Part of this research was also done while the second author visited NTNU and the University of Jyväskylä. He thanks both institutions for their hospitality.

References

[1] A. 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), no. 3, 915–944. Search in Google Scholar

[2] G. Barles and J. Burdeau, The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems, Comm. Partial Differential Equations 20 (1995), no. 1–2, 129–178. Search in Google Scholar

[3] G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 4, 557–579. Search in Google Scholar

[4] G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method, SIAM J. Control Optim. 26 (1988), no. 5, 1133–1148. Search in Google Scholar

[5] G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal. 4 (1991), no. 3, 271–283. Search in Google Scholar

[6] L. Codenotti, M. Lewicka and J. Manfredi, Discrete approximations to the double-obstacle problem and optimal stopping of tug-of-war games, Trans. Amer. Math. Soc. 369 (2017), no. 10, 7387–7403. Search in Google Scholar

[7] H. Hartikainen, A dynamic programming principle with continuous solutions related to the p-Laplacian, 1<p<, Differential Integral Equations 29 (2016), no. 5–6, 583–600. Search in Google Scholar

[8] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover, Mineola, 2006. Search in Google Scholar

[9] H. Ishii, A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 16 (1989), no. 1, 105–135. Search in Google Scholar

[10] E. R. Jakobsen, Encyclopedia of Quantitative Finance. Chapter Monotone Schemes, John Wiley & Sons, New York, 2010. Search in Google Scholar

[11] P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal. 33 (2001), no. 3, 699–717. Search in Google Scholar

[12] B. Kawohl, J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl. (9) 97 (2012), no. 2, 173–188. Search in Google Scholar

[13] E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Anal. 29 (1998), no. 1, 279–292. Search in Google Scholar

[14] M. Lewicka and J. J. Manfredi, The obstacle problem for the p-laplacian via optimal stopping of tug-of-war games, Probab. Theory Related Fields 167 (2017), no. 1–2, 349–378. Search in Google Scholar

[15] M. Lewicka, J. J. Manfredi and D. Ricciotti, Random walks and random tug of war in the Heisenberg group, Math. Ann. (2019). Search in Google Scholar

[16] Q. Liu and A. Schikorra, General existence of solutions to dynamic programming equations, Commun. Pure Appl. Anal. 14 (2015), no. 1, 167–184. Search in Google Scholar

[17] H. Luiro and M. Parviainen, Regularity for nonlinear stochastic games, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 6, 1435–1456. Search in Google Scholar

[18] H. Luiro, M. Parviainen and E. Saksman, On the existence and uniqueness of p-harmonious functions, Differential Integral Equations 27 (2014), no. 3–4, 201–216. Search in Google Scholar

[19] J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for p-harmonic functions, Proc. Amer. Math. Soc. 138 (2010), no. 3, 881–889. Search in Google Scholar

[20] J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of p-harmonious functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 2, 215–241. Search in Google Scholar

[21] A. M. Oberman, A convergent difference scheme for the infinity Laplacian: Construction of absolutely minimizing Lipschitz extensions, Math. Comp. 74 (2005), no. 251, 1217–1230. Search in Google Scholar

[22] A. M. Oberman, Finite difference methods for the infinity Laplace and p-Laplace equations, J. Comput. Appl. Math. 254 (2013), 65–80. Search in Google Scholar

[23] Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 167–210. Search in Google Scholar

[24] Y. Peres and S. Sheffield, Tug-of-war with noise: A game-theoretic view of the p-Laplacian, Duke Math. J. 145 (2008), no. 1, 91–120. Search in Google Scholar

Received: 2019-06-09
Accepted: 2020-02-08
Published Online: 2020-03-19

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