Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


Discrete approximations to the double-obstacle problem and optimal stopping of tug-of-war games

Authors: Luca Codenotti, Marta Lewicka and Juan Manfredi
Journal: Trans. Amer. Math. Soc. 369 (2017), 7387-7403
MSC (2010): Primary 35J92
Published electronically: May 11, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the double-obstacle problem for the $ p$-Laplace operator, $ p\in [2, \infty )$. We prove that for Lipschitz boundary data and Lipschitz obstacles, viscosity solutions are unique and coincide with variational solutions. They are also uniform limits of solutions to discrete min-max problems that can be interpreted as the dynamic programming principle for appropriate tug-of-war games with noise. In these games, both players in addition to choosing their strategies, are also allowed to choose stopping times. The solutions to the double-obstacle problems are limits of values of these games, when the step-size controlling the single shift in the token's position, converges to 0. We propose a numerical scheme based on this observation and show how it works for some examples of obstacles and boundary data.

References [Enhancements On Off] (What's this?)

  • [1] C. Bjorland, L. Caffarelli, and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math. 65 (2012), no. 3, 337-380. MR 2868849,
  • [2] G. Dal Maso, U. Mosco, and M. A. Vivaldi, A pointwise regularity theory for the two-obstacle problem, Acta Math. 163 (1989), no. 1-2, 57-107. MR 1007620,
  • [3] Zohra Farnana, The double obstacle problem on metric spaces, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 1, 261-277. MR 2489026
  • [4] Hans Hartikainen, A dynamic programming principle with continuous solutions related to the $ p$-Laplacian, $ 1 < p < \infty $, Differential Integral Equations 29 (2016), no. 5-6, 583-600. MR 3471974
  • [5] Petri Juutinen, Peter Lindqvist, and Juan 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. MR 1871417,
  • [6] Tero Kilpeläinen and William P. Ziemer, Pointwise regularity of solutions to nonlinear double obstacle problems, Ark. Mat. 29 (1991), no. 1, 83-106. MR 1115077,
  • [7] M. Lewicka and J. Manfredi, The obstacle problem for the $ p-$Laplacian via optimal stopping of tug-of-war games, to appear in Probability Theory and Related Fields.
  • [8] Peter Lindqvist, On the definition and properties of $ p$-superharmonic functions, J. Reine Angew. Math. 365 (1986), 67-79. MR 826152,
  • [9] Hannes Luiro, Mikko Parviainen, and Eero Saksman, On the existence and uniqueness of $ p$-harmonious functions, Differential Integral Equations 27 (2014), no. 3-4, 201-216. MR 3161602
  • [10] Juan J. Manfredi, Mikko Parviainen, and Julio 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. MR 3011990
  • [11] Juan J. Manfredi, Julio D. Rossi, and Stephanie J. Somersille, An obstacle problem for tug-of-war games, Commun. Pure Appl. Anal. 14 (2015), no. 1, 217-228. MR 3299035,
  • [12] Adam M. Oberman, Finite difference methods for the infinity Laplace and $ p$-Laplace equations, J. Comput. Appl. Math. 254 (2013), 65-80. MR 3061067,
  • [13] Yuval Peres, Oded Schramm, Scott Sheffield, and David B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 167-210. MR 2449057,
  • [14] Yuval Peres and Scott Sheffield, Tug-of-war with noise: a game-theoretic view of the $ p$-Laplacian, Duke Math. J. 145 (2008), no. 1, 91-120. MR 2451291,
  • [15] M. Reppen and P. Moosavi, A review of the double obstacle problem, a degree project, KTH Stockholm (2011).
  • [16] S. R. S. Varadhan, Probability theory, Courant Lecture Notes in Mathematics, vol. 7, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001. MR 1852999
  • [17] Fei Wang and Xiao-Liang Cheng, An algorithm for solving the double obstacle problems, Appl. Math. Comput. 201 (2008), no. 1-2, 221-228. MR 2432597,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35J92

Retrieve articles in all journals with MSC (2010): 35J92

Additional Information

Luca Codenotti
Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260

Marta Lewicka
Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260

Juan Manfredi
Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260

Received by editor(s): January 20, 2016
Received by editor(s) in revised form: April 17, 2016, and April 22, 2016
Published electronically: May 11, 2017
Additional Notes: The first and second authors were partially supported by NSF award DMS-1406730
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society