American Institute of Mathematical Sciences

Advanced
January  2015, 14(1): 217-228. doi: 10.3934/cpaa.2015.14.217

An obstacle problem for Tug-of-War games

Juan J. Manfredi 1, , Julio D. Rossi 2, and  Stephanie J. Somersille 3,

1. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, United States
2. 

Departamento de Análisis Matemático, Universidad de Alicante, Ap 99, 03080, Alicante, Spain
3. 

Department of Mathematics, Dartmouth College, Hanover, NH 03755, United States

Received  March 2014 Revised  April 2014 Published  September 2014

We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tug-of-war.
Keywords: Obstacle Problem, infinity laplacian., Tug-of-War games.
Mathematics Subject Classification: Primary: 35J60, 91A05, 49L25, 35J2.
Citation: 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
References:
[1]

T. Antunović, Y. Peres and S. Sheffield and S. Somersille, Tug-of-War and infinity Laplace equation with vanishing Neumann boundary conditions, Communications in Partial Differential Equations, 37 (2012), 1839-1869. doi: 10.1080/03605302.2011.642450.  Google Scholar

[2]

S. N. Armstrong, C. K. Smart and S. J. Somersille, An infinity Laplace equation with gradient term and mixed boundary conditions, Proc. Amer. Math. Soc., 139 (2011), 1763-1776. doi: 10.1090/S0002-9939-2010-10666-4.  Google Scholar

[3]

G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar

[4]

T. Bhattacharya, E. Di Benedetto and J. Manfredi, Limits as $p \to \infty$ of $\Delta_p u_p = f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, (1991), 15-68.  Google Scholar

[5]

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

[6]

V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process, 7 (1998), 376-386. doi: 10.1109/83.661188.  Google Scholar

[7]

M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[8]

A. P. Maitra and W. D. Sudderth, Discrete Gambling and Stochastic Games, Applications of Mathematics 32, Springer-Verlag, 1996. doi: 10.1007/978-1-4612-4002-0.  Google Scholar

[9]

J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization of $p$-harmonic functions, Proc. Amer. Math. Soc., 138 (2010), 881-889. doi: 10.1090/S0002-9939-09-10183-1.  Google Scholar

[10]

J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise, Control Optim. Calc. Var. COCV, 18 (2012), 81-90. doi: 10.1051/cocv/2010046.  Google Scholar

[11]

J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of $p$-harmonious functions, Annali Scuola Normale Sup. Pisa, Clase di Scienze, XI (2012), 215-241.  Google Scholar

[12]

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

[13]

Y. Peres, G. Pete and S. Somersille, Biased Tug-of-War, the biased infinity Laplacian and comparison with exponential cones, Calc. Var. Partial Differential Equations, 38 (2010), 541-564. doi: 10.1007/s00526-009-0298-2.  Google Scholar

[14]

Y. Peres, O. Schramm, S. Sheffield and D. 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

[15]

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

[16]

J. D. Rossi, E. V. Teixeira and J. M. Urbano, Optimal regularity at the free boundary for the infinity obstacle problem,, Preprint., ().   Google Scholar

show all references

References:
[1]

T. Antunović, Y. Peres and S. Sheffield and S. Somersille, Tug-of-War and infinity Laplace equation with vanishing Neumann boundary conditions, Communications in Partial Differential Equations, 37 (2012), 1839-1869. doi: 10.1080/03605302.2011.642450.  Google Scholar

[2]

S. N. Armstrong, C. K. Smart and S. J. Somersille, An infinity Laplace equation with gradient term and mixed boundary conditions, Proc. Amer. Math. Soc., 139 (2011), 1763-1776. doi: 10.1090/S0002-9939-2010-10666-4.  Google Scholar

[3]

G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3.  Google Scholar

[4]

T. Bhattacharya, E. Di Benedetto and J. Manfredi, Limits as $p \to \infty$ of $\Delta_p u_p = f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, (1991), 15-68.  Google Scholar

[5]

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

[6]

V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Process, 7 (1998), 376-386. doi: 10.1109/83.661188.  Google Scholar

[7]

M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[8]

A. P. Maitra and W. D. Sudderth, Discrete Gambling and Stochastic Games, Applications of Mathematics 32, Springer-Verlag, 1996. doi: 10.1007/978-1-4612-4002-0.  Google Scholar

[9]

J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization of $p$-harmonic functions, Proc. Amer. Math. Soc., 138 (2010), 881-889. doi: 10.1090/S0002-9939-09-10183-1.  Google Scholar

[10]

J. J. Manfredi, M. Parviainen and J. D. Rossi, Dynamic programming principle for tug-of-war games with noise, Control Optim. Calc. Var. COCV, 18 (2012), 81-90. doi: 10.1051/cocv/2010046.  Google Scholar

[11]

J. J. Manfredi, M. Parviainen and J. D. Rossi, On the definition and properties of $p$-harmonious functions, Annali Scuola Normale Sup. Pisa, Clase di Scienze, XI (2012), 215-241.  Google Scholar

[12]

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

[13]

Y. Peres, G. Pete and S. Somersille, Biased Tug-of-War, the biased infinity Laplacian and comparison with exponential cones, Calc. Var. Partial Differential Equations, 38 (2010), 541-564. doi: 10.1007/s00526-009-0298-2.  Google Scholar

[14]

Y. Peres, O. Schramm, S. Sheffield and D. 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

[15]

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

[16]

J. D. Rossi, E. V. Teixeira and J. M. Urbano, Optimal regularity at the free boundary for the infinity obstacle problem,, Preprint., ().   Google Scholar

[1]

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
[2]

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
[3]

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
[4]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[5]

Jan Burczak, P. Kaplický. Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2401-2445. doi: 10.3934/cpaa.2016042
[6]

Ali Fuat Alkaya, Dindar Oz. An optimal algorithm for the obstacle neutralization problem. Journal of Industrial & Management Optimization, 2017, 13 (2) : 835-856. doi: 10.3934/jimo.2016049
[7]

Fang Liu. An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2395-2421. doi: 10.3934/cpaa.2018114
[8]

Goro Akagi, Kazumasa Suzuki. On a certain degenerate parabolic equation associated with the infinity-laplacian. Conference Publications, 2007, 2007 (Special) : 18-27. doi: 10.3934/proc.2007.2007.18
[9]

Takeshi Fukao, Masahiro Kubo. Time-dependent obstacle problem in thermohydraulics. Conference Publications, 2009, 2009 (Special) : 240-249. doi: 10.3934/proc.2009.2009.240
[10]

Walter Allegretto, Yanping Lin, Shuqing Ma. Hölder continuous solutions of an obstacle thermistor problem. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 983-997. doi: 10.3934/dcdsb.2004.4.983
[11]

Song Wang. Numerical solution of an obstacle problem with interval coefficients. Numerical Algebra, Control & Optimization, 2020, 10 (1) : 23-38. doi: 10.3934/naco.2019030
[12]

Yinbin Deng, Yi Li, Wei Shuai. Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683
[13]

Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1837-1855. doi: 10.3934/dcdss.2021007
[14]

Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173
[15]

Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905
[16]

Giulio Ciraolo, Antonio Greco. An overdetermined problem associated to the Finsler Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1025-1038. doi: 10.3934/cpaa.2021004
[17]

Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems & Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211
[18]

Laurent Denis, Anis Matoussi, Jing Zhang. The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5185-5202. doi: 10.3934/dcds.2015.35.5185
[19]

Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291
[20]

J.I. Díaz, D. Gómez-Castro. Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem. Conference Publications, 2015, 2015 (special) : 379-386. doi: 10.3934/proc.2015.0379

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences