American Institute of Mathematical Sciences

Advanced
September  2013, 12(5): 1959-1983. doi: 10.3934/cpaa.2013.12.1959

Tug-of-war games and the infinity Laplacian with spatial dependence

Ivana Gómez 1, and  Julio D. Rossi 2,

1. 

Instituto de Matemática Aplicada del Litoral (IMAL), CONICET-UNL, Departamento de Matemática, Facultad de Ingeniería Química, UNL, Güemes 3450, S3000GLN Santa Fe, Argentina
2. 

Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, 1428 – Buenos Aires

Received  January 2012 Revised  October 2012 Published  January 2013

In this paper we look for PDEs that arise as limits of values of tug-of-war games when the possible movements of the game are taken in a family of sets that are not necessarily Euclidean balls. In this way we find existence of viscosity solutions to the Dirichlet problem for an equation of the form $- \langle D^2 v\cdot J_x(D v) ; J_x(Dv)\rangle (x) =0$, that is, an infinity Laplacian with spatial dependence. Here $J_x (Dv(x))$ is a vector that depends on the spatial location and the gradient of the solution.
Keywords: viscosity solutions, Tug-of-war games, infinity Laplacian..
Mathematics Subject Classification: Primary: 35B50, 35J25, 35J70, 49N70, 91A15, 91A2.
Citation: 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
References:
[1]

Tonći Antunović, Yuval Peres, Scott Sheffield and Stephanie Somersille, Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition,, Comm. Partial Differential Equations, 37 (2012), 1839. doi: 10.1080/03605302.2011.642450. Google Scholar

[2]

Scott N. Armstrong and Charles K. Smart, A finite difference approach to the infinity Laplace equation and tug-of-war games,, Trans. Amer. Math. Soc., 364 (2012), 595. doi: 10.1090/S0002-9947-2011-05289-X. Google Scholar

[3]

Scott N. Armstrong and Charles K. Smart, An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions,, Calc. Var. Partial Differential Equations, 37 (2010), 381. doi: 10.1007/s00526-009-0267-9. Google Scholar

[4]

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

[5]

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

[6]

G. Barles and Jérôme Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term,, Comm. Partial Differential Equations, 26 (2001), 2323. doi: 10.1081/PDE-100107824. Google Scholar

[7]

E. N. Barron, L. C. Evans and R. Jensen, The infinity Laplacian, Aronsson's equation and their generalizations,, Trans. Amer. Math. Soc., 360 (2008), 77. doi: 10.1090/S0002-9947-07-04338-3. Google Scholar

[8]

Marino Belloni and Bernd Kawohl, The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as $p\to\infty$,, ESAIM Control Optim. Calc. Var., 10 (2004), 28. doi: 10.1051/cocv:2003035. Google Scholar

[9]

M. Belloni, B. Kawohl and P. Juutinen, The $p$-Laplace eigenvalue problem as $p\to\infty$ in a Finsler metric,, J. Eur. Math. Soc. (JEMS), 8 (2006), 123. doi: 10.4171/JEMS/40. Google Scholar

[10]

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

[11]

Thierry Champion and Luigi De Pascale, Principles of comparison with distance functions for absolute minimizers,, J. Convex. Anal., 14 (2007), 515. Google Scholar

[12]

Fernando Charro, Jesus García Azorero and Julio D. Rossi, A mixed problem for the infinity Laplacian via tug-of-war games,, Calc. Var. Partial Differential Equations, 34 (2009), 307. doi: 10.1007/s00526-008-0185-2. Google Scholar

[13]

Fernando Charro and Ireneo Peral, Limit branch of solutions as $p\to\infty$ for a family of sub-diffusive problems related to the $p$-Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1965. doi: 10.1080/03605300701454792. Google Scholar

[14]

Michael G. Crandall, Gunnar Gunnarsson and Peiyong Wang, Uniqueness of $\infty$-harmonic functions and the eikonal equation,, Comm. Partial Differential Equations, 32 (2007), 1587. doi: 10.1080/03605300601088807. Google Scholar

[15]

Michael G. Crandall, Hitoshi Ishii and Pierre-Louis Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar

[16]

Lawrence C. Evans and Ovidiu Savin, $C^{1,\alpha}$ regularity for infinity harmonic functions in two dimensions,, Calc. Var. Partial Differential Equations, 32 (2008), 325. doi: 10.1007/s00526-007-0143-4. Google Scholar

[17]

Lawrence C. Evans and Charles K. Smart, Everywhere differentiability of infinity harmonic functions,, Calc. Var. Partial Differential Equations, 42 (2011), 289. doi: 10.1007/s00526-010-0388-1. Google Scholar

[18]

E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta_\infty(u)=0$, , NoDEA Nonlinear Differential Equations Appl., 14 (2007), 29. doi: 10.1007/s00030-006-4030-z. Google Scholar

[19]

E. Le Gruyer and J. C. Archer, Harmonious extensions,, SIAM J. Math. Anal., 29 (1998), 279. doi: 10.1137/S0036141095294067. Google Scholar

[20]

Toshihiro Ishibashi and Shigeaki Koike, On fully nonlinear PDEs derived from variational problems of $L^p$ norms,, SIAM J. Math. Anal., 33 (2001), 545. doi: 10.1137/S0036141000380000. Google Scholar

[21]

Robert Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient,, Arch. Rational Mech. Anal., 123 (1993), 51. doi: 10.1007/BF00386368. Google Scholar

[22]

Petri Juutinen and Peter Lindqvist, On the higher eigenvalues for the $\infty$-eigenvalue problem,, Calc. Var. Partial Differential Equations, 23 (2005), 169. doi: 10.1007/s00526-004-0295-4. Google Scholar

[23]

Petri Juutinen, Peter Lindqvist and Juan J. Manfredi, The $\infty$-eigenvalue problem, , Arch. Rational Mech. Anal., 148 (1999), 89. doi: 10.1007/s002050050157. Google Scholar

[24]

Robert V. Kohn and Sylvia Serfaty, A deterministic-control-based approach to motion by curvature,, Comm. Pure Appl. Math., 59 (2006), 344. doi: 10.1002/cpa.20101. Google Scholar

[25]

Ashok P. Maitra and William D. Sudderth, "Discrete Gambling and Stochastic Games,", Applications of Mathematics (New York) 32, (1996). Google Scholar

[26]

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

[27]

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., 11 (2012), 215. doi: 10.2422/2036-2145.201005_003. Google Scholar

[28]

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

[29]

Juan J. Manfredi, Mikko Parviainen and Julio 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. doi: 10.1137/100782073. Google Scholar

[30]

Adam M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions,, Math. Comp., 74 (2005), 1217. doi: 10.1090/S0025-5718-04-01688-6. Google Scholar

[31]

Yuval Peres, Gábor Pete and Stephanie Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones,, Calc. Var. Partial Differential Equations, 38 (2010), 541. doi: 10.1007/s00526-009-0298-2. Google Scholar

[32]

Yuval Peres, Oded Schramm, Scott Sheffield and David B. Wilson, Tug-of-war and the infinity Laplacian,, J. Amer. Math. Soc., 22 (2009), 167. doi: 10.1090/S0894-0347-08-00606-1. Google Scholar

[33]

Yuval Peres and Scott Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian,, Duke Math. J., 145 (2008), 91. doi: 10.1215/00127094-2008-048. Google Scholar

[34]

Julio D. Rossi and Mariel Saez, Optimal regularity for the pseudo infinity Laplacian,, ESAIM Control Optim. Calc. Var., 13 (2007), 294. doi: 10.1051/cocv:2007018. Google Scholar

[35]

Ovidiu Savin, $C^1$ regularity for infinity harmonic functions in two dimensions,, Arch. Ration. Mech. Anal., 176 (2005), 351. doi: 10.1007/s00205-005-0355-8. Google Scholar

[36]

, "Sthocastic Games & Applications," Proceedings of the Nato Advanced Study Institute held in Stony Brook, NY, July 7-17, 1999, Abraham Neyman and Sylvain Sorin (eds.),, NATO Science Series C: Mathematical and Physical Sciences, (2003). Google Scholar

show all references

References:
[1]

Tonći Antunović, Yuval Peres, Scott Sheffield and Stephanie Somersille, Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition,, Comm. Partial Differential Equations, 37 (2012), 1839. doi: 10.1080/03605302.2011.642450. Google Scholar

[2]

Scott N. Armstrong and Charles K. Smart, A finite difference approach to the infinity Laplace equation and tug-of-war games,, Trans. Amer. Math. Soc., 364 (2012), 595. doi: 10.1090/S0002-9947-2011-05289-X. Google Scholar

[3]

Scott N. Armstrong and Charles K. Smart, An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions,, Calc. Var. Partial Differential Equations, 37 (2010), 381. doi: 10.1007/s00526-009-0267-9. Google Scholar

[4]

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

[5]

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

[6]

G. Barles and Jérôme Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term,, Comm. Partial Differential Equations, 26 (2001), 2323. doi: 10.1081/PDE-100107824. Google Scholar

[7]

E. N. Barron, L. C. Evans and R. Jensen, The infinity Laplacian, Aronsson's equation and their generalizations,, Trans. Amer. Math. Soc., 360 (2008), 77. doi: 10.1090/S0002-9947-07-04338-3. Google Scholar

[8]

Marino Belloni and Bernd Kawohl, The pseudo-$p$-Laplace eigenvalue problem and viscosity solutions as $p\to\infty$,, ESAIM Control Optim. Calc. Var., 10 (2004), 28. doi: 10.1051/cocv:2003035. Google Scholar

[9]

M. Belloni, B. Kawohl and P. Juutinen, The $p$-Laplace eigenvalue problem as $p\to\infty$ in a Finsler metric,, J. Eur. Math. Soc. (JEMS), 8 (2006), 123. doi: 10.4171/JEMS/40. Google Scholar

[10]

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

[11]

Thierry Champion and Luigi De Pascale, Principles of comparison with distance functions for absolute minimizers,, J. Convex. Anal., 14 (2007), 515. Google Scholar

[12]

Fernando Charro, Jesus García Azorero and Julio D. Rossi, A mixed problem for the infinity Laplacian via tug-of-war games,, Calc. Var. Partial Differential Equations, 34 (2009), 307. doi: 10.1007/s00526-008-0185-2. Google Scholar

[13]

Fernando Charro and Ireneo Peral, Limit branch of solutions as $p\to\infty$ for a family of sub-diffusive problems related to the $p$-Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1965. doi: 10.1080/03605300701454792. Google Scholar

[14]

Michael G. Crandall, Gunnar Gunnarsson and Peiyong Wang, Uniqueness of $\infty$-harmonic functions and the eikonal equation,, Comm. Partial Differential Equations, 32 (2007), 1587. doi: 10.1080/03605300601088807. Google Scholar

[15]

Michael G. Crandall, Hitoshi Ishii and Pierre-Louis Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar

[16]

Lawrence C. Evans and Ovidiu Savin, $C^{1,\alpha}$ regularity for infinity harmonic functions in two dimensions,, Calc. Var. Partial Differential Equations, 32 (2008), 325. doi: 10.1007/s00526-007-0143-4. Google Scholar

[17]

Lawrence C. Evans and Charles K. Smart, Everywhere differentiability of infinity harmonic functions,, Calc. Var. Partial Differential Equations, 42 (2011), 289. doi: 10.1007/s00526-010-0388-1. Google Scholar

[18]

E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta_\infty(u)=0$, , NoDEA Nonlinear Differential Equations Appl., 14 (2007), 29. doi: 10.1007/s00030-006-4030-z. Google Scholar

[19]

E. Le Gruyer and J. C. Archer, Harmonious extensions,, SIAM J. Math. Anal., 29 (1998), 279. doi: 10.1137/S0036141095294067. Google Scholar

[20]

Toshihiro Ishibashi and Shigeaki Koike, On fully nonlinear PDEs derived from variational problems of $L^p$ norms,, SIAM J. Math. Anal., 33 (2001), 545. doi: 10.1137/S0036141000380000. Google Scholar

[21]

Robert Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient,, Arch. Rational Mech. Anal., 123 (1993), 51. doi: 10.1007/BF00386368. Google Scholar

[22]

Petri Juutinen and Peter Lindqvist, On the higher eigenvalues for the $\infty$-eigenvalue problem,, Calc. Var. Partial Differential Equations, 23 (2005), 169. doi: 10.1007/s00526-004-0295-4. Google Scholar

[23]

Petri Juutinen, Peter Lindqvist and Juan J. Manfredi, The $\infty$-eigenvalue problem, , Arch. Rational Mech. Anal., 148 (1999), 89. doi: 10.1007/s002050050157. Google Scholar

[24]

Robert V. Kohn and Sylvia Serfaty, A deterministic-control-based approach to motion by curvature,, Comm. Pure Appl. Math., 59 (2006), 344. doi: 10.1002/cpa.20101. Google Scholar

[25]

Ashok P. Maitra and William D. Sudderth, "Discrete Gambling and Stochastic Games,", Applications of Mathematics (New York) 32, (1996). Google Scholar

[26]

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

[27]

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., 11 (2012), 215. doi: 10.2422/2036-2145.201005_003. Google Scholar

[28]

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

[29]

Juan J. Manfredi, Mikko Parviainen and Julio 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. doi: 10.1137/100782073. Google Scholar

[30]

Adam M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions,, Math. Comp., 74 (2005), 1217. doi: 10.1090/S0025-5718-04-01688-6. Google Scholar

[31]

Yuval Peres, Gábor Pete and Stephanie Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones,, Calc. Var. Partial Differential Equations, 38 (2010), 541. doi: 10.1007/s00526-009-0298-2. Google Scholar

[32]

Yuval Peres, Oded Schramm, Scott Sheffield and David B. Wilson, Tug-of-war and the infinity Laplacian,, J. Amer. Math. Soc., 22 (2009), 167. doi: 10.1090/S0894-0347-08-00606-1. Google Scholar

[33]

Yuval Peres and Scott Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian,, Duke Math. J., 145 (2008), 91. doi: 10.1215/00127094-2008-048. Google Scholar

[34]

Julio D. Rossi and Mariel Saez, Optimal regularity for the pseudo infinity Laplacian,, ESAIM Control Optim. Calc. Var., 13 (2007), 294. doi: 10.1051/cocv:2007018. Google Scholar

[35]

Ovidiu Savin, $C^1$ regularity for infinity harmonic functions in two dimensions,, Arch. Ration. Mech. Anal., 176 (2005), 351. doi: 10.1007/s00205-005-0355-8. Google Scholar

[36]

, "Sthocastic Games & Applications," Proceedings of the Nato Advanced Study Institute held in Stony Brook, NY, July 7-17, 1999, Abraham Neyman and Sylvain Sorin (eds.),, NATO Science Series C: Mathematical and Physical Sciences, (2003). 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]

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
[3]

Bernd Kawohl, Friedemann Schuricht. First eigenfunctions of the 1-Laplacian are viscosity solutions. Communications on Pure & Applied Analysis, 2015, 14 (1) : 329-339. doi: 10.3934/cpaa.2015.14.329
[4]

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 - A, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683
[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]

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
[7]

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
[8]

Oliver Juarez-Romero, William Olvera-Lopez, Francisco Sanchez-Sanchez. A simple family of solutions for forest games. Journal of Dynamics & Games, 2017, 4 (2) : 87-96. doi: 10.3934/jdg.2017006
[9]

Iryna Pankratova, Andrey Piatnitski. On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 935-970. doi: 10.3934/dcdsb.2009.11.935
[10]

Laura Olian Fannio. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 251-264. doi: 10.3934/dcds.1997.3.251
[11]

Shu-Yu Hsu. Non-existence and behaviour at infinity of solutions of some elliptic equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769
[12]

Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489
[13]

Shige Peng, Mingyu Xu. Constrained BSDEs, viscosity solutions of variational inequalities and their applications. Mathematical Control & Related Fields, 2013, 3 (2) : 233-244. doi: 10.3934/mcrf.2013.3.233
[14]

Graziano Crasta, Benedetto Piccoli. Viscosity solutions and uniqueness for systems of inhomogeneous balance laws. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 477-502. doi: 10.3934/dcds.1997.3.477
[15]

Inwon C. Kim, Helen K. Lei. Degenerate diffusion with a drift potential: A viscosity solutions approach. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 767-786. doi: 10.3934/dcds.2010.27.767
[16]

Boris P. Andreianov, Giuseppe Maria Coclite, Carlotta Donadello. Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5913-5942. doi: 10.3934/dcds.2017257
[17]

Zhen Lei, Yi Zhou. BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 575-583. doi: 10.3934/dcds.2009.25.575
[18]

Pierpaolo Soravia. Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of the Aronsson equation. Mathematical Control & Related Fields, 2012, 2 (4) : 399-427. doi: 10.3934/mcrf.2012.2.399
[19]

Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389
[20]

Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. Communications on Pure & Applied Analysis, 2006, 5 (4) : 793-812. doi: 10.3934/cpaa.2006.5.793

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences