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Stability of degenerate parabolic Cauchy problems
An obstacle problem for Tug-of-War games
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
J. D. Rossi, E. V. Teixeira and J. M. Urbano, Optimal regularity at the free boundary for the infinity obstacle problem, Preprint. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
J. D. Rossi, E. V. Teixeira and J. M. Urbano, Optimal regularity at the free boundary for the infinity obstacle problem, Preprint. |
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