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Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter
1. | Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China |
2. | School of Mathematical Sciences, Fudan University, Shanghai Key Laboratory for Contemporary Applied Mathematics, Shanghai 200433, China |
References:
[1] |
P. Bernard, Smooth critical sub-solutions of the Hamilton-Jacobi equation, Math. Res. Lett., 14 (2007), 503-511.
doi: 10.4310/MRL.2007.v14.n3.a14. |
[2] |
C. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geom., 67 (2004), 457-517. |
[3] |
C. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case, J. Differential Geom., 82 (2009), 229-277. |
[4] |
G. Contreras, Action potential and weak KAM solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458.
doi: 10.1007/s005260100081. |
[5] |
M. Crandall and P. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
[6] |
A. Fathi and J. Mather, Failure of convergence of the Lax-Oleinik semi-group in the time-periodic case, Bull. Soc. Math. France, 128 (2000), 473-483. |
[7] |
A. Fathi, Weak KAM Theorems in Lagrangian Dynamics, Seventh preliminary version, Pisa, 2005. |
[8] |
M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer Verlag, Berlin, New York, 1977. |
[9] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511809187. |
[10] |
R. Mañé, Lagrangian flows: The dynamics of globally minimizing orbits, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 141-153.
doi: 10.1007/BF01233389. |
[11] |
J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.
doi: 10.1007/BF02571383. |
[12] |
J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.
doi: 10.5802/aif.1377. |
[13] |
J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer-Verlag, New York, 1982. |
[14] |
A. Sorrentino, Lecture Notes on Mather's Theory for Lagrangian Systems, preprint, 2010. |
[15] |
K. Wang and J. Yan, A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems, Commun. Math. Phys., 309 (2012), 663-691.
doi: 10.1007/s00220-011-1375-x. |
[16] |
A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346.
doi: 10.1016/0001-8708(71)90020-X. |
show all references
References:
[1] |
P. Bernard, Smooth critical sub-solutions of the Hamilton-Jacobi equation, Math. Res. Lett., 14 (2007), 503-511.
doi: 10.4310/MRL.2007.v14.n3.a14. |
[2] |
C. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, J. Differential Geom., 67 (2004), 457-517. |
[3] |
C. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case, J. Differential Geom., 82 (2009), 229-277. |
[4] |
G. Contreras, Action potential and weak KAM solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458.
doi: 10.1007/s005260100081. |
[5] |
M. Crandall and P. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
[6] |
A. Fathi and J. Mather, Failure of convergence of the Lax-Oleinik semi-group in the time-periodic case, Bull. Soc. Math. France, 128 (2000), 473-483. |
[7] |
A. Fathi, Weak KAM Theorems in Lagrangian Dynamics, Seventh preliminary version, Pisa, 2005. |
[8] |
M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer Verlag, Berlin, New York, 1977. |
[9] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511809187. |
[10] |
R. Mañé, Lagrangian flows: The dynamics of globally minimizing orbits, Bol. Soc. Brasil. Mat. (N.S.), 28 (1997), 141-153.
doi: 10.1007/BF01233389. |
[11] |
J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.
doi: 10.1007/BF02571383. |
[12] |
J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43 (1993), 1349-1386.
doi: 10.5802/aif.1377. |
[13] |
J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer-Verlag, New York, 1982. |
[14] |
A. Sorrentino, Lecture Notes on Mather's Theory for Lagrangian Systems, preprint, 2010. |
[15] |
K. Wang and J. Yan, A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems, Commun. Math. Phys., 309 (2012), 663-691.
doi: 10.1007/s00220-011-1375-x. |
[16] |
A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346.
doi: 10.1016/0001-8708(71)90020-X. |
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