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Backward iteration algorithms for Julia sets of Möbius semigroups
Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions
1. | School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China |
2. | Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China |
3. | School of Mathematical Sciences, Fudan University, Shanghai Key Laboratory for Contemporary Applied Mathematics, Shanghai 200433 |
References:
[1] |
V. I. Arnold, Geometric Methods in The Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983. |
[2] |
V. I. Arnold, Lectures on Partial Differential Equations, Springer, Berlin Heidelberg, 2004.
doi: 10.1007/978-3-662-05441-3. |
[3] |
S. Aubry, The twist map, the extended Frenkel-Kontorova model and the devil's staircase, Phys. D., 7 (1983), 240-258.
doi: 10.1016/0167-2789(83)90129-X. |
[4] |
S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions I: Exact results for the ground states, Phys. D., 8 (1983), 381-422.
doi: 10.1016/0167-2789(83)90233-6. |
[5] |
G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi Mathématiques & Applications, (Berlin) 17, Springer, Paris, 1994. |
[6] |
G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional Variational Problems, An Introduction, (Oxford Lecture Series in Mathematics and Its Applications), Clarendon Press Oxford, 1998. |
[7] |
P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Vol. 58, Springer, 2004. |
[8] |
M. G. Crandall, L. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502.
doi: 10.1090/S0002-9947-1984-0732102-X. |
[9] |
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. (N.S.), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[10] |
G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values, Geom. Funct. Anal., 8 (1998), 788-809.
doi: 10.1007/s000390050074. |
[11] |
M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
[12] |
A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique., Convergence of the solutions of the discounted Hamilton-Jacobi equation, Invent. Math., 105 (2016), 1-27.
doi: 10.1007/s00222-016-0648-6. |
[13] |
A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502.
doi: 10.1137/050621955. |
[14] |
A. Douglis, Solutions in the large for multi-dimensional, non-linear partial differential equations of first order, Ann. Inst. Fourier (Grenoble), 15 (1965), 1-35.
doi: 10.5802/aif.208. |
[15] |
W. E, Aubry-Mather theory and periodic solutions of the forced Burgers equation, Comm. Pure Appl. Math., 52 (1999), 811-828.
doi: 10.1002/(SICI)1097-0312(199907)52:7<811::AID-CPA2>3.0.CO;2-D. |
[16] |
A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043-1046.
doi: 10.1016/S0764-4442(97)87883-4. |
[17] |
A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270.
doi: 10.1016/S0764-4442(98)80144-4. |
[18] |
A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Preliminary Version Number 10, 2008. |
[19] |
A. Fathi and J. N. Mather, Failure of convergence of the Lax-Oleinik semi-group in the time- periodic case, Bull. Soc. Math. France., 128 (2000), 473-483. |
[20] |
N. Ichihara and H. Ishii, Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians, Arch. Ration. Mech. Anal., 194 (2009), 383-419.
doi: 10.1007/s00205-008-0170-0. |
[21] |
J. N. Mather, Existence of quasi periodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467.
doi: 10.1016/0040-9383(82)90023-4. |
[22] |
J. N. Mather, More Denjoy minimal sets for area preserving diffeomorphisms, Comment. Math. Helv., 60 (1985), 508-557.
doi: 10.1007/BF02567431. |
[23] |
J. N. Mather, A criterion for the non-existence of invariant circle, Publ. Math. IHES, 63 (1986), 301-309.
doi: 10.1007/BF02831625. |
[24] |
J. N. Mather, Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian Systems and Related Topics, ed. P.H.Rabinowitz et al. NATO ASI Series C 209, Reidel: Dordrecht, (1987), 177-202. |
[25] |
J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.
doi: 10.1007/BF02571383. |
[26] |
G. Namah and J.-M. Roquejoffre, Remarks on the long time behavior of the solutions of Hamilton-Jacobi equations, Commun. Partial Differ. Equ., 24 (1999), 883-893.
doi: 10.1080/03605309908821451. |
[27] |
K. Wang and J. Yan, A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems, Comm. Math. Phys., 309 (2012), 663-691.
doi: 10.1007/s00220-011-1375-x. |
show all references
References:
[1] |
V. I. Arnold, Geometric Methods in The Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983. |
[2] |
V. I. Arnold, Lectures on Partial Differential Equations, Springer, Berlin Heidelberg, 2004.
doi: 10.1007/978-3-662-05441-3. |
[3] |
S. Aubry, The twist map, the extended Frenkel-Kontorova model and the devil's staircase, Phys. D., 7 (1983), 240-258.
doi: 10.1016/0167-2789(83)90129-X. |
[4] |
S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions I: Exact results for the ground states, Phys. D., 8 (1983), 381-422.
doi: 10.1016/0167-2789(83)90233-6. |
[5] |
G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi Mathématiques & Applications, (Berlin) 17, Springer, Paris, 1994. |
[6] |
G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional Variational Problems, An Introduction, (Oxford Lecture Series in Mathematics and Its Applications), Clarendon Press Oxford, 1998. |
[7] |
P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Vol. 58, Springer, 2004. |
[8] |
M. G. Crandall, L. C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487-502.
doi: 10.1090/S0002-9947-1984-0732102-X. |
[9] |
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. (N.S.), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[10] |
G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values, Geom. Funct. Anal., 8 (1998), 788-809.
doi: 10.1007/s000390050074. |
[11] |
M. G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
[12] |
A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique., Convergence of the solutions of the discounted Hamilton-Jacobi equation, Invent. Math., 105 (2016), 1-27.
doi: 10.1007/s00222-016-0648-6. |
[13] |
A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502.
doi: 10.1137/050621955. |
[14] |
A. Douglis, Solutions in the large for multi-dimensional, non-linear partial differential equations of first order, Ann. Inst. Fourier (Grenoble), 15 (1965), 1-35.
doi: 10.5802/aif.208. |
[15] |
W. E, Aubry-Mather theory and periodic solutions of the forced Burgers equation, Comm. Pure Appl. Math., 52 (1999), 811-828.
doi: 10.1002/(SICI)1097-0312(199907)52:7<811::AID-CPA2>3.0.CO;2-D. |
[16] |
A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043-1046.
doi: 10.1016/S0764-4442(97)87883-4. |
[17] |
A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270.
doi: 10.1016/S0764-4442(98)80144-4. |
[18] |
A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Preliminary Version Number 10, 2008. |
[19] |
A. Fathi and J. N. Mather, Failure of convergence of the Lax-Oleinik semi-group in the time- periodic case, Bull. Soc. Math. France., 128 (2000), 473-483. |
[20] |
N. Ichihara and H. Ishii, Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians, Arch. Ration. Mech. Anal., 194 (2009), 383-419.
doi: 10.1007/s00205-008-0170-0. |
[21] |
J. N. Mather, Existence of quasi periodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467.
doi: 10.1016/0040-9383(82)90023-4. |
[22] |
J. N. Mather, More Denjoy minimal sets for area preserving diffeomorphisms, Comment. Math. Helv., 60 (1985), 508-557.
doi: 10.1007/BF02567431. |
[23] |
J. N. Mather, A criterion for the non-existence of invariant circle, Publ. Math. IHES, 63 (1986), 301-309.
doi: 10.1007/BF02831625. |
[24] |
J. N. Mather, Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian Systems and Related Topics, ed. P.H.Rabinowitz et al. NATO ASI Series C 209, Reidel: Dordrecht, (1987), 177-202. |
[25] |
J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207.
doi: 10.1007/BF02571383. |
[26] |
G. Namah and J.-M. Roquejoffre, Remarks on the long time behavior of the solutions of Hamilton-Jacobi equations, Commun. Partial Differ. Equ., 24 (1999), 883-893.
doi: 10.1080/03605309908821451. |
[27] |
K. Wang and J. Yan, A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems, Comm. Math. Phys., 309 (2012), 663-691.
doi: 10.1007/s00220-011-1375-x. |
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