-
Previous Article
Rates of decay for the wave systems with time dependent damping
- DCDS Home
- This Issue
-
Next Article
Periodic solutions of resonant systems with rapidly rotating nonlinearities
On $C^0$-variational solutions for Hamilton-Jacobi equations
1. | Dipartimento di Matematica Pura ed Applicata, Via Trieste, 63 - 35121 Padova, Italy, Italy |
References:
[1] |
B. Aebischer et al, "Symplectic Geometry," An introduction based on the seminar in Bern, 1992, Progress in Mathematics, 124, Birkhäuser Verlag, Basel, 1994. |
[2] |
M. Bardi and S. Osher, The nonconvex multidimensional Riemann problem for Hamilton-Jacobi equations, SIAM J. Math. Anal., 22 (1991), 344-351.
doi: 10.1137/0522022. |
[3] |
M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations," Systems and Control: Foundations and Applications, Boston, MA: Birkhauser, 1997, xvii+570 pp. |
[4] |
M. Bardi and S. Faggian, Hopf-type estimates and formulas for nonconvex nonconcave Hamilton-Jacobi equations, SIAM J. Math. Anal., 29 (1998), 1067-1086.
doi: 10.1137/S0036141096309629. |
[5] |
G. Barles, "Solutions de viscosité des équations de Hamilton-Jacobi," Springer, Paris, 1994. |
[6] |
S. Benenti, "Symplectic Relations in Analytical Mechanics," Modern developments in analytical mechanics, Vol. I: Geometrical dynamics, Proc. IUTAM-ISIMM Symp., Torino, Italy, (1982), 39-91. |
[7] |
O. Bernardi and F. Cardin, Minimax and viscosity solutions in the convex case, Commun. Pure Appl. Anal., 5 (2006), 793-812.
doi: 10.3934/cpaa.2006.5.793. |
[8] |
G. Capitanio, Caractérisation géométrique des solutions de minimax pour l'équation de Hamilton-Jacobi, Enseign. Math., 49 (2003), 3-34. |
[9] |
F. Cardin, The global finite structure of generic envelope loci for Hamilton-Jacobi equations, J. Math. Phys., 43 (2002), 417-430.
doi: 10.1063/1.1423400. |
[10] |
F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Math. J., 144 (2008), 235-284.
doi: 10.1215/00127094-2008-036. |
[11] |
M. Chaperon, Lois de conservation et géométrie symplectique, C. R. Acad. Sci. Paris Sér. I Math., 312 (1991), 345-348. |
[12] |
M. Chaperon, "Familles génératrices," Cours l'école d'été Erasmus de Samos, Publication Erasmus de l'Université de Thessalonique, 1993. |
[13] |
C. Golé, Optical Hamiltonians and symplectic twist maps, Phys. D, 71 (1994), 185-195.
doi: 10.1016/0167-2789(94)90189-9. |
[14] |
V. Humiliére, "Continuité en topologie symplectique," PhD Thesis, École Polytechnique, 2008. |
[15] |
V. Humiliére, On some completions of the space of Hamiltonian maps, Bull. Soc. Math. France, 136 (2008), 373-404. |
[16] |
T. Joukovskaia, "Singularités de minimax et solutions faibles d'équations aux dérivées partielles," PHD Thesis, Université Paris 7, 1993. |
[17] |
L. D. Landau and E. M. Lifshits, "Theoretical Physics. Vol. I," Mechanics. Fourth edition, Moscow, 1988, 216 pp. |
[18] |
L. D. Landau and E. M. Lifshits, "Theoretical Physics. Vol. III," Quantum mechanics. Eighth edition. Akademie-Verlag, Berlin, 1988, xiv+644 pp. |
[19] |
P. Liebermann and C. M. Marle, "Symplectic Geometry and Analytical Mechanics," D. Reidel Publishing Co., Dordrecht 1987, xvi+526 pp. |
[20] |
P. L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations," Research Notes in Mathematics, 69, Boston - London - Melbourne: Pitman Advanced Publishing Program 1982, 317 pp. |
[21] |
P. L. Lions and M. Nisio, A uniqueness result for the semigroup associated with the Hamilton-Jacobi-Bellman operator, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), 273-276.
doi: 10.3792/pjaa.58.273. |
[22] |
P. L. Lions, Some properties of the viscosity semigroups for Hamilton-Jacobi equations, in "Nonlinear Differential Equations" (Granada, 1984), Res. Notes in Math., 132, Pitman, Boston, MA, (1985), 43-63. |
[23] |
J. N. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems, Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991), Lecture Notes in Math., 1589, Springer, Berlin, (1994), 92-186. |
[24] |
D. McCaffrey, Graph selectors and viscosity solutions on Lagrangian manifolds, ESAIM Control Optim. Calc. Var., 12 (2006), 795-815. |
[25] |
A. Ottolenghi and C. Viterbo, Solutions generalisees pour l'equation de Hamilton-Jacobi dans le cas d'evolution,, Unpublished., ().
|
[26] |
G. P. Paternain, L. Polterovich and K. F. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory, Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday. Mosc. Math. J., 3 (2003), 593-619. |
[27] |
J. C. Sikorav, Sur les immersions lagrangiennes dans un fibré cotangent admettant une phase génératrice globale, C. R. Acad. Sci., Paris, t., 302, Sér. I, (1986), 119-122. |
[28] |
J. C. Sikorav, Problémes d'intersections et de points fixes en géométrie hamiltonienne, Comment. Math. Helv. 62 (1987), 62-73.
doi: 10.1007/BF02564438. |
[29] |
D. Theret, "Utilisation des fonctions génératrices en géométrie symplectique globale," PhD Thesis. Université Paris 7, 1996. |
[30] |
D. Theret, A complete proof of Viterbo's uniqueness theorem on generating functions, Topology and its Applications, 96 (1999), 249-266.
doi: 10.1016/S0166-8641(98)00049-2. |
[31] |
C. Viterbo, Symplectic topology as the geometry of generating functions, Mathematische Annalen, 292 (1992), 685-710.
doi: 10.1007/BF01444643. |
[32] |
C. Viterbo, Solutions d'équations d'Hamilton-Jacobi et géométrie symplectique, Sémin. Équ. Dériv. Partielles, Éc. Polytech., Cent. Math., Palaiseau Sémin, Exp. 22 (1996), 6 pp. |
[33] |
C. Viterbo, Symplectic topology and Hamilton-Jacobi equations, Morse theoretic methods in nonlinear analysis and in symplectic topology, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, (2006), 439-459. |
[34] |
A. Weinstein, "Lectures on Symplectic Manifolds," Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics. Number 29. Providence, R. I.: AMS, 1977. |
show all references
References:
[1] |
B. Aebischer et al, "Symplectic Geometry," An introduction based on the seminar in Bern, 1992, Progress in Mathematics, 124, Birkhäuser Verlag, Basel, 1994. |
[2] |
M. Bardi and S. Osher, The nonconvex multidimensional Riemann problem for Hamilton-Jacobi equations, SIAM J. Math. Anal., 22 (1991), 344-351.
doi: 10.1137/0522022. |
[3] |
M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations," Systems and Control: Foundations and Applications, Boston, MA: Birkhauser, 1997, xvii+570 pp. |
[4] |
M. Bardi and S. Faggian, Hopf-type estimates and formulas for nonconvex nonconcave Hamilton-Jacobi equations, SIAM J. Math. Anal., 29 (1998), 1067-1086.
doi: 10.1137/S0036141096309629. |
[5] |
G. Barles, "Solutions de viscosité des équations de Hamilton-Jacobi," Springer, Paris, 1994. |
[6] |
S. Benenti, "Symplectic Relations in Analytical Mechanics," Modern developments in analytical mechanics, Vol. I: Geometrical dynamics, Proc. IUTAM-ISIMM Symp., Torino, Italy, (1982), 39-91. |
[7] |
O. Bernardi and F. Cardin, Minimax and viscosity solutions in the convex case, Commun. Pure Appl. Anal., 5 (2006), 793-812.
doi: 10.3934/cpaa.2006.5.793. |
[8] |
G. Capitanio, Caractérisation géométrique des solutions de minimax pour l'équation de Hamilton-Jacobi, Enseign. Math., 49 (2003), 3-34. |
[9] |
F. Cardin, The global finite structure of generic envelope loci for Hamilton-Jacobi equations, J. Math. Phys., 43 (2002), 417-430.
doi: 10.1063/1.1423400. |
[10] |
F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Math. J., 144 (2008), 235-284.
doi: 10.1215/00127094-2008-036. |
[11] |
M. Chaperon, Lois de conservation et géométrie symplectique, C. R. Acad. Sci. Paris Sér. I Math., 312 (1991), 345-348. |
[12] |
M. Chaperon, "Familles génératrices," Cours l'école d'été Erasmus de Samos, Publication Erasmus de l'Université de Thessalonique, 1993. |
[13] |
C. Golé, Optical Hamiltonians and symplectic twist maps, Phys. D, 71 (1994), 185-195.
doi: 10.1016/0167-2789(94)90189-9. |
[14] |
V. Humiliére, "Continuité en topologie symplectique," PhD Thesis, École Polytechnique, 2008. |
[15] |
V. Humiliére, On some completions of the space of Hamiltonian maps, Bull. Soc. Math. France, 136 (2008), 373-404. |
[16] |
T. Joukovskaia, "Singularités de minimax et solutions faibles d'équations aux dérivées partielles," PHD Thesis, Université Paris 7, 1993. |
[17] |
L. D. Landau and E. M. Lifshits, "Theoretical Physics. Vol. I," Mechanics. Fourth edition, Moscow, 1988, 216 pp. |
[18] |
L. D. Landau and E. M. Lifshits, "Theoretical Physics. Vol. III," Quantum mechanics. Eighth edition. Akademie-Verlag, Berlin, 1988, xiv+644 pp. |
[19] |
P. Liebermann and C. M. Marle, "Symplectic Geometry and Analytical Mechanics," D. Reidel Publishing Co., Dordrecht 1987, xvi+526 pp. |
[20] |
P. L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations," Research Notes in Mathematics, 69, Boston - London - Melbourne: Pitman Advanced Publishing Program 1982, 317 pp. |
[21] |
P. L. Lions and M. Nisio, A uniqueness result for the semigroup associated with the Hamilton-Jacobi-Bellman operator, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), 273-276.
doi: 10.3792/pjaa.58.273. |
[22] |
P. L. Lions, Some properties of the viscosity semigroups for Hamilton-Jacobi equations, in "Nonlinear Differential Equations" (Granada, 1984), Res. Notes in Math., 132, Pitman, Boston, MA, (1985), 43-63. |
[23] |
J. N. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems, Transition to chaos in classical and quantum mechanics (Montecatini Terme, 1991), Lecture Notes in Math., 1589, Springer, Berlin, (1994), 92-186. |
[24] |
D. McCaffrey, Graph selectors and viscosity solutions on Lagrangian manifolds, ESAIM Control Optim. Calc. Var., 12 (2006), 795-815. |
[25] |
A. Ottolenghi and C. Viterbo, Solutions generalisees pour l'equation de Hamilton-Jacobi dans le cas d'evolution,, Unpublished., ().
|
[26] |
G. P. Paternain, L. Polterovich and K. F. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory, Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday. Mosc. Math. J., 3 (2003), 593-619. |
[27] |
J. C. Sikorav, Sur les immersions lagrangiennes dans un fibré cotangent admettant une phase génératrice globale, C. R. Acad. Sci., Paris, t., 302, Sér. I, (1986), 119-122. |
[28] |
J. C. Sikorav, Problémes d'intersections et de points fixes en géométrie hamiltonienne, Comment. Math. Helv. 62 (1987), 62-73.
doi: 10.1007/BF02564438. |
[29] |
D. Theret, "Utilisation des fonctions génératrices en géométrie symplectique globale," PhD Thesis. Université Paris 7, 1996. |
[30] |
D. Theret, A complete proof of Viterbo's uniqueness theorem on generating functions, Topology and its Applications, 96 (1999), 249-266.
doi: 10.1016/S0166-8641(98)00049-2. |
[31] |
C. Viterbo, Symplectic topology as the geometry of generating functions, Mathematische Annalen, 292 (1992), 685-710.
doi: 10.1007/BF01444643. |
[32] |
C. Viterbo, Solutions d'équations d'Hamilton-Jacobi et géométrie symplectique, Sémin. Équ. Dériv. Partielles, Éc. Polytech., Cent. Math., Palaiseau Sémin, Exp. 22 (1996), 6 pp. |
[33] |
C. Viterbo, Symplectic topology and Hamilton-Jacobi equations, Morse theoretic methods in nonlinear analysis and in symplectic topology, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, (2006), 439-459. |
[34] |
A. Weinstein, "Lectures on Symplectic Manifolds," Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics. Number 29. Providence, R. I.: AMS, 1977. |
[1] |
Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080 |
[2] |
Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121 |
[3] |
Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513 |
[4] |
Giuseppe Marmo, Giuseppe Morandi, Narasimhaiengar Mukunda. The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics. Journal of Geometric Mechanics, 2009, 1 (3) : 317-355. doi: 10.3934/jgm.2009.1.317 |
[5] |
Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683 |
[6] |
Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421 |
[7] |
Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 |
[8] |
Gawtum Namah, Mohammed Sbihi. A notion of extremal solutions for time periodic Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 647-664. doi: 10.3934/dcdsb.2010.13.647 |
[9] |
Gui-Qiang Chen, Bo Su. Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 167-192. doi: 10.3934/dcds.2003.9.167 |
[10] |
David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205 |
[11] |
Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389 |
[12] |
Olga Bernardi, Franco Cardin. Minimax and viscosity solutions of Hamilton-Jacobi equations in the convex case. Communications on Pure and Applied Analysis, 2006, 5 (4) : 793-812. doi: 10.3934/cpaa.2006.5.793 |
[13] |
Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1649-1659. doi: 10.3934/dcds.2016.36.1649 |
[14] |
Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295 |
[15] |
Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461 |
[16] |
Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363 |
[17] |
Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure and Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461 |
[18] |
María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda. Kinematic reduction and the Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2012, 4 (3) : 207-237. doi: 10.3934/jgm.2012.4.207 |
[19] |
Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441 |
[20] |
Gonzalo Dávila. Comparison principles for nonlocal Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022061 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]