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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).
|
[2] |
M. Bardi and S. Osher, The nonconvex multidimensional Riemann problem for Hamilton-Jacobi equations,, SIAM J. Math. Anal., 22 (1991), 344.
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, (1997).
|
[4] |
M. Bardi and S. Faggian, Hopf-type estimates and formulas for nonconvex nonconcave Hamilton-Jacobi equations,, SIAM J. Math. Anal., 29 (1998), 1067.
doi: 10.1137/S0036141096309629. |
[5] |
G. Barles, "Solutions de viscosité des équations de Hamilton-Jacobi,", Springer, (1994).
|
[6] |
S. Benenti, "Symplectic Relations in Analytical Mechanics,", Modern developments in analytical mechanics, (1982), 39.
|
[7] |
O. Bernardi and F. Cardin, Minimax and viscosity solutions in the convex case,, Commun. Pure Appl. Anal., 5 (2006), 793.
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.
|
[9] |
F. Cardin, The global finite structure of generic envelope loci for Hamilton-Jacobi equations,, J. Math. Phys., 43 (2002), 417.
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.
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.
|
[12] |
M. Chaperon, "Familles génératrices,", Cours l'école d'été Erasmus de Samos, (1993). Google Scholar |
[13] |
C. Golé, Optical Hamiltonians and symplectic twist maps,, Phys. D, 71 (1994), 185.
doi: 10.1016/0167-2789(94)90189-9. |
[14] |
V. Humiliére, "Continuité en topologie symplectique,", PhD Thesis, (2008). Google Scholar |
[15] |
V. Humiliére, On some completions of the space of Hamiltonian maps,, Bull. Soc. Math. France, 136 (2008), 373.
|
[16] |
T. Joukovskaia, "Singularités de minimax et solutions faibles d'équations aux dérivées partielles,", PHD Thesis, (1993). Google Scholar |
[17] |
L. D. Landau and E. M. Lifshits, "Theoretical Physics. Vol. I,", Mechanics. Fourth edition, (1988).
|
[18] |
L. D. Landau and E. M. Lifshits, "Theoretical Physics. Vol. III,", Quantum mechanics. Eighth edition. Akademie-Verlag, (1988).
|
[19] |
P. Liebermann and C. M. Marle, "Symplectic Geometry and Analytical Mechanics,", D. Reidel Publishing Co., (1987).
|
[20] |
P. L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", Research Notes in Mathematics, (1982).
|
[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.
doi: 10.3792/pjaa.58.273. |
[22] |
P. L. Lions, Some properties of the viscosity semigroups for Hamilton-Jacobi equations,, in, 132 (1985), 43.
|
[23] |
J. N. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems,, Transition to chaos in classical and quantum mechanics (Montecatini Terme, (1994), 92.
|
[24] |
D. McCaffrey, Graph selectors and viscosity solutions on Lagrangian manifolds,, ESAIM Control Optim. Calc. Var., 12 (2006), 795.
|
[25] |
A. Ottolenghi and C. Viterbo, Solutions generalisees pour l'equation de Hamilton-Jacobi dans le cas d'evolution,, Unpublished., (). Google Scholar |
[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.
|
[27] |
J. C. Sikorav, Sur les immersions lagrangiennes dans un fibré cotangent admettant une phase génératrice globale,, C. R. Acad. Sci., 302 (1986), 119.
|
[28] |
J. C. Sikorav, Problémes d'intersections et de points fixes en géométrie hamiltonienne,, Comment. Math. Helv. \textbf{62} (1987), 62 (1987), 62.
doi: 10.1007/BF02564438. |
[29] |
D. Theret, "Utilisation des fonctions génératrices en géométrie symplectique globale,", PhD Thesis. Université Paris 7, (1996). Google Scholar |
[30] |
D. Theret, A complete proof of Viterbo's uniqueness theorem on generating functions,, Topology and its Applications, 96 (1999), 249.
doi: 10.1016/S0166-8641(98)00049-2. |
[31] |
C. Viterbo, Symplectic topology as the geometry of generating functions,, Mathematische Annalen, 292 (1992), 685.
doi: 10.1007/BF01444643. |
[32] |
C. Viterbo, Solutions d'équations d'Hamilton-Jacobi et géométrie symplectique,, Sémin. Équ. Dériv. Partielles, 22 (1996).
|
[33] |
C. Viterbo, Symplectic topology and Hamilton-Jacobi equations,, Morse theoretic methods in nonlinear analysis and in symplectic topology, (2006), 439.
|
[34] |
A. Weinstein, "Lectures on Symplectic Manifolds,", Conference Board of the Mathematical Sciences, (1977).
|
show all references
References:
[1] |
B. Aebischer et al, "Symplectic Geometry,", An introduction based on the seminar in Bern, (1992).
|
[2] |
M. Bardi and S. Osher, The nonconvex multidimensional Riemann problem for Hamilton-Jacobi equations,, SIAM J. Math. Anal., 22 (1991), 344.
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, (1997).
|
[4] |
M. Bardi and S. Faggian, Hopf-type estimates and formulas for nonconvex nonconcave Hamilton-Jacobi equations,, SIAM J. Math. Anal., 29 (1998), 1067.
doi: 10.1137/S0036141096309629. |
[5] |
G. Barles, "Solutions de viscosité des équations de Hamilton-Jacobi,", Springer, (1994).
|
[6] |
S. Benenti, "Symplectic Relations in Analytical Mechanics,", Modern developments in analytical mechanics, (1982), 39.
|
[7] |
O. Bernardi and F. Cardin, Minimax and viscosity solutions in the convex case,, Commun. Pure Appl. Anal., 5 (2006), 793.
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.
|
[9] |
F. Cardin, The global finite structure of generic envelope loci for Hamilton-Jacobi equations,, J. Math. Phys., 43 (2002), 417.
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.
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.
|
[12] |
M. Chaperon, "Familles génératrices,", Cours l'école d'été Erasmus de Samos, (1993). Google Scholar |
[13] |
C. Golé, Optical Hamiltonians and symplectic twist maps,, Phys. D, 71 (1994), 185.
doi: 10.1016/0167-2789(94)90189-9. |
[14] |
V. Humiliére, "Continuité en topologie symplectique,", PhD Thesis, (2008). Google Scholar |
[15] |
V. Humiliére, On some completions of the space of Hamiltonian maps,, Bull. Soc. Math. France, 136 (2008), 373.
|
[16] |
T. Joukovskaia, "Singularités de minimax et solutions faibles d'équations aux dérivées partielles,", PHD Thesis, (1993). Google Scholar |
[17] |
L. D. Landau and E. M. Lifshits, "Theoretical Physics. Vol. I,", Mechanics. Fourth edition, (1988).
|
[18] |
L. D. Landau and E. M. Lifshits, "Theoretical Physics. Vol. III,", Quantum mechanics. Eighth edition. Akademie-Verlag, (1988).
|
[19] |
P. Liebermann and C. M. Marle, "Symplectic Geometry and Analytical Mechanics,", D. Reidel Publishing Co., (1987).
|
[20] |
P. L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", Research Notes in Mathematics, (1982).
|
[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.
doi: 10.3792/pjaa.58.273. |
[22] |
P. L. Lions, Some properties of the viscosity semigroups for Hamilton-Jacobi equations,, in, 132 (1985), 43.
|
[23] |
J. N. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems,, Transition to chaos in classical and quantum mechanics (Montecatini Terme, (1994), 92.
|
[24] |
D. McCaffrey, Graph selectors and viscosity solutions on Lagrangian manifolds,, ESAIM Control Optim. Calc. Var., 12 (2006), 795.
|
[25] |
A. Ottolenghi and C. Viterbo, Solutions generalisees pour l'equation de Hamilton-Jacobi dans le cas d'evolution,, Unpublished., (). Google Scholar |
[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.
|
[27] |
J. C. Sikorav, Sur les immersions lagrangiennes dans un fibré cotangent admettant une phase génératrice globale,, C. R. Acad. Sci., 302 (1986), 119.
|
[28] |
J. C. Sikorav, Problémes d'intersections et de points fixes en géométrie hamiltonienne,, Comment. Math. Helv. \textbf{62} (1987), 62 (1987), 62.
doi: 10.1007/BF02564438. |
[29] |
D. Theret, "Utilisation des fonctions génératrices en géométrie symplectique globale,", PhD Thesis. Université Paris 7, (1996). Google Scholar |
[30] |
D. Theret, A complete proof of Viterbo's uniqueness theorem on generating functions,, Topology and its Applications, 96 (1999), 249.
doi: 10.1016/S0166-8641(98)00049-2. |
[31] |
C. Viterbo, Symplectic topology as the geometry of generating functions,, Mathematische Annalen, 292 (1992), 685.
doi: 10.1007/BF01444643. |
[32] |
C. Viterbo, Solutions d'équations d'Hamilton-Jacobi et géométrie symplectique,, Sémin. Équ. Dériv. Partielles, 22 (1996).
|
[33] |
C. Viterbo, Symplectic topology and Hamilton-Jacobi equations,, Morse theoretic methods in nonlinear analysis and in symplectic topology, (2006), 439.
|
[34] |
A. Weinstein, "Lectures on Symplectic Manifolds,", Conference Board of the Mathematical Sciences, (1977).
|
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