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June  2011, 31(2): 385-406. doi: 10.3934/dcds.2011.31.385

On $C^0$-variational solutions for Hamilton-Jacobi equations

Olga Bernardi 1, and  Franco Cardin 1,

1. 

Dipartimento di Matematica Pura ed Applicata, Via Trieste, 63 - 35121 Padova, Italy, Italy

Received  March 2010 Revised  April 2011 Published  June 2011

For evolutive Hamilton-Jacobi equations, we propose a refined definition of $C^0$-variational solution, adapted to Cauchy problems for continuous initial data. This weaker framework enables us to investigate the semigroup property for these solutions. In the case of $p$-convex Hamiltonians, when variational solutions are known to be identical to viscosity solutions, we verify directly the semigroup property by using minmax techniques. In the non-convex case, we construct a first explicit evolutive example where minmax and viscosity solutions are different. Provided the initial data allow for the separation of variables, we also detect the semigroup property for convex-concave Hamiltonians. In this case, and for general initial data, we finally give new upper and lower Hopf-type estimates for the variational solutions.
Keywords: Hamilton-Jacobi Theory, semigroup property., weak solutions.
Mathematics Subject Classification: Primary: 70H20, 35D30; Secondary: 37J0.
Citation: Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385
References:
[1]

B. Aebischer et al, "Symplectic Geometry,", An introduction based on the seminar in Bern, (1992).   Google Scholar

[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.  Google Scholar

[3]

M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,", Systems and Control: Foundations and Applications, (1997).   Google Scholar

[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.  Google Scholar

[5]

G. Barles, "Solutions de viscosité des équations de Hamilton-Jacobi,", Springer, (1994).   Google Scholar

[6]

S. Benenti, "Symplectic Relations in Analytical Mechanics,", Modern developments in analytical mechanics, (1982), 39.   Google Scholar

[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.  Google Scholar

[8]

G. Capitanio, Caractérisation géométrique des solutions de minimax pour l'équation de Hamilton-Jacobi,, Enseign. Math., 49 (2003), 3.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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M. Chaperon, Lois de conservation et géométrie symplectique,, C. R. Acad. Sci. Paris Sér. I Math., 312 (1991), 345.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[18]

L. D. Landau and E. M. Lifshits, "Theoretical Physics. Vol. III,", Quantum mechanics. Eighth edition. Akademie-Verlag, (1988).   Google Scholar

[19]

P. Liebermann and C. M. Marle, "Symplectic Geometry and Analytical Mechanics,", D. Reidel Publishing Co., (1987).   Google Scholar

[20]

P. L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", Research Notes in Mathematics, (1982).   Google Scholar

[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.  Google Scholar

[22]

P. L. Lions, Some properties of the viscosity semigroups for Hamilton-Jacobi equations,, in, 132 (1985), 43.   Google Scholar

[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.   Google Scholar

[24]

D. McCaffrey, Graph selectors and viscosity solutions on Lagrangian manifolds,, ESAIM Control Optim. Calc. Var., 12 (2006), 795.   Google Scholar

[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.   Google Scholar

[27]

J. C. Sikorav, Sur les immersions lagrangiennes dans un fibré cotangent admettant une phase génératrice globale,, C. R. Acad. Sci., 302 (1986), 119.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

C. Viterbo, Symplectic topology as the geometry of generating functions,, Mathematische Annalen, 292 (1992), 685.  doi: 10.1007/BF01444643.  Google Scholar

[32]

C. Viterbo, Solutions d'équations d'Hamilton-Jacobi et géométrie symplectique,, Sémin. Équ. Dériv. Partielles, 22 (1996).   Google Scholar

[33]

C. Viterbo, Symplectic topology and Hamilton-Jacobi equations,, Morse theoretic methods in nonlinear analysis and in symplectic topology, (2006), 439.   Google Scholar

[34]

A. Weinstein, "Lectures on Symplectic Manifolds,", Conference Board of the Mathematical Sciences, (1977).   Google Scholar

show all references

References:
[1]

B. Aebischer et al, "Symplectic Geometry,", An introduction based on the seminar in Bern, (1992).   Google Scholar

[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.  Google Scholar

[3]

M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,", Systems and Control: Foundations and Applications, (1997).   Google Scholar

[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.  Google Scholar

[5]

G. Barles, "Solutions de viscosité des équations de Hamilton-Jacobi,", Springer, (1994).   Google Scholar

[6]

S. Benenti, "Symplectic Relations in Analytical Mechanics,", Modern developments in analytical mechanics, (1982), 39.   Google Scholar

[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.  Google Scholar

[8]

G. Capitanio, Caractérisation géométrique des solutions de minimax pour l'équation de Hamilton-Jacobi,, Enseign. Math., 49 (2003), 3.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Chaperon, Lois de conservation et géométrie symplectique,, C. R. Acad. Sci. Paris Sér. I Math., 312 (1991), 345.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[18]

L. D. Landau and E. M. Lifshits, "Theoretical Physics. Vol. III,", Quantum mechanics. Eighth edition. Akademie-Verlag, (1988).   Google Scholar

[19]

P. Liebermann and C. M. Marle, "Symplectic Geometry and Analytical Mechanics,", D. Reidel Publishing Co., (1987).   Google Scholar

[20]

P. L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", Research Notes in Mathematics, (1982).   Google Scholar

[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.  Google Scholar

[22]

P. L. Lions, Some properties of the viscosity semigroups for Hamilton-Jacobi equations,, in, 132 (1985), 43.   Google Scholar

[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.   Google Scholar

[24]

D. McCaffrey, Graph selectors and viscosity solutions on Lagrangian manifolds,, ESAIM Control Optim. Calc. Var., 12 (2006), 795.   Google Scholar

[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.   Google Scholar

[27]

J. C. Sikorav, Sur les immersions lagrangiennes dans un fibré cotangent admettant une phase génératrice globale,, C. R. Acad. Sci., 302 (1986), 119.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

C. Viterbo, Symplectic topology as the geometry of generating functions,, Mathematische Annalen, 292 (1992), 685.  doi: 10.1007/BF01444643.  Google Scholar

[32]

C. Viterbo, Solutions d'équations d'Hamilton-Jacobi et géométrie symplectique,, Sémin. Équ. Dériv. Partielles, 22 (1996).   Google Scholar

[33]

C. Viterbo, Symplectic topology and Hamilton-Jacobi equations,, Morse theoretic methods in nonlinear analysis and in symplectic topology, (2006), 439.   Google Scholar

[34]

A. Weinstein, "Lectures on Symplectic Manifolds,", Conference Board of the Mathematical Sciences, (1977).   Google Scholar

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