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

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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

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

Received March 2010 Revised April 2011 Published June 2011

Abstract
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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).
[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).
[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).
[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).
[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., ().
[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).
[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).
[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).
[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).
[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., ().
[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).
[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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