# American Institute of Mathematical Sciences

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

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.
Citation: Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 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, 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.

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##### 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.
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