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Persistence of the hyperbolic lower dimensional non-twist invariant torus in a class of Hamiltonian systems
On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary
1. | Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padova, Italy |
2. | Dipartimento di Scienze Statistiche, Università di Padova, Via Cesare Battisti 141, 35121 Padova, Italy |
3. | Università degli Studi di Padova, Dipartimento di Matematica Pura ed Applicata, Via Trieste, 63 - 35121 Padova |
References:
[1] |
M. Bardi, A. Cesaroni and L. Rossi, Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control,, \emph{ESAIM Control Optim. Calc. Var.}, ().
doi: http://dx.doi.org/10.1051/cocv/2015033. |
[2] |
G. Barles and J. Burdeau, The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit control problems, Comm. Part. Diff. Eq., 20 (1995), 129-178.
doi: 10.1080/03605309508821090. |
[3] |
G. Barles, A. Porretta and T.T. Tchamba, On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi equations, J. Math. Pures Appl., 94 (2010), 497-519.
doi: 10.1016/j.matpur.2010.03.006. |
[4] |
G. Barles and P.E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations, SIAM J. Math. Anal., 32 (2001), 1311-1323.
doi: 10.1137/S0036141000369344. |
[5] |
H. Berestycki, I. Capuzzo Dolcetta, A. Porretta and L. Rossi, Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators, J. Math. Pures Appl., 103 (2015), 1276-1293.
doi: 10.1016/j.matpur.2014.10.012. |
[6] |
F. Cagnetti, D. Gomes, H. Mitake and H.V. Tran, A new method for large time behavior of degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians, Ann. Inst. H. Poincare Anal. Non Lineaire, 32 (2015), 183-200.
doi: 10.1016/j.anihpc.2013.10.005. |
[7] |
M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[8] |
F. Da Lio, Large time behavior of solutions to parabolic equations with Neumann boundary conditions, J. Math. Anal. Appl., 339 (2008), 384-398.
doi: 10.1016/j.jmaa.2007.06.052. |
[9] |
G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I., 5 (1956), 1-30. |
[10] |
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1988. |
[11] |
H. Ishii and P. Loreti, A class of stochastic optimal control problems with state constraints, Indiana Univ. Math. J., 51 (2002), 1167-1196.
doi: 10.1512/iumj.2002.51.2079. |
[12] |
J.M. Lasry and P.-L. Lions, Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem, Math. Ann., 283 (1989), 583-630.
doi: 10.1007/BF01442856. |
[13] |
O. Ley and V.D. Nguyen, Large time behavior for some nonlinear degenerate parabolic equations, J. Math. Pures Appl., 102 (2014), 293-314.
doi: 10.1016/j.matpur.2013.11.010. |
[14] |
T. Leonori and A. Porretta, Gradient bounds for elliptic problems singular at the boundary, Arch. Ration. Mech. Anal., 202 (2011), 663-705.
doi: 10.1007/s00205-011-0436-9. |
[15] |
G.M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ 1996.
doi: 10.1142/3302. |
[16] |
M.V. Safonov, On the classical solution of Bellman's elliptic equation, Sov. Math. Dokl., 30 (1984), 482-485. |
[17] |
N.S. Trudinger, On regularity and existence of viscosity solutions of nonlinear second order, elliptic equations, Partial differential equations and the calculus of variations, Vol. II, 939-957, Progr. Nonlinear Differential Equations Appl., 2, Birkhauser Boston, Boston, MA, 1989. |
[18] |
G. Tian and X.J. Wang, A priori estimates for fully nonlinear parabolic equations, Int. Math. Res. Notes, 169 (2012), 1-21. |
show all references
References:
[1] |
M. Bardi, A. Cesaroni and L. Rossi, Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control,, \emph{ESAIM Control Optim. Calc. Var.}, ().
doi: http://dx.doi.org/10.1051/cocv/2015033. |
[2] |
G. Barles and J. Burdeau, The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit control problems, Comm. Part. Diff. Eq., 20 (1995), 129-178.
doi: 10.1080/03605309508821090. |
[3] |
G. Barles, A. Porretta and T.T. Tchamba, On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi equations, J. Math. Pures Appl., 94 (2010), 497-519.
doi: 10.1016/j.matpur.2010.03.006. |
[4] |
G. Barles and P.E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations, SIAM J. Math. Anal., 32 (2001), 1311-1323.
doi: 10.1137/S0036141000369344. |
[5] |
H. Berestycki, I. Capuzzo Dolcetta, A. Porretta and L. Rossi, Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators, J. Math. Pures Appl., 103 (2015), 1276-1293.
doi: 10.1016/j.matpur.2014.10.012. |
[6] |
F. Cagnetti, D. Gomes, H. Mitake and H.V. Tran, A new method for large time behavior of degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians, Ann. Inst. H. Poincare Anal. Non Lineaire, 32 (2015), 183-200.
doi: 10.1016/j.anihpc.2013.10.005. |
[7] |
M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[8] |
F. Da Lio, Large time behavior of solutions to parabolic equations with Neumann boundary conditions, J. Math. Anal. Appl., 339 (2008), 384-398.
doi: 10.1016/j.jmaa.2007.06.052. |
[9] |
G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I., 5 (1956), 1-30. |
[10] |
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1988. |
[11] |
H. Ishii and P. Loreti, A class of stochastic optimal control problems with state constraints, Indiana Univ. Math. J., 51 (2002), 1167-1196.
doi: 10.1512/iumj.2002.51.2079. |
[12] |
J.M. Lasry and P.-L. Lions, Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem, Math. Ann., 283 (1989), 583-630.
doi: 10.1007/BF01442856. |
[13] |
O. Ley and V.D. Nguyen, Large time behavior for some nonlinear degenerate parabolic equations, J. Math. Pures Appl., 102 (2014), 293-314.
doi: 10.1016/j.matpur.2013.11.010. |
[14] |
T. Leonori and A. Porretta, Gradient bounds for elliptic problems singular at the boundary, Arch. Ration. Mech. Anal., 202 (2011), 663-705.
doi: 10.1007/s00205-011-0436-9. |
[15] |
G.M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ 1996.
doi: 10.1142/3302. |
[16] |
M.V. Safonov, On the classical solution of Bellman's elliptic equation, Sov. Math. Dokl., 30 (1984), 482-485. |
[17] |
N.S. Trudinger, On regularity and existence of viscosity solutions of nonlinear second order, elliptic equations, Partial differential equations and the calculus of variations, Vol. II, 939-957, Progr. Nonlinear Differential Equations Appl., 2, Birkhauser Boston, Boston, MA, 1989. |
[18] |
G. Tian and X.J. Wang, A priori estimates for fully nonlinear parabolic equations, Int. Math. Res. Notes, 169 (2012), 1-21. |
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