zbMATH — the first resource for mathematics

A Lagrangian approach to weakly coupled Hamilton-Jacobi systems. (English) Zbl 1343.35065

35F21 Hamilton-Jacobi equations
35F50 Systems of nonlinear first-order PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
37K55 Perturbations, KAM for infinite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI arXiv
[1] P. Bernard, Existence of \(C^{1,1}\) critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds, Ann. Sci. Éc. Norm. Supér. (4), 40 (2007), pp. 445–452, http://dx.doi.org/10.1016/j.ansens.2007.01.004 doi:10.1016/j.ansens.2007.01.004.
[2] P. Billingsley, Convergence of Probability Measures, John Wiley, New York, 1999, http://dx.doi.org/10.1002/9780470316962 doi:10.1002/9780470316962. · Zbl 0944.60003
[3] F. Cagnetti, D. A. Gomes, and H. V. Tran, Adjoint methods for obstacle problems and weakly coupled systems of PDE, ESAIM Control Optim. Calc. Var., 19 (2013), pp. 754–779, http://dx.doi.org/10.1051/cocv/2012032 doi:10.1051/cocv/2012032. · Zbl 1273.35090
[4] F. Camilli, O. Ley, P. Loreti, and V. D. Nguyen, Large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations, NoDEA Nonlinear Differential Equations Appl., 19 (2012), pp. 719–749, http://dx.doi.org/10.1007/s00030-011-0149-7 doi:10.1007/s00030-011-0149-7. · Zbl 1254.49016
[5] A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behavior of solutions of Hamilton–Jacobi equations, SIAM J. Math. Anal., 38 (2006), pp. 478–502, http://dx.doi.org/10.1137/050621955 doi:10.1137/050621955. · Zbl 1109.49034
[6] A. Davini and M. Zavidovique, Aubry sets for weakly coupled systems of Hamilton–Jacobi equations, SIAM J. Math. Anal., 46 (2014), pp. 3361–3389, http://dx.doi.org/10.1137/120899960 doi:10.1137/120899960. · Zbl 1327.35063
[7] A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, ghost book, http://www.crm.sns.it/media/person/1235/fathi.pdf. · Zbl 0885.58022
[8] A. Fathi and A. Siconolfi, Existence of \(C^1\) critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), pp. 363–388, http://dx.doi.org/10.1007/s00222-003-0323-6 doi:10.1007/s00222-003-0323-6. · Zbl 1061.58008
[9] A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, Calc. Var. Partial Differential Equations, 22 (2005), pp. 185–228, http://dx.doi.org/10.1007/s00526-004-0271-z doi:10.1007/s00526-004-0271-z.
[10] D. A. Gomes and A. Serra, Systems of weakly coupled Hamilton-Jacobi equations with implicit obstacles, Canad. J. Math., 64 (2012), pp. 1289–1309, http://dx.doi.org/10.4153/cjm-2011-085-7 doi:10.4153/cjm-2011-085-7. · Zbl 1258.35062
[11] H. Ishii, Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), pp. 231–266, http://dx.doi.org/10.1016/j.anihpc.2006.09.002 doi:10.1016/j.anihpc.2006.09.002. · Zbl 1145.35035
[12] H. Ishii and S. Koike, Viscosity solutions for monotone systems of second-order elliptic PDEs, Comm. Partial Differential Equations, 16 (1991), pp. 1095–1128. · Zbl 0742.35022
[13] O. Kallenberg, Foundations of Modern Probability, Springer, New York, 2002, http://dx.doi.org/10.1007/978-1-4757-4015-8 doi:10.1007/978-1-4757-4015-8. · Zbl 0996.60001
[14] A. Klenke, Probability Theory, Springer, Berlin, 2008, http://dx.doi.org/10.1007/978-1-84800-048-3 doi:10.1007/978-1-84800-048-3.
[15] K. Kuttler, Notes for Seminar, Mimeo, www.math.byu.edu/ klkuttle/, 2013.
[16] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 2000.
[17] H. Mitake and H. V. Tran, Remarks on the large time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton–Jacobi equations, Asymptot. Anal., 77 (2012), pp. 43–70. · Zbl 1241.35020
[18] H. Mitake and H. V. Tran, A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton–Jacobi equations, J. Math. Pures Appl. (9), 101 (2014), pp. 76–93. · Zbl 1278.35020
[19] H. Mitake and H. V. Tran, Homogenization of weakly coupled systems of Hamilton–Jacobi equations with fast switching rates, Arch. Ration. Mech. Anal., 211 (2014), pp. 733–769, http://dx.doi.org/10.1007/s00205-013-0685-x doi:10.1007/s00205-013-0685-x. · Zbl 1297.35022
[20] J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, UK, 1997, http://dx.doi.org/10.1017/CBO9780511810633 doi:10.1017/CBO9780511810633.
[21] J. Swart And A. Winter, Markov Processes: Theory and Examples, Mimeo, https://www.uni-due.de/ hm0110/Markovprocesses/sw20.pdf, 2013.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.