zbMATH — the first resource for mathematics

The selection problem for discounted Hamilton-Jacobi equations: some non-convex cases. (English) Zbl 1403.35082
Summary: Here, we study the selection problem for the vanishing discount approximation of non-convex, first-order Hamilton-Jacobi equations. While the selection problem is well understood for convex Hamiltonians, the selection problem for non-convex Hamiltonians has thus far not been studied. We begin our study by examining a generalized discounted Hamilton-Jacobi equation. Next, using an exponential transformation, we apply our methods to strictly quasi-convex and to some non-convex Hamilton-Jacobi equations. Finally, we examine a non-convex Hamiltonian with flat parts to which our results do not directly apply. In this case, we establish the convergence by a direct approach.

35F21 Hamilton-Jacobi equations
35B40 Asymptotic behavior of solutions to PDEs
37J50 Action-minimizing orbits and measures (MSC2010)
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
Full Text: DOI arXiv
[1] E. S. Al-Aidarous, E. O. Alzahrani, H. Ishii and A. M. M. Younas, A convergence result for the ergodic problem for Hamilton-Jacobi equations with Neumann type boundary conditions, \doihref10.1017/S0308210515000517Proc. Royal Soc. Edinburgh Sect. A, 146 (2016), 225-242. · Zbl 1350.35063
[2] S. Armstrong, H. V. Tran and Y. Yu, Stochastic homogenization of a nonconvex Hamilton-Jacobi equation, \doihref10.1007/s00526-015-0833-2Calc. Var. Partial Differential Equations, 54 (2015), 1507-1524. · Zbl 1329.35042
[3] S. Armstrong, H. V. Tran and Y. Yu, Stochastic homogenization of nonconvex Hamilton-Jacobi equations in one space dimension, \doihref10.1016/j.jde.2016.05.010J. Differential Equations, 261 (2016), 2702-2737. · Zbl 1342.35026
[4] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, \doihref10.1007/978-0-8176-4755-1Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. With appendices by M. Falcone and P. Soravia. · Zbl 0890.49011
[5] U. Bessi, Aubry-Mather theory and Hamilton-Jacobi equations, \doihref10.1007/s00220-002-0781-5Comm. Math. Phys., 235 (2003), 495-511.
[6] S. Bolotin, List of open problems, http://www.aimath.org/WWN/dynpde/articles/html/20a/.
[7] F. Cagnetti, D. Gomes and H. V. Tran, Aubry-Mather measures in the non convex setting, \doihref10.1137/100817656SIAM J. Math. Anal., 43 (2011), 2601-2629. · Zbl 1258.35059
[8] 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, \doihref10.1016/j.anihpc.2013.10.005Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 183-200. · Zbl 1312.35020
[9] A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique, Convergence of the solutions of the discounted Hamilton-Jacobi equation, \doihref10.1007/s00222-016-0648-6Invent. Math., 206 (2016), 29-55. · Zbl 1362.35094
[10] L. C. Evans, Adjoint and compensated compactness methods for Hamilton-Jacobi PDE, \doihref10.1007/s00205-010-0307-9Arch. Rat. Mech. Anal., 197 (2010), 1053-1088. · Zbl 1273.70030
[11] A. Fathi, Weak KAM Theorem in Lagrangian Dynamics, Cambridge Univ. PR(Txp), 2016.
[12] A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, \doihref10.1016/S0764-4442(98)80144-4C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270.
[13] A. Fathi, Weak KAM from a PDE point of view: viscosity solutions of the Hamilton-Jacobi equation and Aubry set, \doihref10.1017/S0308210550000064Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1193-1236. · Zbl 1297.35004
[14] A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians, \doihref10.1007/s00526-004-0271-zCalc. Var. Partial Differential Equations, 22 (2005), 185-228.
[15] H. Gao, Random homogenization of coercive Hamilton-Jacobi equations in 1d, \doihref10.1007/s00526-016-0968-9Calc. Var. Partial Differential Equations, 55 (2016), Art. 30, 39pp.
[16] D. A. Gomes, Generalized Mather problem and selection principles for viscosity solutions and Mather measures, \doihref10.1515/ACV.2008.012Adv. Calc. Var., 1 (2008), 291-307. · Zbl 1181.37090
[17] H. Ishii, H. Mitake and H. V. Tran, The vanishing discount problem and viscosity Mather measures, Part 1: the problem on a torus, \doihref10.1016/j.matpur.2016.10.013J. Math. Pures Appl. (9), 108 (2017), 125-149. · Zbl 1375.35182
[18] H. Ishii, H. Mitake and H. V. Tran, The vanishing discount problem and viscosity Mather measures, Part 2: boundary value problems, \doihref10.1016/j.matpur.2016.11.002J. Math. Pures Appl. (9), 108 (2017), 261-305. · Zbl 1375.35183
[19] H. R. Jauslin, H. O. Kreiss and J. Moser, On the forced Burgers equation with periodic boundary conditions, Proc. Sympos. Pure Math., 65, Amer. Math. Soc., Province, RI, 1999. · Zbl 0930.35156
[20] P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, unpublished work (1987).
[21] R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, \doihref10.1088/0951-7715/9/2/002Nonlinearity, 9 (1996), 273-310.
[22] J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, \doihref10.1007/BF02571383Math. Z., 207 (1991), 169-207. · Zbl 0696.58027
[23] H. Mitake and H. V. Tran, Selection problems for a discount degenerate viscous Hamilton-Jacobi equation, \doihref10.1016/j.aim.2016.10.032Adv. Math., 306 (2017), 684-703. · Zbl 1357.35090
[24] K. Soga, Selection problems of \(\mathbb{Z}^2\)-periodic entropy solutions and viscosity solutions, \doihref10.1007/s00526-017-1208-7Calc. Var. Partial Differential Equations, 56 (2017), Art 119.
[25] H. V. Tran, Adjoint methods for static Hamilton-Jacobi equations, \doihref10.1007/s00526-010-0363-xCalc. Var. Partial Differential Equations, 41 (2011), 301-319.
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.