We discuss some implications of linear programming for Mather theory [13, 14, 15] and its finite dimensional approximations. We find that the complementary slackness condition of duality theory formally implies that the Mather set lies in an -dimensional graph and as well predicts the relevant nonlinear PDE for the “weak KAM” theory of Fathi [6, 7, 8, 5].
Classification : 90C05, 35F20
Mots clés : linear programming, duality, weak KAM theory
@article{COCV_2002__8__693_0, author = {Evans, L. C. and Gomes, D.}, title = {Linear programming interpretations of Mather's variational principle}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {693--702}, publisher = {EDP-Sciences}, volume = {8}, year = {2002}, doi = {10.1051/cocv:2002030}, zbl = {1090.90143}, language = {en}, url = {www.numdam.org/item/COCV_2002__8__693_0/} }
Evans, L. C.; Gomes, D. Linear programming interpretations of Mather's variational principle. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 693-702. doi : 10.1051/cocv:2002030. http://www.numdam.org/item/COCV_2002__8__693_0/
[1] Linear Programming in Infinite Dimensional Spaces. Wiley (1987). | MR 893179 | Zbl 0632.90038
and ,[2] Introduction to Linear Optimization. Athena Scientific (1997). | Zbl 0997.90505
and ,[3] Partial differential equations and Monge-Kantorovich mass transfer (survey paper). Available at the website of LCE, at math.berkeley.edu | Zbl 0954.35011
,[4] Some new PDE methods for weak KAM theory. Calc. Var. Partial Differential Equations (to appear). | MR 1986317 | Zbl 1032.37048
,[5] Effective Hamiltonians and averaging for Hamiltonian dynamics I. Arch. Rational Mech. Anal. 157 (2001) 1-33. | MR 1822413 | Zbl 0986.37056
and ,[6] Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 1043-1046. | MR 1451248 | Zbl 0885.58022
,[7] Solutions KAM faibles conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 649-652. | MR 1473840 | Zbl 0943.37031
,[8] Weak KAM theory in Lagrangian Dynamics, Preliminary Version. Lecture Notes (2001).
,[9] Methods of Mathematical Economics. SIAM, Classics in Appl. Math. 37 (2002). | MR 1875314 | Zbl 1075.90001
,[10] Numerical methods and Hamilton-Jacobi equations (to appear).
,[11] Linear Algebra. John Wiley (1997). | MR 1423602 | Zbl 0904.15001
,[12] Homogenization of Hamilton-Jacobi equations. CIRCA (1988) (unpublished).
, and ,[13] Minimal measures. Comment. Math Helvetici 64 (1989) 375-394. | MR 998855 | Zbl 0689.58025
,[14] Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207 (1991) 169-207. | MR 1109661 | Zbl 0696.58027
,[15] Action minimizing orbits in Hamiltonian systems. Transition to Chaos in Classical and Quantum Mechanics, edited by S. Graffi. Sringer, Lecture Notes in Math. 1589 (1994). | MR 1323222 | Zbl 0822.70011
and ,