Linear programming interpretations of Mather's variational principle
ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002), p. 693-702

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 n-dimensional graph and as well predicts the relevant nonlinear PDE for the “weak KAM” theory of Fathi [6, 7, 8, 5].

DOI : https://doi.org/10.1051/cocv:2002030
Classification:  90C05,  35F20
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     pages = {693-702},
     doi = {10.1051/cocv:2002030},
     zbl = {1090.90143},
     language = {en},
     url = {http://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, Volume 8 (2002) pp. 693-702. doi : 10.1051/cocv:2002030. http://www.numdam.org/item/COCV_2002__8__693_0/

[1] E.J. Anderson and P. Nash, Linear Programming in Infinite Dimensional Spaces. Wiley (1987). | MR 893179 | Zbl 0632.90038

[2] D. Bertsimas and J. Tsitsiklis, Introduction to Linear Optimization. Athena Scientific (1997). | Zbl 0997.90505

[3] L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer (survey paper). Available at the website of LCE, at math.berkeley.edu | Zbl 0954.35011

[4] L.C. Evans, Some new PDE methods for weak KAM theory. Calc. Var. Partial Differential Equations (to appear). | MR 1986317 | Zbl 1032.37048

[5] L.C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics I. Arch. Rational Mech. Anal. 157 (2001) 1-33. | MR 1822413 | Zbl 0986.37056

[6] A. Fathi, 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] A. Fathi, 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] A. Fathi, Weak KAM theory in Lagrangian Dynamics, Preliminary Version. Lecture Notes (2001).

[9] J. Franklin, Methods of Mathematical Economics. SIAM, Classics in Appl. Math. 37 (2002). | MR 1875314 | Zbl 1075.90001

[10] D. Gomes, Numerical methods and Hamilton-Jacobi equations (to appear).

[11] P. Lax, Linear Algebra. John Wiley (1997). | MR 1423602 | Zbl 0904.15001

[12] P.-L. Lions, G. Papanicolaou and S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations. CIRCA (1988) (unpublished).

[13] J. Mather, Minimal measures. Comment. Math Helvetici 64 (1989) 375-394. | MR 998855 | Zbl 0689.58025

[14] J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207 (1991) 169-207. | MR 1109661 | Zbl 0696.58027

[15] J. Mather and G. Forni, 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