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Linear programming interpretations of Mather's variational principle

Published online by Cambridge University Press:  15 August 2002

L. C. Evans  and
D. Gomes
L. C. Evans
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA; evans@math.Berkeley.EDU.
D. Gomes
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712, USA.
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Abstract

We discuss some implications of linear programming for Mather theory [13-15] and itsfinite dimensional approximations. We find that the complementaryslackness condition of duality theory formally implies that the Mather set lies in ann-dimensional graph and as well predicts the relevant nonlinear PDE for the “weakKAM” theory of Fathi [5-8].

Keywords

Linear programmingdualityweak KAM theory.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 693 - 702
Copyright
© EDP Sciences, SMAI, 2002

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References

E.J. Anderson and P. Nash, Linear Programming in Infinite Dimensional Spaces. Wiley (1987).
D. Bertsimas and J. Tsitsiklis, Introduction to Linear Optimization. Athena Scientific (1997).
L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer (survey paper). Available at the website of LCE, at math.berkeley.edu
L.C. Evans, Some new PDE methods for weak KAM theory. Calc. Var. Partial Differential Equations (to appear).
Evans, L.C. and Gomes, D., Effective Hamiltonians and averaging for Hamiltonian dynamics I. Arch. Rational Mech. Anal. 157 (2001) 1-33. CrossRef
Fathi, A., Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 1043-1046. CrossRef
Fathi, A., Solutions KAM faibles conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 649-652. CrossRef
A. Fathi, Weak KAM theory in Lagrangian Dynamics, Preliminary Version. Lecture Notes (2001).
J. Franklin, Methods of Mathematical Economics. SIAM, Classics in Appl. Math. 37 (2002).
D. Gomes, Numerical methods and Hamilton-Jacobi equations (to appear).
P. Lax, Linear Algebra. John Wiley (1997).
P.-L. Lions, G. Papanicolaou and S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations. CIRCA (1988) (unpublished).
Mather, J., Minimal measures. Comment. Math Helvetici 64 (1989) 375-394. CrossRef
Mather, J., Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207 (1991) 169-207. CrossRef
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).