A finite dimensional linear programming approximation of Mather's variational problem
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 1094-1109.

We provide an approximation of Mather variational problem by finite dimensional minimization problems in the framework of Γ-convergence. By a linear programming interpretation as done in [Evans and Gomes, ESAIM: COCV 8 (2002) 693-702] we state a duality theorem for the Mather problem, as well a finite dimensional approximation for the dual problem.

DOI: 10.1051/cocv/2009039
Classification: 37J50, 49Q20, 49N60, 74P20, 65K10
Keywords: Mather problem, minimal measures, linear programming, Γ-convergence
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     author = {Granieri, Luca},
     title = {A finite dimensional linear programming approximation {of~Mather's} variational problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1094--1109},
     publisher = {EDP-Sciences},
     volume = {16},
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     year = {2010},
     doi = {10.1051/cocv/2009039},
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     zbl = {1205.37077},
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     url = {http://archive.numdam.org/articles/10.1051/cocv/2009039/}
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Granieri, Luca. A finite dimensional linear programming approximation of Mather's variational problem. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 1094-1109. doi : 10.1051/cocv/2009039. http://archive.numdam.org/articles/10.1051/cocv/2009039/

[1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York, USA (2000). | Zbl

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

[3] V. Bangert, Minimal measures and minimizing closed normal one-currents. GAFA Geom. Funct. Anal. 9 (1999) 413-427. | Zbl

[4] P. Bernard and B. Buffoni, Optimal mass transportation and Mather theory. J. Eur. Math. Soc. (JEMS) 9 (2007) 85-121.

[5] G. Contreras and R. Iturriaga, Global Minimizers of Autonomous Lagrangians. Coloquio Brasileiro de Matematica. IMPA, Rio de Janeiro, Brazil (1999). | Zbl

[6] L. De Pascale, M.S. Gelli and L. Granieri, Minimal measures, one-dimensional currents and the Monge-Kantorovich problem. Calc. Var. 27 (2006) 1-23. | Zbl

[7] L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, in Current Developments in Mathematics, 1997, S.T. Yau Ed., International Press (1998). | Zbl

[8] L.C. Evans, Some new PDE methods for weak KAM theory. Calc. Var. Partial Differ. Eq. 17 (2003) 159-177. | Zbl

[9] L.C. Evans and D. Gomes, Linear programming interpretation of Mather's variational principle. ESAIM: COCV 8 (2002) 693-702. | Numdam | Zbl

[10] A. Fathi, The Weak KAM Theorem in Lagrangian Dynamics, Cambridge Studies in Advanced Mathematics 88. Cambridge University Press, Cambridge, UK (2008).

[11] A. Fathi and A. Siconolfi, Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation. Invent. Math. 155 (2004) 363-388. | Zbl

[12] D. Gomes and A.M. Oberman, Computing the effective Hamiltonian using a variational approach. SIAM J. Control Optim. 43 (2004) 792-812. | Zbl

[13] L. Granieri, Mass Transportation Problems and Minimal Measures. Ph.D. Thesis in Mathematics, Pisa, Italy (2005).

[14] L. Granieri, On action minimizing measures for the Monge-Kantorovich problem. NoDEA 14 (2007) 125-152. | Zbl

[15] J. Jost, Riemannian Geometry and Geometric Analysis. Springer (2002). | Zbl

[16] J. Jost and X. Li-Jost, Calculus of Variations, Cambridge Studies in Advanced Mathematics 64. Cambridge University Press, Cambridge, UK (1998). | Zbl

[17] R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9 (1996) 273-310. | Zbl

[18] J.N. Mather, Minimal measures. Comment. Math. Helv. 64 (1989) 375-394. | Zbl

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

[20] M. Rorro, An approximation scheme for the effective Hamiltonian and applications. Appl. Numer. Math. 56 (2006) 1238-1254. | Zbl

[21] S.M. Sinha, Mathematical Programming. Elsevier (2006).

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