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A finite dimensional linear programming approximation of Mather's variational problem

Published online by Cambridge University Press:  09 October 2009

Luca Granieri
Luca Granieri*
Affiliation:
Dipartimento di Matematica Politecnico di Bari, via Orabona 4, 70125 Bari, Italy. l.granieri@poliba.it, granieriluca@libero.it
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Abstract

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.

Keywords

Mather problemminimal measureslinear programmingΓ-convergence
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 4 , October 2010 , pp. 1094 - 1109
Copyright
© EDP Sciences, SMAI, 2009

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References

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York, USA (2000).
E.J. Anderson and P. Nash, Linear Programming in Infinite Dimensional Spaces. Wiley (1987).
Bangert, V., Minimal measures and minimizing closed normal one-currents. GAFA Geom. Funct. Anal. 9 (1999) 413–427. CrossRef
Bernard, P. and Buffoni, B., Optimal mass transportation and Mather theory. J. Eur. Math. Soc. (JEMS) 9 (2007) 85–121. CrossRef
G. Contreras and R. Iturriaga, Global Minimizers of Autonomous Lagrangians. Coloquio Brasileiro de Matematica. IMPA, Rio de Janeiro, Brazil (1999).
De Pascale, L., Gelli, M.S. and Granieri, L., Minimal measures, one-dimensional currents and the Monge-Kantorovich problem. Calc. Var. 27 (2006) 1–23. CrossRef
L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, in Current Developments in Mathematics, 1997, S.T. Yau Ed., International Press (1998).
Evans, L.C., Some new PDE methods for weak KAM theory. Calc. Var. Partial Differ. Eq. 17 (2003) 159–177. CrossRef
Evans, L.C. and Gomes, D., Linear programming interpretation of Mather's variational principle. ESAIM: COCV 8 (2002) 693–702. CrossRef
A. Fathi, The Weak KAM Theorem in Lagrangian Dynamics, Cambridge Studies in Advanced Mathematics 88. Cambridge University Press, Cambridge, UK (2008).
Fathi, A. and Siconolfi, A., Existence of $ C\sp 1$ critical subsolutions of the Hamilton-Jacobi equation. Invent. Math. 155 (2004) 363–388. CrossRef
Gomes, D. and Oberman, A.M., Computing the effective Hamiltonian using a variational approach. SIAM J. Control Optim. 43 (2004) 792–812. CrossRef
L. Granieri, Mass Transportation Problems and Minimal Measures. Ph.D. Thesis in Mathematics, Pisa, Italy (2005).
Granieri, L., On action minimizing measures for the Monge-Kantorovich problem. NoDEA 14 (2007) 125–152. CrossRef
J. Jost, Riemannian Geometry and Geometric Analysis. Springer (2002).
J. Jost and X. Li-Jost, Calculus of Variations, Cambridge Studies in Advanced Mathematics 64. Cambridge University Press, Cambridge, UK (1998).
Mañé, R., Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity 9 (1996) 273–310. CrossRef
Mather, J.N., Minimal measures. Comment. Math. Helv. 64 (1989) 375–394. CrossRef
Mather, J.N., Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207 (1991) 169–207. CrossRef
Rorro, M., An approximation scheme for the effective Hamiltonian and applications. Appl. Numer. Math. 56 (2006) 1238–1254. CrossRef
S.M. Sinha, Mathematical Programming. Elsevier (2006).