Granieri, Luca On action minimizing measures for the Monge-Kantorovich problem. (English) Zbl 1133.37027 NoDEA, Nonlinear Differ. Equ. Appl. 14, No. 1-2, 125-152 (2007). Summary: In recent years different authors [V. Bangert, Geom. Funct. Anal. 9, No. 3, 413–427 (1999; Zbl 0973.58004), L. de Pascale, M. S. Gelli and L. Granieri, Calc. Var. Partial Differ. Equ. 27, No. 1, 1–23 (2006; Zbl 1096.37033), and L. C. Evans, ibid. 17, No. 2, 159–177 (2003; Zbl 1032.37048)] have noticed and investigated some analogy between Mather’s theory of minimal measures in Lagrangian dynamic and the mass transportation (or Monge-Kantorovich) problem. We replace the closure and homological constraints of Mather’s problem by boundary terms and we investigate the equivalence with the mass transportation problem. An Hamiltonian duality formula for the mass transportation and the equivalence with Brenier’s formulation are also established. Cited in 7 Documents MSC: 37J50 Action-minimizing orbits and measures (MSC2010) 49Q20 Variational problems in a geometric measure-theoretic setting 49Q15 Geometric measure and integration theory, integral and normal currents in optimization Keywords:Mather’s theory of minimal measures; mass transportation problem; Hamiltonian duality formula; optimal transport problems; normal 1-currents Citations:Zbl 0973.58004; Zbl 1096.37033; Zbl 1032.37048 PDFBibTeX XMLCite \textit{L. Granieri}, NoDEA, Nonlinear Differ. Equ. Appl. 14, No. 1--2, 125--152 (2007; Zbl 1133.37027) Full Text: DOI