Abstract

By looking at the Γ-convergence of some easy functionals in terms of underlying Young measures, we emphasize the connection of this issue with the Monge-Kantorovich mass transfer problem. After exploring this relationship, we study a typical example in periodic homogenization to realize the difference between the Monge-Kantorovich problem and the usual cell problem. We also state a version of the Monge-Kantorovich problem for gradients.

Author information

Contact details are reproduced from the original publication and may be historical.

Pablo Pedregal

ETSI Industriales, Universidad de Castilla - La Mancha, 13071 Ciudad Real, Spain

Keywords

  • Young measures
  • relaxed formulation
  • joint measure
  • marginals

Mathematics Subject Classification

49J45; 46N10

References

  1. A. Braides, A. Defranceschi: Homogenization of multiple integrals, Oxford Lectures Series in Mathematics and its Applications 12, Oxford University Press (1998).https://doi.org/10.1093/oso/9780198502463.001.0001Google Scholar
  2. D. Bucur, G. Buttazzo: Variational methods in some shape optimization problems, Volumes of the Scuola Normale Superiore di Pisa (2002).Google Scholar
  3. B. Dacorogna: Direct Methods in the Calculus of Variations, Springer, Berlin (1989).https://doi.org/10.1007/978-3-642-51440-1Google Scholar
  4. G. Dal Maso: Introduction to Γ-Convergence, Birkhäuser, Boston (1993).https://doi.org/10.1007/978-1-4612-0327-8Google Scholar
  5. E. De Giorgi: Sulla convergenza di alcune successioni di integrali del tipo dell’area, Rend. Mat. 8 (1975) 277–294.Google Scholar
  6. E. De Giorgi, T. Franzoni: Su un tipo di convergenza variazionale, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Mat. 58 (1975) 842–850.Google Scholar
  7. W. Gangbo, J. McCann: The geometry of optimal transportation, Acta Math 177 (1996) 113–161.https://doi.org/10.1007/BF02392620Google Scholar
  8. R. Jordan, D. Kinderlehrer, F. Otto: Dynamics of the Fokker-Planck Equation, Phase Transitions 69 (1999) 271–288.https://doi.org/10.1080/01411599908209295Google Scholar
  9. L. Kantorovich: On the translocation of masses, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 37 (1942) 199–201.Google Scholar
  10. L. Kantorovich: On a problem of Monge, Uspekhi Mat. Nauk 3 (1948) 225–226 (in Russian).Google Scholar
  11. G. Monge: Mémoire sur la théorie des déblais et de remblais, in: Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année (1781) 666–704.Google Scholar
  12. P. Pedregal: Parametrized Measures and Variational Principles, Birkhäuser, Basel (1997).https://doi.org/10.1007/978-3-0348-8886-8Google Scholar
  13. P. Pedregal: Relaxation in magnetostriction, Calc. Var 10 (2000) 1–19.https://doi.org/10.1007/PL00013456Google Scholar
  14. P. Pedregal: Γ-convergence through Young measures, SIAM J. Math. Anal. 36(2) (2004) 423–440.https://doi.org/10.1137/S0036141003425696Google Scholar

Suggested citation

P. Pedregal. “Remarks on Γ-Convergence and the Monge-Kantorovich Mass Transfer Problem.” Journal of Convex Analysis 12 (2005), No. 2, 407–415. https://doi.org/10.68381/jca12028

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Published by Heldermann Verlag, 2005. Rights now held by Banach Press.