Minimal measures, one-dimensional currents and the Monge-Kantorovich problem

Calculus of Variations and Partial Differential Equations

Volumn 27, Issue 1, 2006, Pages 1-23

  De Pascale, Luigi ^a   Gelli, Maria Stella ^a   Granieri, Luca ^b  

^a UNIVERSITY OF PISA   (Italy)

^b POLITECNICO DI BARI   (Italy)

Author keywords

Mather's minimal measures; Monge Kantorovich problem; Normal 1 currents; Optimal transport problems

Indexed keywords

EID: 33744950638     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00526-006-0017-1     Document Type: Article

References (37)

1. Ambrosio, L.: Lecture notes on transport problems. In: Mathematical Aspects of Evolving Interlaces. Lecture Notes in Mathematics, vol. 1812, pp. 1-52. Springer, Berlin (2003)
2. Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158(2), 227-260 (2004)
3. Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows in metric spaces and in the space of probability measures. Lectures in mathematics ETH Zürich, Birkhäuser, Basel (2005)
4. Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces, Oxford Lecture Series in Mathematics and Its Applications, vol. 25. Oxford University Press, Oxford (2004)
5. Arnold, V.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, Berlin & New York (1982)
6. Bangert, V.: Minimal measures and minimizing closed normal one-currents. GAFA Geom. Funct. Anal. 9(3), 413-427 (1999)
7. Bernard, P., Buffoni, B.: Optimal mass transportation and Mather theory, Preprint (December 2004) (to appear on J. Eur. Math. Soc. (JEMS))
8. Bouchitté, G., Buttazzo, G.: Characterization of optimal shapes and masses through Monge-Kantorovich equation. J. Eur. Math. Soc. (JEMS) 3, 139-168 (2001)
9. Bouchitté, G., Buttazzo, G., Seppecher, P.: Shape optimization solutions via Monge-Kantorovich equation. C. R. Acad. Sci. Paris 324-I, 1185-1191 (1997)
10. Bouchitté, G., Buttazzo, G., De Pascale, L.: A p-Laplacian approximation for some mass optimization problems. J. Optim. Theory Appl. 18, 1-25 (2003)
11. Brenier, Y.: Extended Monge-Kantorovich Theory, in Optimal Transportation and Applications, Lecture Notes in Mathematics, vol. 1813, pp. 91-121. Springer, Berlin (2003)
12. Benamou, J., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84(3), 375-393 (2000)
13. Contreras, G., Iturriaga, R.: Global minimizers of autonomous Lagrangians. IMPA, Rio de Janeiro (1999)
14. De Pascale, L., Pratelli, A.: Regularity properties for Monge transport density and for solutions of some shape optimization problems. Calc. Var. Partial Differential Equations 14, 249-274 (2002)
15. Di Perna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511-547 (1989)
16. Evans, L.C.: Some new PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations 17(2), 159-177 (2003)
17. Evans, L.C.: Partial differential equations and Monge-Kantorovich mass transfer (survey paper). In: Yau, S.T. (ed.) Current Developments in Mathematics, 1997. International Press (1999)
18. Evans, L.C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137 (1999)
19. Evans, L.C., Gomes, D.: Linear programming interpretation of Mather's variational principle. ESAIM Control Optim. Calc. Var. 8, 693-702 (2002)
20. Fathi, A.: Thèorème KAM faible et thèorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sèr. I Math. 324(9), 1043-1046 (1997)
21. Fathi, A.: Solutions KAM faibles conjuguèes et barrières de Peierls. C. R. Acad. Sci. Paris Sèr. I Math. 325(6), 649-652 (1997)
22. Fathi, A.: Weak Kam Theorem in Lagrangian Dynamics. Cambridge University Press (forthcoming)
23. 1 critical subsolutions of the Hamilton-Jacobi equation. Invent. Math. 155(2), 363-388 (2004)
24. Federer, H.: Geometric Measure Theory. Springer (Berlin) (1969)
25. Gangbo, W., McCann, R.J.: The geometry of optimal transportation. Acta Math. 177, 113-161 (1996)
26. Giaquinta, M., Modica, G., Souček, J.: Cartesian currents in the calculus of variations. I. Cartesian currents. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 37. Springer-Verlag, Berlin (1998)
27. Gomes, D.: Viscosity solutions of Hamilton-Jacobi equations, and asymptotics for Hamiltonian systems. Calc. Var. Partial Differential Equations 14(3), 345-357 (2002)
28. Granieri, L.: On action minimizing measures for the Monge-Kantorovich problem, Preprint (July 2004) (to appear on NoDEA Nonlinear Differential Equations Appl.)
29. Hohloch, S.: Optimale Massebewegung im Monge-Kantorovich Transportproblem, Diplomarbeit (December 2002)
30. Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker-Plank equation. Siam J. Math. Anal. 29, 1-17 (1998)
31. Lang, U., Schroeder, V.: Kirszbraun's theorem and metric spaces of bounded curvature. GAFA Geom. Funct. Anal. 7, 535-560 (1997)
32. Mather, J.: Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207(2), 169-207 (1991)
33. Mather, J.: Minimal measures. Comment. Math. Helv. 64(3), 375-394 (1989)
34. Otto, F.: The geometry of dissipative evolution equations: the porus medium equation. Comm. Partial Differential Equations 26(1/2), 101-174 (2001)
35. Smirnov, S.K.: Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents. St. Petersburg Math. J. 5(4), 841-867 (1994)
36. Sturm, K.T.: Stochastics and Analysis on Metric Spaces (lecture notes in preparation)
37. Villani C, Topics in Mass Transportation, Graduate Studies in Mathematics 58, AMS, Providence, RI