Comptes Rendus
no. 7
Partial differential equations
The Plateau problem from the perspective of optimal transport
[Le problème de Plateau vu dans la perspective du transport optimal]

Présenté par : Haïm Brézis

Haim Brezis1, 2, 3 ; Petru Mironescu4
1 Department of Mathematics, Rutgers University, USA
2 Departments of Mathematics and Computer Science, Technion – I.I.T., 32 000 Haifa, Israel
3 Sorbonne Universités, UPMC Université Paris-6, UMR 7598, Laboratoire Jacques-Louis-Lions, 75005, Paris, France
4 Univ Lyon, Université Claude-Bernard – Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 69622 Villeurbanne, France
Comptes Rendus. Mathématique, Volume 357 (2019) no. 7, pp. 597-612.

Le transport optimal, ainsi que les surfaces minimales, ont été abondamment étudiés au cours de ces dernières décennies. Nous mettons en évidence une analogie surprenante, au niveau méthodologique, entre l'approche de Kantorovich pour le problème de Monge et la minimisation de l'aire dans des problèmes géométriques de type Plateau étudiés par Federer.

Both optimal transport and minimal surfaces have received much attention in recent years. We show that the methodology introduced by Kantorovich on the Monge problem can, surprisingly, be adapted to questions involving least area, e.g., Plateau-type problems as investigated by Federer.

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Accepté le :
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DOI : 10.1016/j.crma.2019.07.007
Haim Brezis 1, 2, 3 ; Petru Mironescu 4

1 Department of Mathematics, Rutgers University, USA
2 Departments of Mathematics and Computer Science, Technion – I.I.T., 32 000 Haifa, Israel
3 Sorbonne Universités, UPMC Université Paris-6, UMR 7598, Laboratoire Jacques-Louis-Lions, 75005, Paris, France
4 Univ Lyon, Université Claude-Bernard – Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 69622 Villeurbanne, France 
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Haim Brezis; Petru Mironescu. The Plateau problem from the perspective of optimal transport. Comptes Rendus. Mathématique, Volume 357 (2019) no. 7, pp. 597-612. doi : 10.1016/j.crma.2019.07.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.07.007/

[1] G. Alberti; S. Baldo; G. Orlandi Functions with prescribed singularities, J. Eur. Math. Soc., Volume 5 (2003) no. 3, pp. 275-311

[2] F. Almgren; W. Browder; E.H. Lieb Co-area, liquid crystals, and minimal surfaces, Tianjin, 1986 (Lecture Notes in Mathematics), Volume vol. 1306, Springer, Berlin (1988), pp. 1-22

[3] L. Ambrosio; N. Fusco; D. Pallara Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000

[4] M. Beckmann A continuous model of transportation, Econometrica, Volume 20 (1952), pp. 643-660

[5] G. Birkhoff Tres observaciones sobre el algebra lineal, Rev. Univ. Nac. Tucumán Ser. A, Mat. Fis. Teor., Volume 5 (1946), pp. 147-150

[6] H. Brezis Liquid crystals and energy estimates for S2-valued maps, Minneapolis, MN, USA, 1985 (J. Ericksen; D. Kinderlehrer, eds.), Springer (1987), pp. 31-52 http://sites.math.rutgers.edu/~brezis/publications.html (See also article 112L1 in)

[7] H. Brezis Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011

[8] H. Brezis Remarks on the Monge–Kantorovich problem in the discrete setting, C. R. Acad. Sci. Paris, Ser. I, Volume 356 (2018) no. 2, pp. 207-213

[9] H. Brezis; J.-M. Coron; E.H. Lieb Harmonic maps with defects, Commun. Math. Phys., Volume 107 (1986) no. 4, pp. 649-705

[10] H. Brezis; P. Mironescu Sobolev Maps with Values Into the Circle, Birkhäuser, 2019 (to appear)

[11] H. Brezis; P. Mironescu; A. Ponce W1,1-maps with values into S1, Geometric Analysis of PDE and Several Complex Variables, Contemporary Mathematics, vol. 368, Amer. Math. Soc., Providence, RI, USA, 2005, pp. 69-100

[12] L.C. Evans Partial differential equations and Monge–Kantorovich mass transfer, Cambridge, MA, USA, 1997, Int. Press, Boston, MA, USA (1999), pp. 65-126

[13] L.C. Evans; R.F. Gariepy Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, USA, 1992

[14] H. Federer Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 153, Springer-Verlag New York Inc., New York, 1969

[15] H. Federer Real flat chains, cochains and variational problems, Indiana Univ. Math. J., Volume 24 (1974–1975), pp. 351-407

[16] H. Federer; W.H. Fleming Normal and integral currents, Ann. of Math. (2), Volume 72 (1960), pp. 458-520

[17] F. Frankl; L. Pontrjagin Ein Knotensatz mit Anwendung auf die Dimensionstheorie, Math. Ann., Volume 102 (1930) no. 1, pp. 785-789

[18] M. Giaquinta; G. Modica; J. Souček 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, Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics, vol. 37, Springer-Verlag, Berlin, 1998

[19] M. Giaquinta; G. Modica; J. Souček Cartesian Currents in the Calculus of Variations. II. Variational Integrals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics, vol. 38, Springer-Verlag, Berlin, 1998

[20] E. Giusti Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, Switzerland, 1984

[21] R.M. Hardt; J.T. Pitts Solving Plateau's problem for hypersurfaces without the compactness theorem for integral currents, Arcata, Calif., 1984 (Proceedings of Symposia in Pure Mathematics), Volume vol. 44, Amer. Math. Soc., Providence, RI, USA (1986), pp. 255-259

[22] R.L. Jerrard; H.M. Soner Functions of bounded higher variation, Indiana Univ. Math. J., Volume 51 (2002) no. 3, pp. 645-677

[23] L.V. Kantorovitch On the translocation of masses, Dokl. Akad. Nauk SSSR, Volume 37 (1942), pp. 227-229 (English translation: J. Math. Sci., 133, 2006, pp. 1381-1382)

[24] R.C. Kirby The Topology of 4-Manifolds, Lecture Notes in Mathematics, vol. 1374, Springer-Verlag, Berlin, 1989

[25] M. Krein; D. Milman On extreme points of regular convex sets, Stud. Math., Volume 9 (1940), pp. 133-138

[26] F.H. Lin, Personal communication, July 2019.

[27] F. Morgan Area-minimizing currents bounded by higher multiples of curves, Rend. Circ. Mat. Palermo, Volume 33 (1984) no. 1, pp. 37-46

[28] A. Poliakovsky On a minimization problem related to lifting of BV functions with values in S1, C. R. Acad. Sci., Sér. 1, Volume 339 (2004) no. 12, pp. 855-860

[29] F. Santambrogio Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and Their Applications, vol. 87, Birkhäuser/Springer, Cham, 2015

[30] H. Seifert Über das Geschlecht von Knoten, Math. Ann., Volume 110 (1935) no. 1, pp. 571-592

[31] C. Villani Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, USA, 2003

[32] C. Villani Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, vol. 338, Springer-Verlag, Berlin, 2009

[33] J. von Neumann (Contributions to the Theory of Games), Volume vol. 2, Princeton University Press (1953), pp. 5-12

[34] B. White The least area bounded by multiples of a curve, Proc. Amer. Math. Soc., Volume 90 (1984) no. 2, pp. 230-232

[35] L.C. Young Some extremal questions for simplicial complexes. V. The relative area of a Klein bottle, Rend. Circ. Mat. Palermo, Volume 12 (1963), pp. 257-274

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