On the existence of dual solutions for Lorentzian cost functions

Kell, Martin; Suhr, Stefan

doi:10.1016/j.anihpc.2019.09.005

Kell, Martin ¹ ; Suhr, Stefan ²

¹ Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
² Ruhr-Universität Bochum, Fakultät für Mathematik, Gebäude NA 4/33, D-44801 Bochum, Germany

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 2, pp. 343-372.

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Résumé

The dual problem of optimal transportation in Lorentz-Finsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal transport is timelike on a set of full measure. In the second part the persistence of absolute continuity along an optimal transportation under obvious assumptions is proven and a solution to the relativistic Monge problem is provided.

MR Zbl

DOI : 10.1016/j.anihpc.2019.09.005

Mots-clés : Optimal transport, Lorentz cost, Dual solution, Relativistic Monge problem

Affiliations des auteurs :

Kell, Martin ¹ ; Suhr, Stefan ²

¹ Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
² Ruhr-Universität Bochum, Fakultät für Mathematik, Gebäude NA 4/33, D-44801 Bochum, Germany

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     title = {On the existence of dual solutions for {Lorentzian} cost functions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {343--372},
     publisher = {Elsevier},
     volume = {37},
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     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2019.09.005/}
}

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Kell, Martin; Suhr, Stefan. On the existence of dual solutions for Lorentzian cost functions. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 2, pp. 343-372. doi : 10.1016/j.anihpc.2019.09.005. http://archive.numdam.org/articles/10.1016/j.anihpc.2019.09.005/

Bibliographie
Cité par

[1] Aazami, A.B.; Javaloyes, M.A. Penrose's singularity theorem in a Finsler spacetime, Class. Quantum Gravity, Volume 33 (2016) no. 2 | DOI | MR | Zbl

[2] Bao, D.; Chern, S.-S.; Shen, Z. An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag, New York, 2000 | DOI | MR | Zbl

[3] Bernard, P.; Buffoni, B. Asymptotic Analysis and Singularities, Adv. Stud. Pure Math., Volume 47 (2007) no. 2, pp. 397–420 | DOI | MR | Zbl

[4] Bernard, P.; Buffoni, B. Optimal mass transportation and Mather theory, J. Eur. Math. Soc. (JEMS), Volume 9 (2007) no. 1, pp. 85–121 | DOI | MR | Zbl

[5] Bernard, P.; Suhr, S. Lyapounov functions of closed cone fields: from Conley theory to time functions, Commun. Math. Phys., Volume 359 (2018) no. 2, pp. 467–498 | DOI | MR | Zbl

[6] Bertrand, J.; Puel, M. The optimal transport problem for relativistic costs, Calc. Var. Partial Differ. Equ., Volume 46 (2013) no. 1–2, pp. 353–374 | MR | Zbl

[7] Bertrand, J.; Pratelli, A.; Puel, M. Kantorovich potentials and continuity of total cost for relativistic cost functions, J. Math. Pures Appl. (9), Volume 110 (2018), pp. 93–122 | DOI | MR | Zbl

[8] Brenier, Y. Optimal Transportion and Applications, Extended Monge-Kantorovich theory (Lecture Notes in Math.), Volume vol. 1813, Springer-Verlag, Berlin (2003), pp. 91–121 (Martina Franca, 2001) | MR | Zbl

[9] Bianchini, S.; Cavalletti, F. The Monge problem for distance cost in geodesic spaces, Commun. Math. Phys., Volume 318 (2013), pp. 615–673 | DOI | MR | Zbl

[10] Caravenna, L. A proof of Sudakov theorem with strictly convex norms, Math. Z., Volume 286 (2011) no. 1–2, pp. 371–407 | MR | Zbl

[11] Cavalletti, F.; Huesmann, M. Existence and uniqueness of optimal transport maps, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 32 (2015) no. 6, pp. 1367–1377 | DOI | Numdam | MR | Zbl

[12] Cavalletti, F.; Mondino, A. Optimal maps in essentially non-branching spaces, Commun. Contemp. Math., Volume 19 (2017) no. 6, pp. 1–27 | DOI | MR | Zbl

[13] Champion, Th.; De Pascale, L. The Monge problem for strictly convex norms in $R^{n}$ , J. Eur. Math. Soc., Volume 12 (2010), pp. 1355–1369 | DOI | MR | Zbl

[14] De Pascale, L.; Rigot, S. Monge's transport problem in the Heisenberg group, Adv. Calc. Var., Volume 4 (2011) no. 2, pp. 195–227 | DOI | MR | Zbl

[15] Eckstein, M.; Miller, T. Causal evolution of wave packets, Phys. Rev. A, Volume 95 (2017) | DOI

[16] Fremlin, D.H. Measure Theory, vol. 4, Torres Fremlin, Colchester, 2006 | MR | Zbl

[17] Gigli, N. Optimal maps in non-branching spaces with Ricci curvature bounded from below, Geom. Funct. Anal., Volume 22 (2012) no. 4, pp. 990–999 | DOI | MR | Zbl

[18] Kell, M. Transport maps, non-branching sets of geodesics and measure rigidity, Adv. Math., Volume 320 (2017), pp. 520–573 | DOI | MR | Zbl

[19] Louet, J.; Pratelli, A.; Zeisler, F. On the continuity of the total cost in the mass transport problem with relativistic cost functions | arXiv

[20] McCann, R. A convexity principle for interacting gases, Adv. Math., Volume 128 (1997), pp. 153–179 | DOI | MR | Zbl

[21] McCann, R. Displacement concavity of Boltzmann's entropy characterizes positive energy in general relativity | arXiv

[22] McCann, R.; Puel, M. Constructing a relativistic heat flow by transport time steps, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 26 (2009) no. 6, pp. 2539–2580 | Numdam | MR | Zbl

[23] Miller, T. Polish spaces of causal curves, J. Geom. Phys., Volume 116 (2017), pp. 295–315 | DOI | MR | Zbl

[24] Minguzzi, E. An equivalence of Finslerian relativistic theories, Rep. Math. Phys., Volume 77 (2016) no. 1, pp. 45–55 | DOI | MR | Zbl

[25] Sturm, K.-Th. On the geometry of metric measure spaces. I, Acta Math., Volume 196 (2006) no. 1, pp. 65–131 | MR | Zbl

[26] Sudakov, V.N., Number in Russian Series Statements, Volume vol. 141 (1979), pp. 1–178 | MR | Zbl

[27] Suhr, S. Optimal transportation for Lorentzian cost functions, Münster J. Math., Volume 11 (2018), pp. 13–47 | MR | Zbl

[28] Villani, C. Optimal Transport, Old and New, Springer-Verlag, Berlin, Heidelberg, 2009 | DOI | MR | Zbl

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