Regularity via duality in calculus of variations and degenerate elliptic PDEs

Journal of Mathematical Analysis and Applications

Volumn 457, Issue 2, 2018, Pages 1649-1674

  Santambrogio, Filippo ^a  

^a UNIVERSITÉ PARIS SUD   (France)

Author keywords

Convex duality; Fractional Sobolev spaces; p Laplacian; Traffic congestion

Indexed keywords

EID: 85010543374     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2017.01.030     Document Type: Article

References (30)

1. Adams, R.A., Sobolev Spaces. Pure Appl. Math., vol. 65, 1975, Academic Press, New York–London.
2. Ambrosio, L., Figalli, A., On the regularity of the pressure field of Brenier's weak solutions to incompressible Euler equations. Calc. Var. Partial Differential Equations 31:4 (2008), 497–509.
3. Ambrosio, L., Figalli, A., Geodesics in the space of measure-preserving maps and plans. Arch. Ration. Mech. Anal. 194 (2009), 421–462.
4. Baillon, J.-B., Carlier, G., From discrete to continuous Wardrop equilibria. Net. Heterogenous Media 7 (2012), 219–241.
5. Beckmann, M., A continuous model of transportation. Econometrica 20 (1952), 643–660.
6. Benamou, J.-D., Brenier, Y., A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84 (2000), 375–393.
7. Bojarski, B., Iwaniec, T., p-harmonic equation and quasiregular mappings. Partial Differential Equations, Warsaw, 1984, 1987, Banach Center Publications, 19.
8. Bouchitté, G., Buttazzo, G., Characterization of optimal shapes and masses through Monge–Kantorovich equation. J. Eur. Math. Soc. 3:2 (2001), 139–168.
9. 1 regularity of orthotropic p-harmonic functions in the plane. preprint, available at http://cvgmt.sns.it/paper/3113/, 2016.
10. Brasco, L., Carlier, G., On certain anisotropic elliptic equations arising in congested optimal transport: local gradient bounds. Adv. Calc. Var. 7 (2014), 379–407.
11. Brasco, L., Carlier, G., Santambrogio, F., Congested traffic dynamics, weak flows and very degenerate elliptic equations. J. Math. Pures Appl. 93:6 (2010), 652–671.
12. L. Brasco, C. Leone, G. Pisante, A. Verde, Sobolev and Lipschitz regularity for local minimizers of widely degenerate anisotropic functionals, Nonlinear Anal., available online, in press.
13. Brasco, L., Santambrogio, F., A sharp estimate à la Calderón–Zygmund for the p-Laplacian. Commun. Contemp. Math., 2017 in press.
14. Brenier, Y., Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Comm. Pure Appl. Math. 52:4 (1999), 411–452.
15. Brezis, H., Functional Analysis. Theory and Applications. 1983, Masson, Paris (in French).
16. Burger, M., Bregman distances in inverse problems and partial differential equation. Advances in Mathematical Modeling, Optimization and Optimal Control Springer Optimization and Its Applications, vol. 109, 2016, 3–33.
17. Cardaliaguet, P., Mészáros, A.R., Santambrogio, F., First order mean field games with density constraints: pressure equals price. SIAM J. Control Optim. 54:5 (2016), 2672–2709.
18. Carlier, G., Jimenez, C., Santambrogio, F., Optimal transportation with traffic congestion and Wardrop equilibria. SIAM J. Control Optim. 47 (2008), 1330–1350.
19. Ekeland, I., Temam, R., Convex Analysis and Variational Problems. Classics in Mathematics, 1999, Society for Industrial and Applied Mathematics.
20. Hatchi, R., Wardrop equilibria: rigorous derivation of continuous limits from general networks models. preprint arXiv:1506.08960.
21. Ladyzhenskaya, O.A., Ural'tseva, N.N., Linear and Quasilinear Elliptic Equations. 1968, Academic Press, New York–London.
22. Lasry, J.-M., Lions, P.-L., Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343:10 (2006), 679–684.
23. Lasry, J.-M., Lions, P.-L., Mean-field games. Jpn. J. Math. 2 (2007), 229–260.
24. Lindqvist, P., Notes on the p-Laplace equation. available online at the page http://www.math.ntnu.no/~lqvist/p-laplace.pdf.
25. Mingione, G., The singular set of solutions of non-differentiable elliptic systems. Arch. Ration. Mech. Anal. 166 (2003), 287–301.
26. A. Prosinski, F. Santambrogio, Global-in-time regularity via duality for congestion-penalized mean field games, preprint, available at cvgmt.sns.it.
27. Santambrogio, F., Optimal Transport for Applied Mathematicians. Progress in Nonlinear Differential Equations and Their Applications, vol. 87, 2015, Birkhäuser, Basel.
28. n. Benilan, P., Robert, J., (eds.) Journées d'Analyse Non Linéaire Lecture Notes in Mathematics, vol. 665, 1978, Springer, 205–227.
29. Uhlenbeck, K., Regularity for a class of non-linear elliptic systems. Acta Math. 138 (1977), 219–240.
30. Villani, C., Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften, vol. 338, 2009, Springer-Verlag, Berlin.