Sobolev regularity for the first order Hamilton–Jacobi equation

Calculus of Variations and Partial Differential Equations

Volumn 54, Issue 3, 2015, Pages 3037-3065

  Cardaliaguet, Pierre ^a   Porretta, Alessio ^b   Tonon, Daniela ^a  

^a UNIVERSITÉ PARIS DAUPHINE   (France)

^b UNIVERSITY OF ROME TOR VERGATA   (Italy)

Author keywords

35B65; 49L25

Indexed keywords

EID: 84944351238     PISSN: 09442669     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00526-015-0893-3     Document Type: Article

References (22)

1. Ambrosio, L., Ascenzi, O.: Hölder continuity of solutions of one-dimensional Lagrange problems of calculus of variations. Ric. Mat. 40(2), 311–319 (1991)
2. Barles, G.: A short proof of the C^{0,\alpha }C0,α-regularity of viscosity subsolutions for superquadratic viscous Hamilton–Jacobi equations and applications. Nonlinear Anal. 73(1), 31–47 (2010)
3. Cannarsa, P., Cardaliaguet, P.: Hölder estimates in space-time for viscosity solutions of Hamilton–Jacobi equations. Commun. Pure Appl. Math. 63(5), 590–629 (2010)
4. Capuzzo Dolcetta, I., Leoni, F., Porretta, A.: Hölder estimates for degenerate elliptic equations with coercive Hamiltonians. Trans. Am. Math. Soc. 362(9), 4511–4536 (2010)
5. Cardaliaguet, P.: A note on the regularity of solutions of Hamilton–Jacobi equations with superlinear growth in the gradient variable. ESAIM Control Optim. Calc. Var. 15(2), 367–376 (2009)
6. Cardaliaguet, P., Rainer, C.: Hölder regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton–Jacobi equations with super-quadratic growth in the gradient. SIAM J. Control Optim. 49, 555–573 (2011)
7. Cardaliaguet, P.: Weak solutions for first order mean field games with local coupling (preprint)
8. Cardaliaguet, P., Graber, J.: Mean field games systems of first order. ESAIM: COCV 21(3), 690–722 (2015)
9. Cardaliaguet, P., Silvestre, L.: Hölder continuity to Hamilton–Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side. Commu. Partial Differ. Equ. 37(9), 1668–1688 (2012)
10. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)
11. Dacorogna, B., Marcellini, P.: Implicit Partial Differential Equations. Birkhauser, Boston (1999)
12. Dall’Aglio, A., Porretta, A.: Local and global regularity of weak solutions of elliptic equations with superquadratic Hamiltonian. Trans. Am. Math. Soc 367(5), 3017–3039 (2015)
13. DiBenedetto, E.: Degenerate Parabolic Equations. Springer, Berlin (1993)
14. Giaquinta, M., Modica, G.: Regularity results for some classes of higher order nonlinear elliptic systems. J. Reine Angew. Math. 311(312), 145–169 (1979)
15. Gehring, F.W.: The L^pLp-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130(1), 265–277 (1973)
16. Huang, M., Caines, P.E., Malhamé, R.P.: Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized \varepsilon ε-Nash equilibria. IEEE Trans. Autom. Control 52(9), 1560–1571 (2007)
17. Kinnunen, J., Lewis, J.L.: Higher integrability for parabolic systems of p-Laplacian type. Duke Math. J. 102(2), 253–272 (2000)
18. Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. I-le cas stationnaire. C. R. Math. 343(9), 619–625 (2006)
19. Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. II-horizon fini et contrôle optimal. C. R. Math. 343(10), 679–684 (2006)
20. Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)
21. Lions, P.-L.: Théorie des jeux de champ moyen et applications (mean field games). Cours du College de France. http://www.college-de-france.fr/default/EN/all/equder/audiovideo.jsp2009 (2007)
22. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions, vol. 2. Princeton University Press, New Jersey (1970)