References
- Hopf E . Generalized solutions of non-linear equations of first order. J Math Mech. 1965;14:951–973.
- Lax PD . Hyperbolic systems of conservation laws II. Commun Pure Appl Math. 1957;10:537–566.
- Crandall MG , Lions PL . Viscosity solutions of Hamilton--Jacobi equations. Trans Amer Math Soc. 1983;277:1–42.
- Bardi M , Evans LC . On Hopf’s formulas for solutions of Hamilton--Jacobi equations. Nonlinear Anal TMA. 1984;8(11):1373–1381.
- Bardi M , Dolcetta IC . Optimal control and viscosity solutions of Hamilton--Jacobi equations. Boston: Birkhäuser; 1997.
- Cannarsa P , Sinestrari C . Semiconcave functions, Hamilton--Jacobi equations and optimal control. Boston: Birkhäuser; 2004.
- Albano P , Cannarsa P . Propagation of singularities for solutions of nonlinear first order partial differential equations. Arch Ration Mech Anal. 2002;162:1–23.
- Barron EN , Cannarsa P , Jensen R , et al . Regularity of Hamilton--Jacbi equations when forward is backward. Indiana Univ Math J. 1999;48:385–409.
- Fleming WH . The Cauchy Problem for a nonlinear first order partial differential equations. J Differ Equ. 1969;5:515–530.
- Cardaliaguet P . Introduction to differential games. Lecture notes. Brest: Université de Bretagne Occidentale; 2010.
- Evans LC . Envelopes and nonconvex Hamilton-Jacobi equations. Calc Vari Partial Differ Equ. 2014;50(1–2):257–282.
- Bardi M , Faggian S . Hopf-type estimates and formulas for nonconvex nonconcave Hamilton--Jacobi equations. SIAM J Math Anal. 1998;29:1067–1086.
- Evans LC . Adjoint and compensated compactness methods for Hamilton--Jacobi PDE. Arch Ration Mech Anal. 2010;197(3):1053–1088.
- Hoang N . Hopf-Lax formula and generalized characteristics. Appl Anal. 2017;96(2):261–277. DOI:10.1080/00036811.2015.1124422
- Hoang N . Regularity of generalized solutions of Hamilton--Jacobi equations. Nonlinear Anal. 2004;59:745–757.
- Van TD , Hoang N , Tsuji M . On Hopf’s formula for Lipschitz solutions of the Cauchy problem for Hamilton--Jacobi equations. Nonlinear Anal. 1997;29(10):1145–1159.
- Lions JP , Rochet J-C . Hopf formula and multitime Hamilton--Jacobi equation. Proc AMS. 1986;96(1):79–84.
- Alvarez O , Barron EN , Ishii H . Hopf-Lax formulas for semicontinuous data. Indiana Univ Math J. 1999;48(3):993–1036.
- Van TD , Tsuji M , Duy NTS . The characteristic method and its generalizations for first order nonlinear PDEs. London: Chapman & Hall/CRC; 2000.
- Azé D , Penot J-P . Uniformly convex and uniformly smooth convex functions. Annales de la Faculté des Sciences de Toulouse, Université Paul Sabatier. 1995;4(4):705–730.
- Rockafellar T . Convex analysis. Princeton (NJ): Princeton University Press; 1970.