zbMATH — the first resource for mathematics

On existence and uniqueness of solutions of Hamilton-Jacobi equations. (English) Zbl 0603.35016
Aus dem Vorwort der Verff: Our main purpose is to establish some uniqueness and existence theorems for Hamilton-Jacobi equations. We will focus on the Cauchy problem \[ u_ t+H(x,t,u,Du)=0\quad in\quad {\mathbb{R}}^ N\times (0,T),\quad u(x,0)=\phi (x), \] and the stationary problem \(u+H(x,u,Du)=0\) in \({\mathbb{R}}^ N\), in which H is a real-valued function of its arguments, x denotes points on \({\mathbb{R}}^ N\), Du stands for the spatial gradient \((u_{x1},...,u_{xN})\) of u, which is itself a real-valued function of either (x,t) or x as appropriate.
Reviewer: J.Wloka

35G25 Initial value problems for nonlinear higher-order PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B50 Maximum principles in context of PDEs
Full Text: DOI
[1] Barles, G., Contrôle impulsionnel déterministe, inequations quasivariationelles et équations de Hamilton-Jacobi du premier ordre, ()
[2] Crandall, M.G.; Evans, L.C.; Lions, P.L., Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. am. math. soc., 282, 487-502, (1984) · Zbl 0543.35011
[3] Crandall, M.G.; Lions, P.L., Viscosity solutions of Hamilton-Jacobi equations, Trans. am. math. soc., 277, 1-42, (1983) · Zbl 0599.35024
[4] Crandall, M.G.; Lions, P.L., Solutions de viscosité non bornées des équations de Hamilton-Jacobi du premier ordre, C. r. hebd. séanc. acad. sci. Paris, 298, 217-220, (1984) · Zbl 0565.49022
[5] Crandall M. G. & Lions P. L., submitted for publication.
[6] Crandall, M.G.; Newcomb, R., Viscosity solutions of Hamilton-Jacobi equations at the boundary, Proc. am. math. soc., 94, 283-290, (1985) · Zbl 0575.35008
[7] Crandall, M.G.; Souganidis, P.E., (), North-Holland Mathematics Studies 92
[8] Ishii, H., Remarks on the existence of viscosity solutions of Hamilton-Jacobi equations, (), 5-24 · Zbl 0546.35042
[9] Ishii, H., Uniqueness of unbounded solutions of Hamilton-Jacobi equations, Indiana univ. math. J., 33, 721-748, (1984) · Zbl 0551.49016
[10] Lions, P.L., Generalized solutions of Hamilton-Jacobi equations, Research notes in mathematics, 69, (1982), Pitman Boston · Zbl 1194.35459
[11] Lions, P.L., Existence results for first-order Hamilton-Jacobi equations, Ric. mat. napoli, 32, 1-23, (1983) · Zbl 0552.70012
[12] S{\scouganidis} P. E., Existence of viscosity solutions of Hamilton-Jacobi equations, J. diff. Eqns (to appear).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.