×

zbMATH — the first resource for mathematics

Some properties of viscosity solutions of Hamilton-Jacobi equations. (English) Zbl 0543.35011
This article is concerned with viscosity solutions for the Hamilton- Jacobi equations \(F(x,u(x),Du(x))=0\) for \(x\in \Omega\) where \(\Omega\) is an open subset of \(R^ n,\), F is a continuous, real valued function on \(\Omega \times R\times R^ n\) and Du is the gradient of u. The concept of viscosity solution has been introduced in previous works of M. G. Crandall and P. L. Lions along with existence and uniqueness results. Here two equivalent notions of viscosity solutions are studied and used to give simple proofs for some earlier important results on existence, uniqueness and piecewise regularity of viscosity solutions. Finally, the relationship between viscosity solutions and nonlinear semigroup theory is studied.
Reviewer: V.Barbu

MSC:
35F20 Nonlinear first-order PDEs
47H20 Semigroups of nonlinear operators
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A15 Variational methods applied to PDEs
35B65 Smoothness and regularity of solutions to PDEs
PDF BibTeX Cite
Full Text: DOI
References:
[1] Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. · Zbl 0328.47035
[2] Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1 – 42. · Zbl 0599.35024
[3] Michael G. Crandall and Pierre-Louis Lions, Condition d’unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 3, 183 – 186 (French, with English summary). · Zbl 0469.49023
[4] Ennio De Giorgi, Antonio Marino, and Mario Tosques, Problems of evolution in metric spaces and maximal decreasing curve, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 68 (1980), no. 3, 180 – 187 (Italian, with English summary). · Zbl 0465.47041
[5] Lawrence C. Evans, On solving certain nonlinear partial differential equations by accretive operator methods, Israel J. Math. 36 (1980), no. 3-4, 225 – 247. · Zbl 0454.35038
[6] -, Some max-min methods for the Hamilton-Jacobi equation, Indiana Univ. Math. J. (to appear). · Zbl 0543.35012
[7] Avner Friedman, The Cauchy problem for first order partial differential equations, Indiana Univ. Math. J. 23 (1974), 27 – 40. · Zbl 0243.35014
[8] Pierre-Louis Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. · Zbl 0497.35001
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.