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


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
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