A well posedness result for generalized solutions of Hamilton-Jacobi equations

Abstract

We study the Dirichlet problem for stationary Hamilton-Jacobi equations $$ \begin{cases} H(x, u(x), \nabla u(x)) = 0 & \ \textrm{in} \ \Omega \\ u(x)=\varphi(x) & \ \textrm{on} \ \partial \Omega. \end{cases} $$ We consider a Caratheodory hamiltonian $H=H(x,u,p)$, with a Sobolev-type (but not continuous) regularity with respect to the space variable $x$, and prove existence and uniqueness of a Lipschitz continuous maximal generalized solution which, in the continuous case, turns out to be the classical viscosity solution. In addition, we prove a continuous dependence property of the solution with respect to the boundary datum $\varphi$, completing in such a way a well posedness theory.