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

We study the Dirichlet problem for stationary Hamilton-Jacobi equations {H(x,u(x),∇u(x))=0u(x)=φ(x) in Ω on ∂Ω. 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 φ, completing in such a way a well posedness theory.