Hamilton-Jacobi equations with measurable dependence on the state variable

Abstract

We study the Hamilton-Jacobi equation \[H(x,Du)=0 , \] where $H(x,p)$ is assumed to be measurable in $x$, quasiconvex and continuous in $p$. The notion of viscosity solution is adapted to the measurable setting making use of suitable measure--theoretic devices. We obtain integral representation formulae generalizing the ones valid for continuous equations, comparison principles and uniqueness results. We examine stability properties of the new definition and present two approximation procedures: the first one is based on a regularization of the Hamiltonian by mollification and in the second one the approximating sequence is made up by minimizers of certain variational integrals.