Nonsmooth multi-time Hamilton-Jacobi systems

We establish existence of a solution for systems of Hamilton-Jacobi equations of the form (1.1). A previous result-see [3]-valid for C-1 Hamiltonians is here extended to the case where Hamiltonians are locally Lipschitz continuous. The main tool for dealing with this kind of non-smoothness consists in the interpretation of the existence issue in terms of commutativity of the minimum problems originating the Hamiltonians involved in (1.1). In turn, a sufficient condition for such commutativity is based on a notion of Lie bracket for nonsmooth vector-fields introduced in [20]. Besides existence, we establish uniqueness-actually, a comparison result-, regularity and four different representations of the solution. Moreover, we prove a front-propagation property in the vector-valued time (t(1), . . , t(N)). The paper also contains results concerning semigroup properties of the solution and the additivity of a suitable defined exponential map.