On $C^0$-variational solutions for Hamilton-Jacobi equations

Abstract. For evolutive Hamilton-Jacobi equations, we propose a refined def- inition of C0-variational solution, adapted to Cauchy problems for continuous initial data. This weaker framework enables us to investigate the semigroup property for these solutions. In the case of p-convex Hamiltonians, when vari- ational solutions are known to be identical to viscosity solutions, we verify directly the semigroup property by using minmax techniques. In the non- convex case, we construct a first explicit evolutive example where minmax and viscosity solutions are different. Provided the initial data allow for the sepa- ration of variables, we also detect the semigroup property for convex-concave Hamiltonians. In this case, and for general initial data, we finally give new upper and lower Hopf-type estimates for the variational solutions