We consider Lipschitz continuous solutions to evolutive Hamilton-Jacobi equations. Under a condition of strict convexity of the Hamiltonian, we show that there exists a notion of strong trace of the gradient of the solution. This result is based on a Liouville-type result of classification of global solutions on the half space. Under zero Dirichlet boundary condition, we show that the solution only depends on the normal variable. As a consequence, we show that the existence of a pointwise tangential gradient implies existence of a pointwise normal gradient. For the Liouville-type result, and when the Hamiltonian is not convex, we give a counterexample with a solution which is not one-dimensional. We give two applications. On the one hand, for the classical stationary Dirichlet problem on a bounded domain, we show the existence of a closed subset of the boundary of the domain, where Taylor expansion of the solution is uniform. On the other hand, for Hamilton-Jacobi equations on a network, we show that the space derivative of the solution has a trace at each node, which satisfies a natural germ condition.