We consider Lipschitz continuous viscosity solutions to an evolutive Hamilton-Jacobi equation. The equation arises outside a closed set Γ. Under a condition of strict convexity of the Hamiltonian, we show that there exists a notion of strong trace of the derivatives of the solution on the Lipschitz boundary Γ of codimension d ≥ 2. The very special case d = 1 is done in a separated work. This result is based on a Liouville-type result of classification of global solutions with zero Dirichlet condition on the boundary Γ, where Γ is an affine subspace. We show in particular that such solutions only depend on the normal variable to Γ. As a consequence, we show more generally that the existence of a pointwise tangential gradient along Γ implies the existence of pointwise directional derivatives in all directions. This result also holds true for Hamiltonians depending on the time-space variables, under an additional Dini condition involving certain moduli of continuity. We also give a counterexample for d = 2 in the stationary case, where the Hamiltonian is only continuous in the space variable, and where the solution has no directional derivatives in any directions normal to Γ. Such phenomenon does not hold for d = 1.