Behavior near the Boundary for Solutions of the Wave Equation

We present two regularity results concerning the solutions of the wave equation with homogeneous Dirichlet boundary conditions. The first one gives the precise behavior of the L2 norms of the solutions of finite energy in a neighborhood of the boundary. The second one is a regularity property on the normal derivative of the limit of a sequence of "very weak" solutions, when a condition on the behavior of the L2 norms of these solutions near the boundary is fulfilled. These results are motivated by the passage from internal control to boundary control in exact controllability, and by the study of limits of wave equations with singular right hand sides.