Boundary sidewise observability of the wave equation

Abstract

The wave equation on a bounded domain of ${\mathbb{R}}^n$ with nonhomogeneous boundary Dirichlet data or sources supported on a subset of the boundary is considered. We analyze the problem of observing the source out of boundary measurements done away from its support. We first notice that due to the existence of solutions that are arbitrarily concentrated near the source, for any given integer $N$, these observability inequalities may not hold even if we allow a loss of $N$ derivatives. We then establish observability inequalities in Sobolev norms, under a suitable microlocal geometric condition on the support of the source and the measurement set, for sources fulfilling pseudo-differential conditions that exclude these concentration phenomena. The proof relies on microlocal arguments and is essentially based on the use of microlocal defect measures.