On the lack of observability for wave equations: a Gaussian beam approach

Abstract

This paper is devoted to study the property of bservability for wave equations guaranteeing that the total energy of solutions may be estimated by means of the energy concentrated on a subset of the domain or of the boundary. We prove that this property fails in three different situations. First, we consider the wave equation with piecewise smooth coefficients when the observation is made in the exterior boundary. We also present a wave equation with highly oscillating Hölder continuous coefficients for which observability fails from any open set that does not contain the origin. Finally, lack of observability is proved for the constant coefficient wave equation when the observation is made from an interior hypersurface. All the counterexamples presented here are constructed using highly localized solutions known as Gaussian beams.