Concentration and Lack of Observability¶of Waves in Highly Heterogeneous Media

Abstract

We construct rapidly oscillating Hölder continuous coefficients for which the corresponding 1-dimensional wave equation lacks the classical observability property guaranteeing that the total energy of solutions may be bounded above by the energy localized in an open subset of the domain where the equation holds, if the observation time is large enough. The coefficients we build oscillate arbitrarily fast around two accumulation points. This allows us to build quasi-eigenfunctions for the corresponding eigenvalue problem that concentrate the energy away from the observation region as much as we wish. This example may be extended to several space dimensions by separation of variables and illustrates why the well-known controllability and dispersive properties for wave equations with smooth coefficients fail in the class of Hölder continuous coefficients. In particular we show that for such coefficients no Strichartz-type estimate holds.