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Concentration and lack of observability of waves in highly heterogeneous media. (English) Zbl 1016.35003
Arch. Ration. Mech. Anal. 164, No. 1, 39-72 (2002); addendum ibid. 185, No. 3, 365-377 (2007).
The authors construct oscillating Hölder continuous coefficients for which the corresponding one-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 they build oscillate arbitrarily fast around two accumulation points. This allows them to build quasi-eigenfunctions for the corresponding eigenvalue problem that concentrate the energy away from the observation region as much as they 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, the authors show that for such coefficients no Strichartz-type estimate holds.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
93B07 Observability
35L05 Wave equation
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