Concentration and lack of observability of waves in highly heterogeneous media

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 eq...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2002-08, Vol.164 (1), p.39-72
Main Authors: CASTRO, C, ZUAZUA, E
Format: Article
Language:English
Subjects:
Exact sciences and technology
Fundamental areas of phenomenology (including applications)
Physics
Solid mechanics
Structural and continuum mechanics
Studies
Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...)
Vibrations and mechanical waves
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Summary: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.[PUBLICATION ABSTRACT]
ISSN:0003-9527
1432-0673
DOI:10.1007/s002050200202