dc.contributor.author Fanelli, F. dc.contributor.author Zuazua, E. dc.date.accessioned 2016-06-13T13:33:47Z dc.date.available 2016-06-13T13:33:47Z dc.date.issued 2014-12-31 dc.identifier.issn 0294-1449 dc.identifier.uri http://hdl.handle.net/20.500.11824/199 dc.description.abstract In this paper we prove observability estimates for 1-dimensional wave equations with non-Lipschitz coefficients. For coefficients in the Zygmund class we prove a "classical" estimate, which extends the well-known observability results in the energy space for BV regularity. When the coefficients are instead log-Lipschitz or log-Zygmund, we prove observability estimates "with loss of derivatives": in order to estimate the total energy of the solutions, we need measurements on some higher order Sobolev norms at the boundary. This last result represents the intermediate step between the Lipschitz (or Zygmund) case, when observability estimates hold in the energy space, and the H√∂lder one, when they fail at any finite order (as proved in [9]) due to an infinite loss of derivatives. We also establish a sharp relation between the modulus of continuity of the coefficients and the loss of derivatives in the observability estimates. In particular, we will show that under any condition which is weaker than the log-Lipschitz one (not only H√∂lder, for instance), observability estimates fail in general, while in the intermediate instance between the Lipschitz and the log-Lipschitz ones they can hold only admitting a loss of a finite number of derivatives. This classification has an exact counterpart when considering also the second variation of the coefficients. dc.format application/pdf dc.language.iso eng en_US dc.rights Reconocimiento-NoComercial-CompartirIgual 3.0 España en_US dc.rights.uri http://creativecommons.org/licenses/by-nc-sa/3.0/es/ en_US dc.title Weak observability estimates for 1-D wave equations with rough coefficients dc.type info:eu-repo/semantics/article en_US dc.identifier.doi 10.1016/j.anihpc.2013.10.004 dc.relation.publisherversion http://www.sciencedirect.com/science/article/pii/S0294144913001297 dc.rights.accessRights info:eu-repo/semantics/openAccess en_US dc.type.hasVersion info:eu-repo/semantics/acceptedVersion en_US dc.journal.title Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis en_US
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