Weak observability estimates for 1-D wave equations with rough coefficients

Fanelli F, Zuazua E (2015)

Publication Type: Journal article

Publication year: 2015

Journal

Annales de l'Institut Henri Poincaré - Analyse Non Linéaire Elsevier Masson / Institute Henri Poincaré

Book Volume: 32

Pages Range: 245-277

Journal Issue: 2

DOI: 10.1016/j.anihpc.2013.10.004

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 Hölder 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 Hölder, 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.

Authors with CRIS profile

Enrique Zuazua Iriondo

Involved external institutions

Basque Center of Applied Mathematics (BCAM) ES Spain (ES) Ikerbasque (Basque Foundation for Science) ES Spain (ES)

How to cite

APA:

Fanelli, F., & Zuazua, E. (2015). Weak observability estimates for 1-D wave equations with rough coefficients. Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 32(2), 245-277. https://dx.doi.org/10.1016/j.anihpc.2013.10.004

MLA:

Fanelli, Francesco, and Enrique Zuazua. "Weak observability estimates for 1-D wave equations with rough coefficients." Annales de l'Institut Henri Poincaré - Analyse Non Linéaire 32.2 (2015): 245-277.

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