[Estimées d'observabilité pour l'équation des ondes avec des coefficients continus]
Le but de cette note est de démontrer des estimées d'observabilité pour l'équation des ondes avec une densité continue dans le domaine, et qui satisfait une condition de type multiplicateur seulement au sens des distributions. Notre argument est essentiellement basé sur le fait que l'on peut alors construire des approximations convenables d'une telle fonction de densité par des fonctions régulières pour lesquelles les équations des ondes correspondantes sont uniformément observables. La preuve se termine alors par un passage à la limite relativement standard.
The goal of this note is to prove observability estimates for the wave equation with a density which is only continuous in the domain, and satisfies some multiplier-type condition only in the sense of distributions. Our main argument is that one can construct suitable approximations of such density by a sequence of smooth densities whose corresponding wave equations are uniformly observable. The end of the argument then consists in a rather standard passage to the limit.
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@article{CRMATH_2017__355_5_499_0,
author = {Dehman, Belhassen and Ervedoza, Sylvain},
title = {Observability estimates for the wave equation with rough coefficients},
journal = {Comptes Rendus. Math\'ematique},
pages = {499--514},
publisher = {Elsevier},
volume = {355},
number = {5},
year = {2017},
doi = {10.1016/j.crma.2017.03.011},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2017.03.011/}
}
TY - JOUR AU - Dehman, Belhassen AU - Ervedoza, Sylvain TI - Observability estimates for the wave equation with rough coefficients JO - Comptes Rendus. Mathématique PY - 2017 SP - 499 EP - 514 VL - 355 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2017.03.011/ DO - 10.1016/j.crma.2017.03.011 LA - en ID - CRMATH_2017__355_5_499_0 ER -
%0 Journal Article %A Dehman, Belhassen %A Ervedoza, Sylvain %T Observability estimates for the wave equation with rough coefficients %J Comptes Rendus. Mathématique %D 2017 %P 499-514 %V 355 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2017.03.011/ %R 10.1016/j.crma.2017.03.011 %G en %F CRMATH_2017__355_5_499_0
Dehman, Belhassen; Ervedoza, Sylvain. Observability estimates for the wave equation with rough coefficients. Comptes Rendus. Mathématique, Tome 355 (2017) no. 5, pp. 499-514. doi : 10.1016/j.crma.2017.03.011. http://www.numdam.org/articles/10.1016/j.crma.2017.03.011/
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☆ The research of the first author was supported by the Tunisian Ministry for Higher Education and Scientific Research within the LAB-STI 02 program. This work was elaborated while the first author was visiting the Institut Camille-Jordan of ‘Université Claude-Bernard – Lyon-1’. He wishes to thank all the colleagues for their warm hospitality.
