Stability of the determination of a coefficient for  wave equations in an infinite waveguide

Yavar Kian

doi:10.3934/ipi.2014.8.713

Article Contents

2014, Volume 8, Issue 3: 713-732. Doi: 10.3934/ipi.2014.8.713

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Stability of the determination of a coefficient for wave equations in an infinite waveguide

Yavar Kian^1,

1.
CPT, UMR CNRS 7332, Aix Marseille Université, Campus de Luminy, Case 907, 13288 Marseille, cedex 9

Received: August 31, 2013

Revised: May 31, 2014

Published: July 2014

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Abstract

We consider the stability of the inverse problem consisting of the determination of a coefficient of order zero $q$, appearing in the Dirichlet initial-boundary value problem for a wave equation $\partial_t^2u-\Delta u+q(x)u=0$ in $(0,T)\times\Omega$, with $\Omega=\omega\times\mathbb{R}$ an unbounded cylindrical waveguide and $\omega$ a bounded smooth domain of $\mathbb{R}^2$, from boundary observations. The observation is given by the Dirichlet to Neumann map associated to the wave equation. Using suitable geometric optics solutions, we prove a Hölder stability estimate in the determination of $q$ from the Dirichlet to Neumann map. Moreover, provided that the coefficient $q$ is lying in a set of functions $\mathcal A$, where, for any $q_1,q_2\in\mathcal A$, $|q_1-q_2|$ attains its maximum in a fixed bounded subset of $\overline{\Omega}$, we extend this result to the same inverse problem with measurements on a bounded subset of the lateral boundary $(0,T)\times\partial\Omega$.

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Mathematics Subject Classification: 35R30.

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