American Institute of Mathematical Sciences

August  2014, 8(3): 713-732. doi: 10.3934/ipi.2014.8.713

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

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

Received  September 2013 Revised  June 2014 Published  August 2014

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 $(0,T)\times\partial\Omega$.
Citation: Yavar Kian. Stability of the determination of a coefficient for wave equations in an infinite waveguide. Inverse Problems & Imaging, 2014, 8 (3) : 713-732. doi: 10.3934/ipi.2014.8.713
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