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Determining a boundary coefficient in a dissipative wave equation: uniqueness and directional Lipschitz stability

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Abstract

We are concerned with the problem of determining the damping boundary coefficient appearing in a dissipative wave equation from a single boundary measurement. We prove that uniqueness holds at the origin provided that the initial condition is appropriately chosen. We show that the choice of the initial condition leading to uniqueness is related to a fine version of the unique continuation property for elliptic operators. We also establish a Lipschitz directional stability estimate at the origin, which is obtained by a linearization process.

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Acknowledgements

We would like to thank Luc Robbiano who indicated to us the principle of the proof of Lemma 2.1.

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Authors and Affiliations

  1. UR Analysis and Control of PDEs, UR 13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019, Monastir, Tunisia

    Kaïs Ammari

  2. IECL, UMR CNRS 7502, Université de Lorraine, Boulevard des Aiguillettes, BP 70239, 54506, Vandoeuvre Les Nancy Cedex, France

    Mourad Choulli

  3. Ile du Saulcy, 57 045, Metz Cedex 01, France

    Mourad Choulli

Authors
  1. Kaïs Ammari
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  2. Mourad Choulli
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Correspondence to Kaïs Ammari.

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Communicated by Abdelaziz Rhandi.

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Ammari, K., Choulli, M. Determining a boundary coefficient in a dissipative wave equation: uniqueness and directional Lipschitz stability. Semigroup Forum 95, 527–538 (2017). https://doi.org/10.1007/s00233-016-9845-2

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  • DOI: https://doi.org/10.1007/s00233-016-9845-2

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