American Institute of Mathematical Sciences

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September  2007, 8(2): 389-416. doi: 10.3934/dcdsb.2007.8.389

Detecting perfectly insulated obstacles by shape optimization techniques of order two

Lekbir Afraites 1, , Marc Dambrine 1, , Karsten Eppler 2, and  Djalil Kateb 1,

1. 

Laboratoire de Mathématiques Appliquées de Compiègne, Université de Technologie de Compiègne, 60205 Compiègne Cedex, France, France, France
2. 

Fakultät Mathematik und Naturwissenschaften, Fachrichtung Mathematik, Technische Universität Dresden, 01069 Dresden, Germany

Received  September 2006 Revised  March 2007 Published  June 2007

The paper extends investigations of identification problems by shape optimization methods for perfectly conducting inclusions to the case of perfectly insulating material. The Kohn and Vogelius criteria as well as a tracking type objective are considered for a variational formulation. In case of problems in dimension two, the necessary condition implies immediately a perfectly matching situation for both formulations. Similar to the perfectly conducting case, the compactness of the shape Hessian is shown and the ill-posedness of the identification problem follows. That is, the second order quadratic form is no longer coercive. We illustrate the general results by some explicit examples and we present some numerical results.
Keywords: second order methods, shape calculus, necessary condition of stability., Geometric inverse problem.
Mathematics Subject Classification: Primary: 49Q10, 34A55; Secondary: 49Q1.
Citation: Lekbir Afraites, Marc Dambrine, Karsten Eppler, Djalil Kateb. Detecting perfectly insulated obstacles by shape optimization techniques of order two. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 389-416. doi: 10.3934/dcdsb.2007.8.389
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