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Unique continuation property near a corner and its fluid-structure controllability consequences

Published online by Cambridge University Press:  28 March 2008

Axel Osses
Affiliation:
Departamento de Ingenería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), FCFM Universidad de Chile, Casilla 170/3 - Correo 3, Santiago, Chile; axosses@dim.uchile.cl
Jean-Pierre Puel
Affiliation:
Laboratoire de Mathématiques de Versailles, UMR 8100, Université de Versailles St-Quentin, 45 avenue des États-Unis, 78035 Versailles cedex, France; Jean-Pierre.Puel@math.uvsq.fr
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Abstract

We study a non standard unique continuation property for the biharmonic spectral problem $\Delta^2 w=-\lambda\Delta w$ in a 2D corner with homogeneous Dirichlet boundary conditions and a supplementary third order boundary condition on one side of the corner. We prove that if the corner has an angle $0<\theta_0<2\pi$, $\theta_0\not=\pi$ and $\theta_0\not=3\pi/2$, a unique continuation property holds. Approximate controllability of a 2-D linear fluid-structure problem follows from this property, with a control acting on the elastic side of a corner in a domain containing a Stokes fluid. The proof of the main result is based in a power series expansion of the eigenfunctions near the corner, the resolution of a coupled infinite set of finite dimensional linear systems, and a result of Kozlov, Kondratiev and Mazya, concerning the absence of strong zeros for the biharmonic operator [Math. USSR Izvestiya34 (1990) 337–353]. We also show how the same methodology used here can be adapted to exclude domains with corners to have a local version of the Schiffer property for the Laplace operator.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2008

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References

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Kozlov, V.A., Kondratiev, V.A. and Mazya, V.G., On sign variation and the absence of strong zeros of solutions of elliptic equations. Math. USSR Izvestiya 34 (1990) 337353. CrossRef
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V.A. Kozlov, V.G. Mazya and J. Rossmann, Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs 52. AMS, Providence (1997).
A. Osses and J.-P. Puel, Approximate controllability for a hydro-elastic model in a rectangular domain, in Optimal Control of partial Differential Equations (Chemnitz, 1998), Internat. Ser. Numer. Math. 133, Birkhäuser, Basel (1999) 231–243.
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