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

Osses, Axel; Puel, Jean-Pierre

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 279-294.

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 $0<{\theta }_0<2\pi$, ${\theta }_0\ne \pi$ and ${\theta }_0\ne 3\pi /2$, 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 Izvestiya 34 (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.

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