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

¹ 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
² 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

Received: 19 September 2005
Revised: 27 January 2006
Revised: 27 September 2007

Abstract

We study a non standard unique continuation property for the biharmonic spectral problem 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 , and , 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.