Document ESAIM: COCV
DOI: 10.1051/cocv:2008024
Unique continuation property near a corner and its fluid-structure controllability consequences
Axel Osses1 and Jean-Pierre Puel21 Corresponding author: 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 September 19, 2005. Revised January 27, 2006 and September 27, 2007. Published online March 28, 2008.)
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.
Mathematics Subject Classification. 35B60, 35B37
Key words: Continuation of solutions of PDE, fluid-structure control, domains with corners
© EDP Sciences, SMAI 2008