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

Abstract

We study a non standard unique continuation property for the biharmonic spectral prob- lem Δ 2 w = −λΔw in a 2D corner with homogeneous Dirichlet boundary conditions and a supplemen- tary third order boundary condition on one side of the corner. We prove that if the corner has an angle 0 <θ 0 < 2π, θ0 � π and θ0 � π/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. Mathematics Subject Classification. 35B60, 35B37.