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.

Keywords: continuation of solutions of PDE, fluid-structure control, domains with corners

@article{COCV_2009__15_2_279_0, author = {Osses, Axel and Puel, Jean-Pierre}, title = {Unique continuation property near a corner and its fluid-structure controllability consequences}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {279--294}, publisher = {EDP-Sciences}, volume = {15}, number = {2}, year = {2009}, doi = {10.1051/cocv:2008024}, zbl = {1176.35042}, mrnumber = {2513087}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2008024/} }

TY - JOUR AU - Osses, Axel AU - Puel, Jean-Pierre TI - Unique continuation property near a corner and its fluid-structure controllability consequences JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 DA - 2009/// SP - 279 EP - 294 VL - 15 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008024/ UR - https://zbmath.org/?q=an%3A1176.35042 UR - https://www.ams.org/mathscinet-getitem?mr=2513087 UR - https://doi.org/10.1051/cocv:2008024 DO - 10.1051/cocv:2008024 LA - en ID - COCV_2009__15_2_279_0 ER -

%0 Journal Article %A Osses, Axel %A Puel, Jean-Pierre %T Unique continuation property near a corner and its fluid-structure controllability consequences %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 279-294 %V 15 %N 2 %I EDP-Sciences %U https://doi.org/10.1051/cocv:2008024 %R 10.1051/cocv:2008024 %G en %F COCV_2009__15_2_279_0

Osses, Axel; Puel, Jean-Pierre. Unique continuation property near a corner and its fluid-structure controllability consequences. ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 2, pp. 279-294. doi : 10.1051/cocv:2008024. http://www.numdam.org/articles/10.1051/cocv:2008024/

[1] Some results about Schiffer's conjectures. Inverse Problems 15 (1999) 647-658. | MR | Zbl

and ,[2] Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman Advanced Publishing Program, Boston-London-Melbourne (1985). | MR | Zbl

,[3] On sign variation and the absence of strong zeros of solutions of elliptic equations. Math. USSR Izvestiya 34 (1990) 337-353. | MR | Zbl

, and ,[4] Non-homogeneous Boundary Value Problems and applications. Springer-Verlag, Berlin (1972). | Zbl

and ,[5] Approximate controllability of a hydro-elastic coupled system. ESAIM: COCV 1 (1995) 1-15. | Numdam | MR | Zbl

and ,[6] Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs 52. AMS, Providence (1997). | MR | Zbl

, and ,[7] 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. | MR | Zbl

and ,[8] Approximate controllability of a linear model in solid-fluid interaction. ESAIM: COCV 4 (1999) 497-513. | Numdam | MR | Zbl

and ,[9] A partial solution of the Pompeiu problem. Math. Anal. 223 (1976) 183-190. | MR | Zbl

,[10] Analyticity of the boundary of Lipschitz domains without the Pompeiu property. Indiana Univ. Math. J. 30 (1981) 357-369. | MR | Zbl

,*Cited by Sources: *