Stabilization of Schrödinger equation in exterior domains
ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 3, pp. 570-579.

We prove uniform local energy estimates of solutions to the damped Schrödinger equation in exterior domains under the hypothesis of the Exterior Geometric Control. These estimates are derived from the resolvent properties.

DOI : 10.1051/cocv:2007024
Classification : 47A10, 35A27, 35A21, 35A25
Mots-clés : cut-off resolvent, local energy decay, stabilization 
@article{COCV_2007__13_3_570_0,
     author = {Aloui, Lassaad and Khenissi, Moez},
     title = {Stabilization of {Schr\"odinger} equation in exterior domains},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {570--579},
     publisher = {EDP-Sciences},
     volume = {13},
     number = {3},
     year = {2007},
     doi = {10.1051/cocv:2007024},
     mrnumber = {2329177},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2007024/}
}
TY  - JOUR
AU  - Aloui, Lassaad
AU  - Khenissi, Moez
TI  - Stabilization of Schrödinger equation in exterior domains
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2007
SP  - 570
EP  - 579
VL  - 13
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2007024/
DO  - 10.1051/cocv:2007024
LA  - en
ID  - COCV_2007__13_3_570_0
ER  - 
%0 Journal Article
%A Aloui, Lassaad
%A Khenissi, Moez
%T Stabilization of Schrödinger equation in exterior domains
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2007
%P 570-579
%V 13
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2007024/
%R 10.1051/cocv:2007024
%G en
%F COCV_2007__13_3_570_0
Aloui, Lassaad; Khenissi, Moez. Stabilization of Schrödinger equation in exterior domains. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 3, pp. 570-579. doi : 10.1051/cocv:2007024. http://archive.numdam.org/articles/10.1051/cocv:2007024/

[1] L. Aloui and M. Khenissi, Stabilisation de l'équation des ondes dans un domaine extérieur. Rev. Math. Iberoamericana 28 (2002) 1-16. | Zbl

[2] N. Burq, Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Act. Math. 1 (1998) 1-29. | Zbl

[3] N. Burq, Semi-classical estimates for the resolvent in non trapping geometries. Int. Math. Res. Not. 5 (2002) 221-241.

[4] A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time decay of the wave functions. Duke Math. J. 46 (1979) 583-612. | Zbl

[5] M. Khenissi, Équation des ondes amorties dans un domaine extérieur. Bull. Soc. Math. France 131 (2003) 211-228. | Numdam | Zbl

[6] R.B. Melrose and J. Sjostrand, Singularities of boundary value problems I. Comm. Pure Appl. Math. 31 (1978) 593-617. | Zbl

[7] C.S. Morawetz, Decay for solution of the exterior problem for the wave equation. Comm. Pure Appl. Math. 28 (1975) 229-264. | Zbl

[8] J. Ralston, Solution of the wave equation with localized energy. Comm. Pure Appl. Math. 22 (1969) 807-823. | Zbl

[9] J. Rauch, Local decay of scattering solutions of Schrödinger-type equation. Comm. Math. Phys. 61 (1978) 149-168. | Zbl

[10] M. Reed and B. Simon, Methods of modern mathematical physics, Vol. I: Functional Analysis. New York, Academic Press (1972). | Zbl

[11] Y. Tsutsumi, Local energy decay of solutions to the free Schrödinger equation in exterior domains. J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 31 (1984) 97-108. | Zbl

[12] B. Vainberg, On the analytical properties of the resolvent for certain class of operator-pencils. Math. USSR-Sb. 6 (1968) 241-273. | Zbl

[13] B. Vainberg, On the exterior elliptic problems polynomially depending on a spectral parameters, and asymptotic behaviour for large time of solutions of non stationary problems. Math. USSR-Sb. 21 (1973) 221-239. | Zbl

[14] B. Vainberg, On the short wave asymptotic behaviour of solutions of stationary problems and asymptotic behaviour as t→+∞ of solutions of non-stationary problems. Russian Math. Surveys 30 (1975) 1-58. | Zbl

[15] B. Vainberg, Asymptotic methods in equations of mathematical physics. Gordon and Breach, New York (1988). | MR | Zbl

[16] G. Vodev, On the uniform decay of the local energy. Serdica Math. J. 25 (1999) 191-206. | EuDML | Zbl

[17 ] H. Wilcox, Scattering Theory for the d'Alembert Equation in Exterior Domains. Lect. Notes Math. 442, Springer-Verlag (1975). | Zbl

Cité par Sources :