[Condition de radiation et unicité pour l'onde élastique sortante dans un demi-plane avec frontière libre]
Dans cette Note, nous exhibons une condition de radiation explicite, du type Sommerfeld, qui nous permet de montrer (dans le domaine fréquenciel) l'unicité des solutions du problème d'onde élastique sortante dans un demi-plan avec frontière libre. Cette expression est obtenue par une analyse asymptotique rigoureuse de la fonction de Green's associée. La difficulté principale est que la condition de bord de frontière libre permet la propagation d'une onde de Rayleigh qui n'est pas négligeable dans le champ lointain. Nous donnons également un résultat d'existence pour ce problème.
In this Note we deduce an explicit Sommerfeld-type radiation condition which is convenient to prove the uniqueness for the time-harmonic outgoing wave problem in an isotropic elastic half-plane with free boundary condition. The expression is obtained from a rigorous asymptotic analysis of the associated Green's function. The main difficulty is that the free boundary condition allows the propagation of a Rayleigh wave which cannot be neglected in the far field expansion. We also give the existence result for this problem.
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@article{CRMATH_2009__347_21-22_1321_0,
author = {Dur\'an, Mario and Muga, Ignacio and N\'ed\'elec, Jean-Claude},
title = {Radiation condition and uniqueness for the outgoing elastic wave in a half-plane with free boundary},
journal = {Comptes Rendus. Math\'ematique},
pages = {1321--1324},
publisher = {Elsevier},
volume = {347},
number = {21-22},
year = {2009},
doi = {10.1016/j.crma.2009.10.005},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2009.10.005/}
}
TY - JOUR AU - Durán, Mario AU - Muga, Ignacio AU - Nédélec, Jean-Claude TI - Radiation condition and uniqueness for the outgoing elastic wave in a half-plane with free boundary JO - Comptes Rendus. Mathématique PY - 2009 SP - 1321 EP - 1324 VL - 347 IS - 21-22 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2009.10.005/ DO - 10.1016/j.crma.2009.10.005 LA - en ID - CRMATH_2009__347_21-22_1321_0 ER -
%0 Journal Article %A Durán, Mario %A Muga, Ignacio %A Nédélec, Jean-Claude %T Radiation condition and uniqueness for the outgoing elastic wave in a half-plane with free boundary %J Comptes Rendus. Mathématique %D 2009 %P 1321-1324 %V 347 %N 21-22 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2009.10.005/ %R 10.1016/j.crma.2009.10.005 %G en %F CRMATH_2009__347_21-22_1321_0
Durán, Mario; Muga, Ignacio; Nédélec, Jean-Claude. Radiation condition and uniqueness for the outgoing elastic wave in a half-plane with free boundary. Comptes Rendus. Mathématique, Tome 347 (2009) no. 21-22, pp. 1321-1324. doi : 10.1016/j.crma.2009.10.005. http://archive.numdam.org/articles/10.1016/j.crma.2009.10.005/
[1] Théorème d'unicité pour un problème d'ondes élastiques, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 6, pp. 525-530
[2] Computing Green's function of elasticity in a half-plane with impedance boundary condition, C. R. Mec., Volume 334 (2006) no. 12, pp. 725-731
[3] The Helmholtz equation in a locally perturbed half-plane with passive boundary, IMA J. Appl. Math., Volume 71 (2006) no. 6, pp. 853-876
[4] The Helmholtz equation in a locally perturbed half-space with non-absorbing boundary, Arch. Ration. Mech. Anal., Volume 191 (2009) no. 1, pp. 143-172
[5] Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998
[6] Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity (Kupradze, V.D., ed.), North-Holland Series in Applied Mathematics and Mechanics, vol. 25, North-Holland Publishing Co., Amsterdam, 1979 (Russian ed)
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