Partial Differential Equations
The Helmholtz equation with impedance in a half-space
Comptes Rendus. Mathématique, Volume 341 (2005) no. 9, pp. 561-566.

In this Note we obtain existence and uniqueness results for the Helmholtz equation in the half-space R+3 with an impedance or Robin boundary condition. Basically, we follow the procedure we have already used in the bi-dimensional case (the half-plane). Thus, we compute the associated Green's function with the help of a double Fourier transform and we analyze its far field in order to obtain radiation conditions that allow us to prove the uniqueness of an outgoing solution. Again, these radiation conditions are somewhat unusual due to the appearance of a surface wave guided by the boundary. An integral representation of the solution is presented by means of the Green's function and the boundary data.

Dans cette Note, nous démontrons un résultat d'existence et d'unicité de la solution de l'équation de Helmholtz dans un demi-espace avec une condition d'impédance. Le domaine est non borné et sa frontière également. Les conditions de radiation sont différentes des conditions usuelles pour un problème extérieur, ceci étant lié à la présence d'ondes de surface. Nous calculons la fonction de Green du demi-espace et nous étudions son comportement à l'infini. Ceci conduit à l'expression des conditions de radiation qui permettent de démontrer l'unicité. L'utilisation de la représentation intégrale donne le résultat d'existence.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.09.021
Durán, Mario 1; Muga, Ignacio 2; Nédélec, Jean-Claude 3

1 Facultad de Ingeniería, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
2 Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile
3 CMAP, École polytechnique, 91128 Palaiseau cedex, France
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Durán, Mario; Muga, Ignacio; Nédélec, Jean-Claude. The Helmholtz equation with impedance in a half-space. Comptes Rendus. Mathématique, Volume 341 (2005) no. 9, pp. 561-566. doi : 10.1016/j.crma.2005.09.021. http://www.numdam.org/articles/10.1016/j.crma.2005.09.021/

[1] Durán, M.; Muga, I.; Nédélec, J.-C. The Helmholtz equation with impedance in a half-plane, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 483-488

[2] L.C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, American Mathematical Society, Providence, RI, 1998

[3] Karamyan, G. The inverse scattering problem for the acoustic equation in a half-space, Inverse Problems, Volume 18 (2002), pp. 1673-1686

[4] Karamyan, G. Inverse scattering in a half-space with passive boundary, Commun. Partial Differential Equations, Volume 28 (2003) no. 9–10, pp. 1627-1641

[5] Lassas, M.; Cheney, M.; Uhlmann, G. Uniqueness for a wave propagation inverse problem in a half-space, Inverse Problems, Volume 14 (1998), pp. 679-684

[6] Nédélec, J.-C. Acoustic and Electromagnetic Equations, Springer-Verlag, Berlin/New York, March 15, 2001

[7] O. Poisson, Calculs des pôles de résonance associés à la diffraction d'ondes acoustiques et élastiques en dimension 2, Doctorate thesis of L'université de Paris IX Dauphine, Mai 1992

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