In this Note we obtain existence and uniqueness results for the Helmholtz equation in the half-space 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.
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Mario Durán 1; Ignacio Muga 2; Jean-Claude Nédélec 3
@article{CRMATH_2005__341_9_561_0, author = {Mario Dur\'an and Ignacio Muga and Jean-Claude N\'ed\'elec}, title = {The {Helmholtz} equation with impedance in a half-space}, journal = {Comptes Rendus. Math\'ematique}, pages = {561--566}, publisher = {Elsevier}, volume = {341}, number = {9}, year = {2005}, doi = {10.1016/j.crma.2005.09.021}, language = {en}, }
Mario Durán; Ignacio Muga; Jean-Claude Nédélec. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.09.021/
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