Abstract
We obtain uniqueness and existence results of an outgoing solution for the Helmholtz equation in a half-space, or in a compact local perturbation of it, with an impedance boundary condition. It is worth noting that these kinds of domains have unbounded boundaries which lead to a non-classical exterior problem. The established radiation condition is somewhat different from the usual Sommerfeld’s one, due to the appearance of surface waves (in the case of a non-absorbing boundary). A half-space Green’s function framework is used to carry out our computations. This is an extended and detailed version of the previous article “The Helmholtz equation with impedance in a half-space,” Duran et al. (CR Acad Sci Paris Ser I 341:561–566, 2005).
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References
Abramowitz M. and Stegun I.A. (1965). Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York
Amrouche Ch. (2002). The Neumann problem in the half-space. C.R. Acad. Sci. Paris Ser. I 335: 151–156
Chandler-Wilde S.N. (1997). The impedance boundary value problem for the Helmholtz equation in a half-plane. Math. Methods Appl. Sci. 20: 813–840
Chandler-Wilde S.N. and Peplow A.T. (2005). A boundary integral equation formulation for the Helmholtz equation in a locally perturbed half-plane. Zamm Zeitschrift Fur Angewandte Mathematik und Mechanik 85(2): 79–88
Durán M., Muga I. and Nédélec J.-C. (2006). The Helmholtz equation in a locally perturbed half-plane with passive boundary. IMA J. Appl. Math. 71: 853–876
Durán M., Muga I. and Nédélec J.-C. (2005). The Helmholtz equation with impedance in a half-space. C. R. Acad. Sci. Paris Ser. I 341: 561–566
Evans L.C. (1998). Partial Differential Equations. American Mathematical Society, Providence
Gradshteyn I.S. and Ryzhik I.M. (1980). Table of integrals, series and products. Academic Press, New York
Karamyan G. (2003). Inverse scattering in a half-space with passive boundary. Commun. Partial Diff. Equ. 28(9-10): 1627–1641
Odeh F.M. (1963). Uniqueness theoremes for the Helmholtz equation in domains with infinite boundaries. J. Math. Mech. 12(6): 857–867
Nédélec J.-C. (2001). Acoustic and Electromagnetic Equations. Springer, New York
Rudin W. (1973). Functional Analysis. Mc-Graw-Hill, New York
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Communicated by C. A. Stuart
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Durán, M., Muga, I. & Nédélec, JC. The Helmholtz Equation in a Locally Perturbed Half-Space with Non-Absorbing Boundary. Arch Rational Mech Anal 191, 143–172 (2009). https://doi.org/10.1007/s00205-008-0135-3
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DOI: https://doi.org/10.1007/s00205-008-0135-3