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Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case. (English) Zbl 1084.35102
The authors derive generalized impedance boundary conditions (up to order 3) for a scattering problem involving the Helmholtz equation in obstacle domains. The framework is considered within the context of time harmonic acoustic waves in three dimensions, where an exact model is initially given for the transmission problem that couples the propagations into a highly absorbing interior domain and an exterior domain.
Existence and uniqueness of approximate solutions (in the \(H^1\) space), stability and error estimates are obtained for the corresponding boundary value problems with respect to the medium absorption.
The main arguments of the paper are based on the use of the Gaussian and mean curvature of the boundary, the Laplace-Beltrami operator, an asymptotic ansatz, a modified trace theorem, stability estimates, and Green’s formula.

MSC:
35Q60 PDEs in connection with optics and electromagnetic theory
35J25 Boundary value problems for second-order elliptic equations
78A45 Diffraction, scattering
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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[1] Ammari H., SIAM J. Appl. Math. 59 pp 1322–
[2] Artola M., C. R. Acad. Sci. Paris Sér. I Math. 313 pp 381–
[3] M. Artola and M. Cessenat, The Leontovich Conditions in Electromagnetism. Les Grands Systèmes des Sciences et de la Technologie, RMA Res. Notes Appl. Math. 28 (Masson, 1994) pp. 11–21.
[4] DOI: 10.1137/S0036139999353826 · Zbl 0983.35138 · doi:10.1137/S0036139999353826
[5] DOI: 10.1006/jmaa.1998.6153 · Zbl 0923.35179 · doi:10.1006/jmaa.1998.6153
[6] Antoine X., Asymp. Anal. 26 pp 257–
[7] Bendali A., SIAM J. Appl. Math. 58 pp 1664–
[8] DOI: 10.1090/S0025-5718-1977-0436612-4 · doi:10.1090/S0025-5718-1977-0436612-4
[9] DOI: 10.1016/S0377-0427(01)00508-8 · Zbl 0998.78008 · doi:10.1016/S0377-0427(01)00508-8
[10] Hoppe D. J., Impedance Boundary Conditions in Electromagnetics (1995)
[11] Liu J., IEEE Trans. Antennas Propagat. 49 pp 1794– · Zbl 1001.78021 · doi:10.1109/8.982462
[12] Rytov S. M., J. Phys. USSR. 2 pp 233–
[13] DOI: 10.1049/PBEW041E · doi:10.1049/PBEW041E
[14] DOI: 10.1002/(SICI)1099-1476(19990510)22:7<587::AID-MMA54>3.0.CO;2-B · Zbl 0923.35114 · doi:10.1002/(SICI)1099-1476(19990510)22:7<587::AID-MMA54>3.0.CO;2-B
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