×

Existence, uniqueness and regularity for solutions of the conical diffraction problem. (English) Zbl 1010.78008

Summary: This paper is devoted to the analysis of two Helmholtz equations in \(\mathbb{R}^2\) coupled via quasiperiodic transmission conditions on a set of piecewise smooth interfaces. The solution of this system is quasiperiodic in one direction and satisfies outgoing wave conditions with respect to the other direction. It is shown that Maxwell’s equations for the diffraction of a time-harmonic oblique incident plane wave by periodic interfaces can be reduced to problems of this kind. The analysis is based on a strongly elliptic variational formulation of the differential problem in a bounded periodic cell involving nonlocal boundary operators. We obtain existence and uniqueness results for solutions corresponding to electromagnetic fields with locally finite energy. Special attention is paid to the regularity and leading asymptotics of solutions near the edges of the interface.

MSC:

78A45 Diffraction, scattering
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1137/0732053 · Zbl 0853.65134 · doi:10.1137/0732053
[2] DOI: 10.1007/s002110050227 · Zbl 0866.65075 · doi:10.1007/s002110050227
[3] DOI: 10.1137/S0036139995279408 · Zbl 0872.65108 · doi:10.1137/S0036139995279408
[4] DOI: 10.1002/mma.1670170502 · Zbl 0817.35109 · doi:10.1002/mma.1670170502
[5] DOI: 10.1080/713820571 · doi:10.1080/713820571
[6] DOI: 10.1017/S0308210500021132 · Zbl 0789.35042 · doi:10.1017/S0308210500021132
[7] DOI: 10.2307/2001542 · Zbl 0727.35131 · doi:10.2307/2001542
[8] DOI: 10.1016/0022-247X(91)90104-8 · Zbl 0738.35095 · doi:10.1016/0022-247X(91)90104-8
[9] DOI: 10.1051/m2an:1999155 · Zbl 0937.78003 · doi:10.1051/m2an:1999155
[10] DOI: 10.1016/0022-247X(85)90118-0 · Zbl 0597.35021 · doi:10.1016/0022-247X(85)90118-0
[11] DOI: 10.1017/S0956792500001169 · Zbl 0806.35175 · doi:10.1017/S0956792500001169
[12] Dobson D. C., Model. Math. Anal. Numer. 28 pp 419– (1994) · Zbl 0820.65087 · doi:10.1051/m2an/1994280404191
[13] DOI: 10.1002/(SICI)1099-1476(19980925)21:14<1297::AID-MMA997>3.0.CO;2-C · Zbl 0913.65121 · doi:10.1002/(SICI)1099-1476(19980925)21:14<1297::AID-MMA997>3.0.CO;2-C
[14] Friedman A., Part 7 pp 67– (1995)
[15] Kondratiev V. A., Trans. Moscow Math. Soc. 16 pp 227– (1967)
[16] DOI: 10.1364/JOSAA.13.000779 · doi:10.1364/JOSAA.13.000779
[17] DOI: 10.1364/JOSA.71.000811 · doi:10.1364/JOSA.71.000811
[18] DOI: 10.1137/0522104 · Zbl 0756.35004 · doi:10.1137/0522104
[19] Weisel J., Mitteilungen Mathem. Seminar Giessen 138 pp 1– (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.