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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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An inverse problem for scattering by a doubly periodic structure
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by Gang Bao and Zhengfang Zhou PDF
Trans. Amer. Math. Soc. 350 (1998), 4089-4103 Request permission

Abstract:

Consider scattering of electromagnetic waves by a doubly periodic structure $S=\{x_3=f(x_1, x_2)\}$ with $f(x_1+n_1\Lambda _1, x_2+n_2\Lambda _2)=f(x_1, x_2)$ for integers $n_1$, $n_2$. Above the structure, the medium is assumed to be homogeneous with a constant dielectric coefficient. The medium is a perfect conductor below the structure. An inverse problem arises and may be described as follows. For a given incident plane wave, the tangential electric field is measured away from the structure, say at $x_3=b$ for some large $b$. To what extent can one determine the location of the periodic structure that separates the dielectric medium from the conductor? In this paper, results on uniqueness and stability are established for the inverse problem. A crucial step in our proof is to obtain a lower bound for the first eigenvalue of the following problem in a convex domain $\Omega$: \[ \left \{ \begin {array}{l} - \triangle u = \lambda u \quad \text {in} \quad \Omega , \ \nabla \cdot u = 0 \quad \text {in} \quad \Omega , \ n \times u = 0 \quad \text {on} \quad \partial \Omega . \end {array} \right . \]
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Additional Information
  • Gang Bao
  • Affiliation: Department of Mathematics University of Florida Gainesville, Florida 32611
  • Email: bao@math.ufl.edu
  • Zhengfang Zhou
  • Affiliation: Department of Mathematics Michigan State University East Lansing, Michigan 48824
  • Email: zfzhou@math.msu.edu
  • Received by editor(s): September 18, 1996
  • Additional Notes: The research of the first author was partially supported by NSF grant DMS 95-01099, NSF University-Industry Cooperative Research Programs grant DMS 97-05139, and a Research Development Award (University of Florida)
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 4089-4103
  • MSC (1991): Primary 35R30; Secondary 35P15, 78A45
  • DOI: https://doi.org/10.1090/S0002-9947-98-02227-2
  • MathSciNet review: 1487607