# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48 .

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## An inverse problem for scattering by a doubly periodic structureHTML articles powered by AMS MathViewer

by Gang Bao and Zhengfang Zhou
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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