This paper is devoted to the study of the behavior of the unique solution$u_\delta \in H^{1}_{0}(\Omega)$, as $\delta \to 0$, to the equation\begin{equation*} \dive(\epss_\delta A \nabla u_{\delta}) + k^2 \epss_0 \Sigmau_{\delta} = \epss_0 f \mbox{in} \Omega, \end{equation*} where $\Omega$ is asmooth connected bounded open subset of $\mR^d$ with $d=2$ or 3, $f \inL^2(\Omega)$, $k$ is a non-negative constant, $A$ is a uniformly ellipticmatrix-valued function, $\Sigma$ is a real function bounded above and below bypositive constants, and $\epss_\delta$ is a complex function whose {\bf thereal part takes the value 1 and -1}, and the imaginary part is positive andconverges to 0 as $\delta$ goes to 0. This is motivated from a result in\cite{NicoroviciMcPhedranMilton94} and the concept of complementary suggestedin \cite{LaiChenZhangChanComplementary, PendryNegative, PendryRamakrishna}.After introducing the reflecting complementary media, complementary mediagenerated by reflections, we characterize $f$ for which$\|u_\delta\|_{H^1(\Omega)}$ remains bounded as $\delta$ goes to 0. For such an$f$, we also show that $u_\delta$ converges weakly in $H^1(\Omega)$ and providea formula to compute the limit.
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机译：本文致力于研究唯一解$ u_ \ delta在H ^ {1} _ {0}（\ Omega）$中的行为，如$ \ delta至0 $，到方程\ begin { equation *} \ dive（\ epss_ \ delta A \ nabla u _ {\ delta}）+ k ^ 2 \ epss_0 \ Sigmau _ {\ delta} = \ epss_0 f \ mbox {in} \ Omega，\ end {equation *}其中$ \ Omega $是$ \ mR ^ d $的光滑连通有界开放子集，其中$ d = 2 $或3，$ f \ inL ^ 2（\ Omega）$，$ k $是非负常数，$ A $是一个统一的椭圆矩阵值函数，$ \ Sigma $是一个由正常数上下限制的实函数，而$ \ epss_ \ delta $是一个复杂函数，其{\ bf实数部分取值1和-1}，并且虚部为正，当$ \ delta $变为0时收敛到0。这是由\ cite {NicoroviciMcPhedranMilton94}中的结果和\ cite {LaiChenZhangChanComplementary，PendryNegative，PendryRamakrishna}中的补充建议的概念引起的。引入反射后互补媒体，由反射产生的互补媒体，我们具有特色erize $ f $ for $ \ | u_ \ delta \ | __ {H ^ 1（\ Omega）} $仍然有界，因为$ \ delta $变为0。对于这样的$ f $，我们还显示$ u_ \ delta $在$ H ^ 1（\ Omega）$中微弱收敛，并提供一个计算限制的公式。
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