Asymptotic behavior of solutions to the Helmholtz equations with sign changing coefficients

Abstract

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 \Sigma u_{\delta} = \epss_0 f \mbox{in} \Omega, \end{equation*} where $\Omega$ is a smooth connected bounded open subset of $\mR^d$ with $d=2$ or 3, $f \in L^2(\Omega)$, $k$ is a non-negative constant, $A$ is a uniformly elliptic matrix-valued function, $\Sigma$ is a real function bounded above and below by positive constants, and $\epss_\delta$ is a complex function whose {\bf the real part takes the value 1 and -1}, and the imaginary part is positive and converges to 0 as $\delta$ goes to 0. This is motivated from a result in \cite{NicoroviciMcPhedranMilton94} and the concept of complementary suggested in \cite{LaiChenZhangChanComplementary, PendryNegative, PendryRamakrishna}. After introducing the reflecting complementary media, complementary media generated 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 provide a formula to compute the limit.