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

TL;DR

This paper investigates the asymptotic behavior of solutions to Helmholtz equations with sign-changing coefficients, characterizing conditions for boundedness and convergence as a parameter approaches zero, motivated by negative index materials.

Contribution

It introduces the concept of reflecting complementary media and characterizes functions for which solutions remain bounded, providing a formula for their weak limits.

Findings

Boundedness of solutions under specific conditions

Weak convergence of solutions as the parameter tends to zero

Explicit formula for the limit solution

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.