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*} \mathrm {div}(s_\delta A \nabla u_{\delta }) + k^2 s_0 \Sigma u_{\delta } = s_0 f \mbox { in } \Omega , \end{equation*} where $\Omega$ is a smooth connected bounded open subset of $\mathbb {R}^d$ 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 $s_\delta$ is a complex function whose real part takes the values $1$ and $-1$ and whose imaginary part is positive and converges to $0$ as $\delta$ goes to 0. This is motivated from a result of Nicorovici, McPhedran, and Milton; another motivation is the concept of complementary media. 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.