Asymptotic behavior of solutions to the Helmholtz equations with sign changing coefficients
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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.References
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Additional Information
- Hoai-Minh Nguyen
- Affiliation: Chair of Analysis and Applied Mathematics, École Polytechnique Féderale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland – School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- Email: hoai-minh.nguyen@epfl.ch, hmnguyen@math.umn.edu
- Received by editor(s): September 23, 2013
- Published electronically: November 24, 2014
- Additional Notes: This research was supported by NSF grant DMS-1201370 and by the Alfred P. Sloan Foundation
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 6581-6595
- MSC (2010): Primary 35B40, 35Q60; Secondary 78A40, 78M35
- DOI: https://doi.org/10.1090/S0002-9947-2014-06305-8
- MathSciNet review: 3356948