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


Author: Hoai-Minh Nguyen
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
Published electronically: November 24, 2014
MathSciNet review: 3356948
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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

$\displaystyle \mathrm {div}(s_\delta A \nabla u_{\delta }) + k^2 s_0 \Sigma u_{\delta } = s_0 f$$\displaystyle \mbox { in } \Omega ,$    

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 $ \Vert u_\delta \Vert _{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.

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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

DOI: https://doi.org/10.1090/S0002-9947-2014-06305-8
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
Article copyright: © Copyright 2014 American Mathematical Society