Abstract
We study the limiting absorption principle and the well-posedness of Maxwell equations with anisotropic sign-changing coefficients in the time-harmonic domain. The starting point of the analysis is to obtain Cauchy problems associated with two Maxwell systems using a change of variables. We then derive a priori estimates for these Cauchy problems using two different approaches. The Fourier approach involves the complementing conditions for the Cauchy problems associated with two elliptic equations, which were studied in a general setting by Agmon, Douglis, and Nirenberg. The variational approach explores the variational structure of the Cauchy problems of the Maxwell equations. As a result, we obtain general conditions on the coefficients for which the limiting absorption principle and the well-posedness hold. Moreover, these new conditions are of a local character and easy to check. Our work is motivated by and provides general sufficient criteria for the stability of electromagnetic fields in the context of negative-index metamaterials.
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Notes
The complementing conditions holds on \(\partial O_k \cap S\) a priori. However, since \(\varphi _k = 0\) in \(B_{r_k}(x_k) {\setminus } B_{r_k/2}(x_k)\), one can modify \(\varepsilon , \, \mu , \, {{\hat{\varepsilon }} }, \, {{\hat{\mu }}}\) in \(B_{r_k}(x_k) {\setminus } B_{2r_k/3}(x_k)\) so that the complementing conditions hold on the whole boundary and the systems are unchanged, e.g. \(\varepsilon > {{\hat{\varepsilon }} }\) and \(\mu > {{\hat{\mu }}}\) in \(B_{r_k}(x_k) {\setminus } B_{2r_k/3}(x_k)\) (see Proposition 2.1).
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Proof of Proposition 3.1
Proof of Proposition 3.1
We only consider the case \((\mu , {{\hat{\mu }}})\) does not satisfy the complementing conditions at some point \(x_0 \in \partial \Omega \). The other case can be dealt similarly. Then, from [2], there exist sequences \(\big ( ( u_{n}, \hat{u}_{n} ) \big )\subset H^{2}(\Omega ),\)\(\big ( (f_{n}, {\hat{f}}_{n}) \big ) \subset L^{2}(\Omega ),\)\( \big ( p_{n} \big ) \subset H^{\frac{3}{2}}(\partial \Omega )\) and \(\big ( q_{n} \big ) \subset H^{\frac{1}{2}}(\partial \Omega )\) such that
\( \big ( \left\Vert f_{n}\right\Vert _{L^{2}(\Omega )} \big ), \, \big (\Vert {\hat{f}}_{n}\Vert _{L^{2}(\Omega )} \big ), \big (\left\Vert p_{n}\right\Vert _{H^{\frac{3}{2}}(\partial \Omega )} \big ), \, \big (\left\Vert q_{n}\right\Vert _{H^{\frac{1}{2}}(\partial \Omega )} \big )\) and \( \big ( \left\Vert \left( u_{n}, \hat{u}_{n} \right) \right\Vert _{H^{1}(\Omega )} \big ) \) are bounded and \(\lim _{n \rightarrow + \infty }\left\Vert \left( u_{n}, \hat{u}_{n} \right) \right\Vert _{H^{2}(\Omega )} = + \infty .\)
Let \(\phi _{n} \in H^{2}\) and \(\hat{\phi }_{n} \in H^{2}\) be the unique solution of
By the standard theory of elliptic equations, one has
for some positive constant C independent of n. Set
We have, in \(\Omega \),
Since \(u_{n} - \hat{u}_{n} = p_{n} = \phi _{n}\) on \(\partial \Omega ,\) and \({\hat{\varphi }}_n = 0\) on \(\partial \Omega \), we deduce that
Set
and, in \(\Omega \),
One can easily check that \((E_n, {{\hat{E}}}_n, H_n, {{\hat{H}}}_n)\) and \((J_{e, n}, {{\hat{J}} }_{e, n}, J_{m, n}, {{\hat{J}} }_{e, n})\) satisfies all the required properties. \(\quad \square \)
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Nguyen, HM., Sil, S. Limiting Absorption Principle and Well-Posedness for the Time-Harmonic Maxwell Equations with Anisotropic Sign-Changing Coefficients. Commun. Math. Phys. 379, 145–176 (2020). https://doi.org/10.1007/s00220-020-03805-1
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DOI: https://doi.org/10.1007/s00220-020-03805-1