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Limiting Absorption Principle and Well-Posedness for the Time-Harmonic Maxwell Equations with Anisotropic Sign-Changing Coefficients

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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

  1. 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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Correspondence to Hoai-Minh Nguyen.

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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

$$\begin{aligned} \begin{array}{c} \text {div} \left( \mu \nabla u_{n} \right) = f_{n}, \quad \quad \text {div} \left( {{\hat{\mu }}}\nabla \hat{u}_{n} \right) = {\hat{f}}_{n} \quad \text { in } \Omega , \\ u_{n} - \hat{u}_{n} = p_{n} \quad \text { on } \partial \Omega , \\ \left( \mu \nabla u_{n} - {{\hat{\mu }}}\nabla \hat{u}_{n} \right) \cdot \nu = q_{n} \quad \text { on } \partial \Omega , \end{array} \end{aligned}$$

\( \big ( \left\Vert f_{n}\right\Vert _{L^{2}(\Omega )} \big ), \, \big (\Vert {\hat{f}}_{n}\Vert _{L^{2}(\Omega )} \big ), \big (\left\Vert p_{n}\right\Vert _{H^{\frac{3}{2}}(\partial \Omega )} \big ), \, \big (\left\Vert q_{n}\right\Vert _{H^{\frac{1}{2}}(\partial \Omega )} \big )\) 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

$$\begin{aligned} \left\{ \begin{aligned} \text {div} \left( \mu \nabla \phi _{n} \right)&= f_{n}&\text { in } \Omega , \\ \phi _{n}&= p_{n}&\text { on } \partial \Omega . \end{aligned}\right. \qquad \text { and } \qquad \left\{ \begin{aligned} \text {div} \left( {{\hat{\mu }}}\nabla \hat{\phi }_{n} \right)&= {\hat{f}}_{n}&\text { in } \Omega , \\ \hat{\phi }_{n}&= 0&\text { on } \partial \Omega . \end{aligned}\right. \end{aligned}$$

By the standard theory of elliptic equations, one has

$$\begin{aligned} \left\Vert \phi _{n} \right\Vert _{H^{2}} \le C \left( \left\Vert f_{n} \right\Vert _{L^{2}} +\left\Vert p_{n} \right\Vert _{H^{\frac{3}{2}}(\partial \Omega )} \right) \quad \text { and } \quad \left\Vert \hat{\phi }_{n} \right\Vert _{H^{2}} \le C \left\Vert {\hat{f}}_{n} \right\Vert _{L^{2}}, \end{aligned}$$

for some positive constant C independent of n. Set

$$\begin{aligned} H_{n} = \nabla u_{n} - \nabla \phi _{n} \quad \text{ and } \quad {{\hat{H}}}_{n} = \nabla \hat{u}_{n} - \nabla \hat{\phi }_{n}, \text{ in } \Omega . \end{aligned}$$

We have, in \(\Omega \),

$$\begin{aligned} \text {curl} H_{n} = 0 = \text {curl} {{\hat{H}}}_{n} \quad \text { and } \quad \text {div} \left( \mu H_{n} \right) = 0 = \text {div} \big ( {{\hat{\mu }}}{{\hat{H}}}_{n} \big ). \end{aligned}$$

Since \(u_{n} - \hat{u}_{n} = p_{n} = \phi _{n}\) on \(\partial \Omega ,\) and \({\hat{\varphi }}_n = 0\) on \(\partial \Omega \), we deduce that

$$\begin{aligned} \nu \times \big ( H_{n} - {{\hat{H}}}_{n} \big ) = 0 \qquad \text { on } \partial \Omega . \end{aligned}$$

Set

$$\begin{aligned} E_n = {\hat{E}}_n = 0 \text{ in } \Omega \end{aligned}$$

and, in \(\Omega \),

$$\begin{aligned} J_{e,n} = -i\omega \mu H_{n}, \quad {{\hat{J}} }_{e,n} = -i\omega {{\hat{\mu }}}{{\hat{H}}}_{n}, \quad J_{m,n} =0 , \quad \text{ and } \quad {{\hat{J}} }_{m,n} = 0. \end{aligned}$$

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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