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Low-frequency asymptotics for dissipative Maxwell’s equations in bounded domains. (English) Zbl 0704.35007

We consider a dissipative boundary value problem from electromagnetic theory: \[ (M)\quad curl E_{\omega}-i\omega \mu H_{\omega}=\hat K_{\omega}\text{ in } \Omega \subset {\mathbb{R}}^ 3 \]
\[ curl H_{\omega}+i\omega \epsilon E_{\omega}-\sigma E_{\omega}-\sigma E_{\omega}=\hat I_{\omega}\text{ in } \Omega \] \(\nu\times E_{\omega}=0\) on \(\partial \Omega\), \(\nu\):normal. The tensors \(\epsilon\),\(\mu\) of dielectricity and permeability are uniformly positive definite while the tensor \(\sigma\) of conductivity is uniformly positive definite in some strict subregion \(D\subset \subset \Omega\) and vanishes in \(\Omega\setminus D\). We assume that the bounded domain \(\Omega\) is simply connected and that \(\partial (\Omega \setminus D)\) consists of the two connected components \(\partial D\) and \(\partial\Omega\). The equations (M) are supposed to hold in the sense of distributions. Hence these equations imply transmission conditions on \(\partial D.\)
The aim of the paper is to obtain the limits of \(E_{\omega}\), \(H_{\omega}\) as \(\omega\) tends to 0 from the behaviour of \(\hat K_{\omega}\), \(\hat I_{\omega}\) at \(\omega=0.\)
The conditions which one has to impose upon the data are - classically speaking \[ \hat I_{\omega}=I+i\omega J_{\omega},\quad \hat K_{\omega}=K+i\omega L_{\omega} \] where \(J_{\omega}\to J_ 0\), \(L_{\omega}\to L_ 0\) in \(L^ 2(\Omega)\), div\(K=0\) in \(\Omega\), \(\nu K=0\) on \(\partial \Omega\), div\(I=0\) in \(\Omega\setminus D\), \(\int_{\partial D}\nu Id_ 0=0.\)
Removing \(E_{\omega}\) from the system (M) one obtains a second order system for \(H_{\omega}\) which allows some controlled (with respect to \(\omega\)) coerciveness estimate. From this we conclude existence, uniqueness and convergence of order \(O(\omega^{1/2}+\| J_{\omega}- J_ 0\| +\| L_{\omega}-L_ 0\|)\) to the solution of a certain electromagnetostatic problem.
Reviewer: N.Weck

MSC:

35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
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