## 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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