Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Homogenization of a stationary periodic Maxwell system in a bounded
  domain in the case of constant magnetic permeability   In a bounded domain $\mathcal{O}\subset\mathbb{R}^3$ of class $C^{1,1}$, we
consider a stationary Maxwell system with the boundary conditions of perfect
conductivity. It is assumed that the magnetic permeability is given by a
constant positive $(3\times 3)$-matrix $\mu_0$ and the dielectric permittivity
is of the form $\eta({\mathbf x}/ \varepsilon)$, where $\eta({\mathbf x})$ is a
$(3 \times 3)$-matrix-valued function with real entries, periodic with respect
to some lattice, bounded and positive definite. Here $\varepsilon >0$ is the
small parameter. Suppose that the equation involving the curl of the magnetic
field intensity is homogeneous, and the right-hand side $\mathbf r$ of the
second equation is a divergence-free vector-valued function of class $L_2$. It
is known that, as $\varepsilon \to 0$, the solutions of the Maxwell system,
namely, the electric field intensity ${\mathbf u}_\varepsilon$, the electric
displacement vector ${\mathbf w}_\varepsilon$, the magnetic field intensity
${\mathbf v}_\varepsilon$, and the magnetic displacement vector ${\mathbf
z}_\varepsilon$ weakly converge in $L_2$ to the corresponding homogenized
fields ${\mathbf u}_0$, ${\mathbf w}_0$, ${\mathbf v}_0$, ${\mathbf z}_0$ (the
solutions of the homogenized Maxwell system with effective coefficients). We
improve the classical results. It is shown that ${\mathbf v}_\varepsilon$ and
${\mathbf z}_\varepsilon$ converge to ${\mathbf v}_0$ and ${\mathbf z}_0$,
respectively, in the $L_2$-norm, the error terms do not exceed $C \varepsilon
\| {\mathbf r}\|_{L_2}$. We also find approximations for ${\mathbf
v}_\varepsilon$ and ${\mathbf z}_\varepsilon$ in the energy norm with error
$C\sqrt{\varepsilon} \|{\mathbf r}\|_{L_2}$. For ${\mathbf u}_\varepsilon$ and
${\mathbf w}_\varepsilon$ we obtain approximations in the $L_2$-norm with error
$C\sqrt{\varepsilon} \| {\mathbf r}\|_{L_2}$.

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The categories of this paper are: stat
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
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The categories of this paper are: hep-lat
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The categories of this paper are: math-ph
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th