论文标题
在恒定渗透率的情况下，非组织周期性麦克斯韦系统的均质化
Homogenization of nonstationary periodic Maxwell system in the case of constant permeability
论文作者
论文摘要
在$ l_2（{\ Mathbb r}^3; {\ Mathbb c}^3）$中，我们考虑一个自adjoint Opitiator $ {\ Mathcal l} _ \ Varepsilon $，$ \ VAREPSILON> 0 η（\ MathBf {X}/\ Varepsilon）^{ - 1} \ operatorName {curl}μ_0^{ - 1/2} - 1/2} - μ_0^{1/2} \ nabla μ_0^{1/2} $，其中$μ_0$是一个恒定的正矩阵，矩阵值函数$η（\ Mathbf {x}）$，而实用函数$ν（\ sathbf {x}）$与某些lattice，正面确定和界限。我们研究操作员值函数的行为$ \ cos（τ{\ Mathcal l} _ \ varepsilon^{1/2}）$和$ {\ Mathcal l} _ \ varepsilon^{ - 1/2} { - 1/1/2} \ sin（-1/1/1/1/1/1/1/1/2} \ sin（ $τ\ in {\ mathbb r} $和小$ \ varepsilon $。结果表明，这些运算符将操作员$ {\ MATHCAL L}^0 $的相应运算符值函数收集到从Sobolev Space $ H^S $（合适的$ S $）到$ L_2 $的运算符的规范中。这里$ {\ Mathcal L}^0 $是具有恒定系数的有效运算符。另外，在$（h^s \ to H^1）$的校正中的近似值 - 运算符$ {\ Mathcal l} _ \ varepsilon^{ - 1/2} \ sin（τ{\ Mathcal l} _ \ varepsilon^{1/2}）我们证明了误差估计并研究了有关操作员规范类型以及估计值对$τ$的依赖性的结果。如果磁性渗透性等于$μ_0$，则将结果应用于非平稳麦克斯韦系统的cauchy问题的均质化，并且介电渗透效率由矩阵$η（\ Mathbf {x}/\ varepsilon）给出。
In $L_2({\mathbb R}^3;{\mathbb C}^3)$, we consider a selfadjoint operator ${\mathcal L}_\varepsilon$, $\varepsilon >0$, given by the differential expression $μ_0^{-1/2}\operatorname{curl} η(\mathbf{x}/\varepsilon)^{-1} \operatorname{curl} μ_0^{-1/2} - μ_0^{1/2}\nabla ν(\mathbf{x}/\varepsilon) \operatorname{div} μ_0^{1/2}$, where $μ_0$ is a constant positive matrix, a matrix-valued function $η(\mathbf{x})$ and a real-valued function $ν(\mathbf{x})$ are periodic with respect to some lattice, positive definite and bounded. We study the behavior of the operator-valued functions $\cos (τ{\mathcal L}_\varepsilon^{1/2})$ and ${\mathcal L}_\varepsilon^{-1/2} \sin (τ{\mathcal L}_\varepsilon^{1/2})$ for $τ\in {\mathbb R}$ and small $\varepsilon$. It is shown that these operators converge to the corresponding operator-valued functions of the operator ${\mathcal L}^0$ in the norm of operators acting from the Sobolev space $H^s$ (with a suitable $s$) to $L_2$. Here ${\mathcal L}^0$ is the effective operator with constant coefficients. Also, an approximation with corrector in the $(H^s \to H^1)$-norm for the operator ${\mathcal L}_\varepsilon^{-1/2} \sin (τ{\mathcal L}_\varepsilon^{1/2})$ is obtained. We prove error estimates and study the sharpness of the results regarding the type of the operator norm and regarding the dependence of the estimates on $τ$. The results are applied to homogenization of the Cauchy problem for the nonstationary Maxwell system in the case where the magnetic permeability is equal to $μ_0$, and the dielectric permittivity is given by the matrix $η(\mathbf{x}/\varepsilon)$.
