论文标题
用$ {\ Mathbb r}^d $定期系数的双曲程方程的均质化：结果的清晰度
Homogenization of hyperbolic equations with periodic coefficients in ${\mathbb R}^d$: sharpness of the results
论文作者
论文摘要
在$ l_2（{\ Mathbb r}^d; {\ Mathbb c}^n）$中，一个自偏椭圆相关的第二阶二阶差异操作员$ {\ mathcal a} _ \ varepsilon $。假定运算符$ {\ Mathcal a} _ \ Varepsilon $的系数是定期的，并且取决于$ {\ Mathbf x}/\ Varepsilon $，其中$ \ varepsilon> 0 $是一个小参数。我们找到了操作员的近似值$ \ cos（{\ Mathcal a} _ \ varepsilon^{1/2}τ）$和$ {\ Mathcal a} _ \ varepsilon^{ - 1/1/2} \ sin（{\ Mathcal a} _} _ _ v varepsilon^under nord nord nord in nord in nord从sobolev space $ h^s（{\ mathbb r}^d）$到$ l_2（{\ mathbb r}^d）$（带有合适的$ s $）。我们还发现操作员的矫正器$ {\ mathcal a} _ \ varepsilon^{ - 1/2} \ sin（{\ Mathcal a} _ \ varepsilon^{1/2}τ）in $（h^s \ to h^1）研究了有关结果相对于操作员规范的类型以及估计值对$τ$的依赖性的问题。结果适用于研究双曲程方程的解决方案的行为$ \partial_τ^2 {\ Mathbf u} _ \ varepsilon = - {\ Mathcal a} _ \ varepsilon {\ Varepsilon {
In $L_2({\mathbb R}^d;{\mathbb C}^n)$, a selfadjoint strongly elliptic second order differential operator ${\mathcal A}_\varepsilon$ is considered. It is assumed that the coefficients of the operator ${\mathcal A}_\varepsilon$ are periodic and depend on ${\mathbf x}/\varepsilon$, where $\varepsilon >0$ is a small parameter. We find approximations for the operators $\cos ( {\mathcal A}_\varepsilon^{1/2}τ)$ and ${\mathcal A}_\varepsilon^{-1/2}\sin ( {\mathcal A}_\varepsilon^{1/2}τ)$ in the norm of operators acting from the Sobolev space $H^s({\mathbb R}^d)$ to $L_2({\mathbb R}^d)$ (with suitable $s$). We also find approximation with corrector for the operator ${\mathcal A}_\varepsilon^{-1/2}\sin ( {\mathcal A}_\varepsilon^{1/2}τ)$ in the $(H^s \to H^1)$-norm. The question about the sharpness of the results with respect to the type of the operator norm and with respect to the dependence of estimates on $τ$ is studied. The results are applied to study the behavior of the solutions of the Cauchy problem for the hyperbolic equation $\partial_τ^2 {\mathbf u}_\varepsilon = - {\mathcal A}_\varepsilon {\mathbf u}_\varepsilon + {\mathbf F}$.
