Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates

摘要

Let $ mathcal {O}subset mathbb{R}^d$ be a bounded domain of class $ C^{1,1}$. In $ L_2(mathcal {O};mathbb{C}^n)$, a selfadjoint matrix second order elliptic differential operator $ B_{D,arepsilon }$, $ 0 0$, is studied as $ arepsilon ightarrow 0$. Approximations for the exponential $ e^{-B_{D,arepsilon }t}$ are obtained in the operator norm on $ L_2(mathcal {O};mathbb{C}^n)$ and in the norm of operators acting from $ L_2(mathcal {O};mathbb{C}^n)$ to the Sobolev space $ H^1(mathcal {O};mathbb{C}^n)$. The results are applied to homogenization of solutions of the first initial boundary-value problem for parabolic systems.