Homogenization of parabolic and elliptic periodic operators in $L_2(\mathbb {R}^d)$ with the first and second correctors taken into account
Elizaveta Vasilevskaya, Tatiana Aleksandrovna Suslina · St Petersburg Mathematical Journal · 2013
In the space $L_2(\mathbb {R}^d;{\mathbb C}^n)$, a wide class of matrix elliptic second order differential operators (DO’s) $\mathcal {A}_\varepsilon$ is studied; the $\mathcal {A}_\varepsilon$ are assumed to admit a factorization of the form $\mathcal {A}_\varepsilon = \mathcal {X}_\varepsilon ^* \mathcal {X}_\varepsilon$, where $\mathcal {X}_\varepsilon$ is a homogeneous first order DO. The coefficients of these operators are periodic and depend on $\mathbf {x}/\varepsilon$, $\varepsilon >0$. The behavior of the operator exponential $e^{-\mathcal {A}_\varepsilon \tau }$, $\tau >0$, and of the resolvent $({\mathcal {A}}_\varepsilon +I)^{-1}$ for small $\varepsilon$ is investigated. An approximation for the exponential $e^{-\mathcal {A}_\varepsilon \tau }$ in the operator norm in $L_2(\mathbb {R}^d; \mathbb {C}^n)$ with an error term of order $\tau ^{-3/2}\varepsilon ^3$ is obtained. For the resolvent $({\mathcal {A}}_\varepsilon +I)^{-1}$, approximation in the norm of operators acting from $H^1(\mathbb {R}^d; \mathbb {C}^n)$ to $L_2( \mathbb {R}^d; \mathbb {C}^n)$ is found with an error term of order $\varepsilon ^3$. In these approximations, the first and second order correctors are taken into account.