On operator-type homogenization estimates for elliptic equations with lower order terms

Abstract:

In the space $\mathbb {R}^d$, a divergent-type second order elliptic equation in a nonselfadjoint form is studied. The coefficients of the equation oscillate with a period $\varepsilon {\to } 0$. They can be unbounded in lower order terms. In this case, they are subordinate to some integrability conditions over the unit periodicity cell. An $L^2$-estimate of order of $O(\varepsilon )$ is proved for the difference of solutions of the original and homogenized problems. The estimate is of operator type. It can be stated as an estimate for the difference of the corresponding resolvents in the operator ($L^2(\mathbb {R}^d){\to } L^2(\mathbb {R}^d)$)-norm. Also, an $H^1$-approximation is found for the original solution with error estimate of order of $O(\varepsilon )$. This estimate, also of operator type, implies that an appropriate approximation of order of $O(\varepsilon )$ is found for the original resolvent in the operator ($L^2(\mathbb {R}^d){\to } H^1(\mathbb {R}^d)$)-norm.

The results are obtained with the help of the so-called shift method, first suggested by V. V. Zhikov.