On resolvent approximations of elliptic differential operators with periodic coefficients

Abstract

We consider resolvents $(A_{ϵ}+1{)}^{-1}$ of elliptic second-order differential operators $A_{ϵ}=-\text{div}a\left(x/ϵ\right)\mathrm{\nabla }$ in ${\mathbb{R}}^d$ with ε-periodic measurable matrix $a\left(x/ϵ\right)$ and study the asymptotic behaviour of $(A_{ϵ}+1{)}^{-1}$, as the period ε goes to zero. We provide a construction for the leading terms of the ‘operator asymptotics’ of $(A_{ϵ}+1{)}^{-1}$ in the sense of $L^2$-operator-norm convergence and prove order ${ϵ}^2$ remainder estimates. We apply the modified method of the first approximation with the usage of Steklov's smoothing. The class of operators covered by our analysis includes uniformly elliptic families with bounded coefficients and also with unbounded coefficients from the John–Nirenberg space BMO (bounded mean oscillation).

Keywords:

• Error estimate
• operator norm resolvent approximations
• corrector

MSC2010:

• 35J05

Disclosure statement

No potential conflict of interest was reported by the author.