On Resolvent Approximations of Elliptic Differential Operators with Locally Periodic Coefficients

Abstract

We study the asymptotic behavior of the resolvents $$(A_{\varepsilon}+1)^{-1}$$ of elliptic second-order differential operators $$A_{\varepsilon}=-{\textrm{div}}a^{\varepsilon}(x)\nabla$$ in $$\mathbb{R}^{d}$$ with rapidly oscillating coefficients, as the small parameter $$\varepsilon$$ tends to zero. The matrix $$a^{\varepsilon}(x)=a(x,x/\varepsilon)$$ has the two-scale structure: it depends on the fast variable $$x/\varepsilon$$ and on the slow variable $$x$$ , with periodicity only in the fast variable. We provide a construction for the leading terms in the ‘‘operator asymptotics’’ of $$(A_{\varepsilon}+1)^{-1}$$ in the sense of $$L^{2}$$ -operator-norm convergence with order $$\varepsilon^{2}$$ remainder estimates. We apply the modified method of the first approximation with the usage of the shift proposed by V.V. Zhikov in Dokl. Math., 72:1 (2005).