Operator estimates for homogenization of higher-order elliptic operators with periodic coefficients

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In L 2 ( R d ; C n ) L_2(\mathbb {R}^d;\mathbb {C}^n) , a strongly elliptic differential operator A ε {\mathcal {A}}_\varepsilon of order 2 p 2p is studied. Its coefficients are periodic and depend on x / ε \mathbf {x}/\varepsilon . The resolvent ( A ε + I ) − 1 ({\mathcal {A}}_\varepsilon +I)^{-1} is approximated in the operator norm on L 2 ( R d ; C n ) L_2(\mathbb {R}^d;\mathbb {C}^n) : ( A ε + I ) − 1 = ( A 0 + I ) − 1 + ∑ j = 1 2 p − 1 ε j K j , ε + O ( ε 2 p ) . \begin{equation*} ({\mathcal {A}}_\varepsilon +I)^{-1} = ({\mathcal {A}}^0+I)^{-1} + \sum _{j=1}^{2p-1} \varepsilon ^{j} {\mathcal {K}}_{j,\varepsilon } + O(\varepsilon ^{2p}). \end{equation*} Here A 0 {\mathcal {A}}^0 is an effective operator with constant coefficients, and K j , ε {\mathcal {K}}_{j,\varepsilon } , j = 1 , … , 2 p − 1 j=1,\dots ,2p-1 , are appropriate correctors.