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

In L 2 ( R d ; C n ) L_2(R^d;C^n) , a strongly elliptic differential operator A ε {A}_ε of order 2 p 2p is studied. Its coefficients are periodic and depend on x / ε x/ε . The resolvent ( A ε + I ) − 1 ({A}_ε +I)⁻¹ is approximated in the operator norm on L 2 ( R d ; C n ) L_2(R^d;C^n) : ( A ε + I ) − 1 = ( A 0 + I ) − 1 + ∑ j = 1 2 p − 1 ε j K j , ε + O ( ε 2 p ) . {equation*} ({A}_ε +I)⁻¹ = ({A}^0+I)⁻¹ + ∑ ⱼ₌₁²ᵖ⁻¹ ε ʲ {K}j,ε + O(ε ²ᵖ). {equation*} Here A 0 {A}^0 is an effective operator with constant coefficients, and K j , ε {K}j,ε , j = 1 , … , 2 p − 1 j=1, ,2p-1 , are appropriate correctors.