On operator estimates in homogenization of nonlocal operators of convolution type

Abstract

The paper studies a bounded symmetric operator Aε in L2(Rd) with(Aεu)(x)=ε−d−2∫Rda((x−y)/ε)μ(x/ε,y/ε)(u(x)−u(y))dy; here ε is a small positive parameter. It is assumed that a(x) is a non-negative L1(Rd) function such that a(−x)=a(x) and the moments Mk=∫Rd|x|ka(x)dx, k=1,2,3, are finite. It is also assumed that μ(x,y) is Zd-periodic both in x and y function such that μ(x,y)=μ(y,x) and 0<μ−⩽μ(x,y)⩽μ+<∞. Our goal is to study the limit behaviour of the resolvent (Aε+I)−1, as ε→0. We show that, as ε→0, the operator (Aε+I)−1 converges in the operator norm in L2(Rd) to the resolvent (A0+I)−1 of the effective operator A0 being a second order elliptic differential operator with constant coefficients of the form A0=−divg0∇. We then obtain sharp in order estimates of the rate of convergence.