Homogenization of nonlocal convolution type operators: Approximation for the resolvent with corrector

2023/11/28 by Piatnitski, Andrei, Sloushch, Vladimir, Suslina, Tatiana +1 · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Primary 35B27 #Secondary 45E10

paper · doi:10.48550/arxiv.2311.16574

Abstract

In L₂(ℝ^d), we consider a selfadjoint bounded operator \mathbb A_ε, ε >0, of the form (\mathbb A_ε u) (x) = ε^-d-2 ∫_ℝ^d a((x - y )/ ε ) μ(x /ε, y /ε) ( u(x) - u(y) ) dy. It is assumed that a(x) is a nonnegative function of class L₁(ℝ^d) such that \hboxa(-x) = a(x) and μ(x,y) is ℤ^d-periodic in each variable and such that μ(x,y) = μ(y,x) and 0< μ_- \leqslant μ(x,y) \leqslant μ₊< ∞. Moreover, it is assumed that the moments M_k (a)= ∫_ℝ^d | x |^k a(x) dx, k=1,2,3,4, are finite. We obtain approximation of the resolvent (\mathbb A_ε + I)^-1 for small ε in the operator norm on L₂(ℝ^d) with error of order O(ε²).

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• Homogenization of non-symmetric convolution type operators

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