Periodic and stochastic homogenization of general nonlocal operators with oscillating coefficients

TL;DR

This paper studies the homogenization of nonlocal operators with oscillating coefficients, including fractional Laplacians, in periodic and random settings, extending results to nonlinear equations.

Contribution

It establishes homogenization theorems for nonlocal operators with oscillating coefficients using $B3$-convergence and compactness, covering both linear and nonlinear cases.

Findings

Proved homogenization theorems for periodic and random nonlocal operators.

Extended homogenization results to general nonlinear nonlocal equations.

Utilized $B3$-convergence and compactness methods in proofs.

Abstract

This paper investigates homogenization problems for the nonlocal operators with rapidly oscillating coefficients in the cases of periodic and random statistically homogeneous micro-structures. These operators involve the fractional Laplacian and some operators compared to it. Based on the $\Gamma$-convergence method and compactness arguments, we prove the homogenization theorems for these nonlocal operators with product-type and symmetric coefficient-structured kernels respectively. Furthermore, these results are extended to general nonlinear nonlocal equations.