Homogenization of Green and Neumann Functions

TL;DR

This paper investigates the asymptotic behavior of Green and Neumann functions for elliptic operators with periodic coefficients, providing expansions and convergence rates relevant for boundary value problems.

Contribution

It introduces new asymptotic expansions and convergence results for Green and Neumann functions in the context of oscillating periodic coefficients.

Findings

Asymptotic expansions of Green and Neumann functions derived.

Near optimal convergence rates in $W^{1,p}$ for boundary value solutions.

Explicit asymptotic behavior of Poisson kernels and Dirichlet-to-Neumann maps.

Abstract

For a family of second-order elliptic operators with rapidly oscillating periodic coefficients, we study the asymptotic behavior of the Green and Neumann functions, using Dirichlet and Neumann correctors. As a result we obtain asymptotic expansions of Poisson kernels and the Dirichlet-to-Neumann maps as well as near optimal convergence rates in $W^{1,p}$ for solutions with Dirichlet or Neumann boundary conditions.