Homogenization of the Neumann problem for elliptic systems with periodic coefficients

TL;DR

This paper proves that solutions to elliptic systems with periodic coefficients and Neumann boundary conditions can be approximated by an effective operator as the period tends to zero, with explicit error estimates.

Contribution

It establishes the convergence of the Neumann problem's resolvent for elliptic systems with periodic coefficients and provides sharp error bounds, including interior estimates, without regularity assumptions on coefficients.

Findings

Resolvent converges in operator norm as epsilon approaches zero.

Error estimate of order epsilon for resolvent approximation.

Interior approximation with error of order epsilon.

Abstract

Let ${\mathcal O} \subset {\mathbb R}^d$ be a bounded domain with the boundary of class $C^{1,1}$. In $L_2({\mathcal O};{\mathbb C}^n)$, a matrix elliptic second order differential operator ${\mathcal A}_{N,\varepsilon}$ with the Neumann boundary condition is considered. Here $\varepsilon>0$ is a small parameter, the coefficients of ${\mathcal A}_{N,\varepsilon}$ are periodic and depend on ${\mathbf x} /\varepsilon$. There are no regularity assumptions on the coefficients. It is shown that the resolvent $({\mathcal A}_{N,\varepsilon}+\lambda I)^{-1}$ converges in the $L_2({\mathcal O};{\mathbb C}^n)$-operator norm to the resolvent of the effective operator ${\mathcal A}_N^0$ with constant coefficients, as $\varepsilon \to 0$. A sharp order error estimate $|({\mathcal A}_{N,\varepsilon}+\lambda I)^{-1} - ({\mathcal A}_{N}^0 +\lambda I)^{-1}|_{L_2\to L_2} \le C\varepsilon$ is obtained. Approximation for the resolvent $({\mathcal A}_{N,\varepsilon}+\lambda I)^{-1}$ in the norm of operators acting from $L_2({\mathcal O};{\mathbb C}^n)$ to the Sobolev space $H^1({\mathcal O};{\mathbb C}^n)$ with an error $O(\sqrt{\varepsilon})$ is found. Approximation is given by the sum of the operator $({\mathcal A}^0_N +\lambda I)^{-1}$ and the first order corrector. In a strictly interior subdomain ${\mathcal O}'$ a similar approximation with an error $O(\varepsilon)$ is obtained.