Homogenization of elliptic problems: error estimates in dependence of the spectral parameter

TL;DR

This paper provides error estimates for homogenization of strongly elliptic differential operators with periodic coefficients, analyzing how the spectral parameter influences approximation accuracy in various operator norms.

Contribution

It introduces new error estimates for the homogenization of elliptic operators, explicitly depending on the spectral parameter and boundary conditions.

Findings

Error estimates depend on the spectral parameter and epsilon

Approximate resolvents in different norms with explicit error bounds

Results apply to Dirichlet and Neumann boundary conditions

Abstract

We consider a strongly elliptic differential expression of the form $b(D)^* g(x/\varepsilon) b(D)$, $\varepsilon >0$, where $g(x)$ is a matrix-valued function in ${\mathbb R}^d$ assumed to be bounded, positive definite and periodic with respect to some lattice; $b(D)=\sum_{l=1}^d b_l D_l$ is the first order differential operator with constant coefficients. The symbol $b(\xi)$ is subject to some condition ensuring strong ellipticity. The operator given by $b(D)^* g(x/\varepsilon) b(D)$ in $L_2({\mathbb R}^d;{\mathbb C}^n)$ is denoted by $A_\varepsilon$. Let ${\mathcal O} \subset {\mathbb R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2({\mathcal O};{\mathbb C}^n)$, we consider the operators $A_{D,\varepsilon}$ and $A_{N,\varepsilon}$ given by $b(D)^* g(x/\varepsilon) b(D)$ with the Dirichlet or Neumann boundary conditions, respectively. For the resolvents of the operators $A_\varepsilon$, $A_{D,\varepsilon}$, and $A_{N,\varepsilon}$ in a regular point $\zeta$ we find approximations in different operator norms with error estimates depending on $\varepsilon$ and the spectral parameter $\zeta$.