Convergence Rates in L^2 for Elliptic Homogenization Problems

TL;DR

This paper investigates the convergence rates of solutions and eigenvalues in L^2 for elliptic systems with rapidly oscillating coefficients, providing new results even for smooth domains.

Contribution

It establishes new convergence rate results in L^2 for elliptic homogenization problems, extending previous uniform estimates to eigenvalues and less regular domains.

Findings

Derived L^2 convergence rates for solutions of elliptic systems

Established convergence rates for Dirichlet, Neumann, and Steklov eigenvalues

Results are new even for smooth domains

Abstract

We study rates of convergence of solutions in L^2 and H^{1/2} for a family of elliptic systems {L_\epsilon} with rapidly oscillating oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {L_\epsilon}. Most of our results, which rely on the recently established uniform estimates for the L^2 Dirichlet and Neumann problems in \cite{12,13}, are new even for smooth domains.