Convergence Rates in L^2 for Elliptic Homogenization Problems

Abstract

We study rates of convergence of solutions in L^2 and H^{1/2} for a family of elliptic systems {L_ε} 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_ε}. 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.

25 pages

References in corpus (2)

• Interior error estimate for periodic homogenization
• Error estimate and unfolding for periodic homogenization

Cited by in corpus (69)

• Homogenization of the elliptic Dirichlet problem: operator error estimates in $L_2$
• Lipschitz regularity for elliptic equations with random coefficients
• Homogenization and norm resolvent convergence for elliptic operators in a strip perforated along a curve
• Convergence Rates in Periodic Homogenization of Systems of Elasticity
• Resolvent estimates for high-contrast elliptic problems with periodic coefficients
• Uniform Lipschitz Estimates in Bumpy Half-Spaces
• Norm-resolvent convergence of one-dimensional high-contrast periodic problems to a Kronig-Penney dipole-type model
• Precised approximations in elliptic homogenization beyond the periodic setting
• Boundary Korn Inequality and Neumann Problems in Homogenization of Systems of Elasticity
• Optimal convergence rates for elliptic homogenization problems in nondivergence-form: analysis and numerical illustrations
• Uniform boundary regularity in almost-periodic homogenization
• Slow convergence in periodic homogenization problems for divergence type elliptic operators
• Quantitative Estimates on Periodic Homogenization of Nonlinear Elliptic Operators
• Homogenization of the stationary Maxwell system with periodic coefficients in a bounded domain
• Eigenvalue homogenization for quasilinear elliptic operators
• Estimates of eigenvalues and eigenfunctions in elliptic homogenization with rapidly oscillating potentials
• Sharp operator-norm asymptotics for thin elastic plates with rapidly oscillating periodic properties
• Uniform Regularity Estimates in Parabolic Homogenization
• Boundary Layers in Periodic Homogenization of Neumann Problems
• Homogenization of elliptic problems: error estimates in dependence of the spectral parameter
• Commutator Estimates for the Dirichlet-to-Neumann Map in Lipschitz Domains
• On homogenization of the first initial-boundary value problem for periodic hyperbolic systems
• The Methods of Layer Potentials for General Elliptic Homogenization Problems in Lipschitz Domains
• Convergence rate for quasilinear eigenvalue homogenization
• Eigenvalue homogenization for quasilinear elliptic equations with different boundary conditions
• Eigenvalue homogenization for quasilinear elliptic operators in one space dimension
• Homogenization of initial boundary value problems for parabolic systems with periodic coefficients
• Homogenization of the Neumann problem for higher-order elliptic equations with periodic coefficients
• Quantitative Estimates in Homogenization of Parabolic Systems of Elasticity in Lipschitz Cylinders
• Weighted $L^2$ Estimates for Elliptic Homogenization in Lipschitz Domains
• Characterizations of diffusion matrices in homogenization of elliptic equations in nondivergence-form
• Quantitative estimates for homogenization of nonlinear elliptic operators in perforated domains
• Convergence Rates of Neumann problems for Stokes Systems
• Convergence Rates in Parabolic Homogenization with Time-Dependent Periodic Coefficients
• Lipschitz estimates in almost-periodic homogenization
• Convergence rates on periodic homogenization of p-Laplace type equations
• Uniform Regularity Estimates in Homogenization Theory of Elliptic System with Lower Order terms
• Limiting distribution of elliptic homogenization error with periodic diffusion and random potential
• Convergence rates in a weighted Fucik problem
• Homogenization of locally periodic parabolic operators with non-self-similar scales
• Sharp Convergence Rates for Darcy's Law
• Convergence rates for general elliptic homogenization problems in a bounded Lipschitz domain
• Applications of Fourier analysis in homogenization of Dirichlet problem II. $L^p$ estimates
• $L^p$ Neumann Problems in Homogenization of General Elliptic Systems
• Nodal Sets and Doubling Conditions in Elliptic Homogenization
• Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates
• Convergence Rates and Interior Estimates in Homogenization of Higher Order Elliptic Systems
• Homogenization of Boundary Value Problems in Perforated Lipschitz Domains
• suboptimal error estimates for homogenization of linear elasticity systems on perforated domains
• Resolvent estimates in homogenisation of periodic problems of fractional elasticity
• Convergence rates and $W^{1,p}$ estimates in homogenization theory of Stokes systems in Lipschitz domains
• Uniform Boundary Estimates in Homogenization of Higher Order Elliptic Systems
• Quantitative Estimates on Reiterated Homogenization of Linear Elliptic operators Using Fourier Transform Methods
• Eigenvalues homogenization for the fractional $p-$Laplacian operator
• Optimal Boundary Estimates for Stokes Systems in Homogenization Theory
• Quantitative Homogenization of Elliptic PDE with Random Oscillatory Boundary Data
• Random Sampling and Efficient Algorithms for Multiscale PDEs
• Convergence rate for the homogenization of stationary diffusions in dilutely perforated domains with reflecting boundaries
• Estimates of Eigenvalues and Eigenfunctions in Periodic Homogenization
• Error Estimates of Reiterated Stokes Systems via Fourier Transform Methods
• Leading and second order homogenization of an elastic scattering problem for highly oscillating anisotropic medium
• Convergence rates in homogenization of higher order parabolic systems
• A Concurrent Global-local Numerical Method for Multiscale PDEs
• Homogenization of the Neumann problem for elliptic systems with periodic coefficients
• Two-parametric error estimates in homogenization of second order elliptic systems in ${\mathbb{R}}^d$ including lower order terms
• Recent progress in the theory of homogenization with oscillating Dirichlet data
• Homogenization approach for the transmission eigenvalue problem for periodic media and application to the inverse problem
• High-contrast random composites: homogenisation framework and new spectral phenomena
• A Nitsche Hybrid multiscale method with non-matching grids