Homogenization of the Neumann problem for higher-order elliptic equations with periodic coefficients

Abstract

Let $\mathcal{O}\subset {\mathbb{R}}^d$ be a bounded domain of class $C^{2p}$. In $L_2(\mathcal{O};{\mathbb{C}}^n)$, we study a selfadjoint strongly elliptic operator $A_{N,\epsilon }$ of order $2p$ given by the expression $b(\mathbf{D}{)}^{\ast }g(\mathbf{x}/\epsilon )b(\mathbf{D})$, $\epsilon >0$, with the Neumann boundary conditions. Here $g(\mathbf{x})$ is a bounded and positive definite $(m\times m)$-matrix-valued function in ${\mathbb{R}}^d$, periodic with respect to some lattice; $b(\mathbf{D})={\sum }_{|\alpha |=p}{b}_{\alpha }{\mathbf{D}}^{\alpha }$ is a differential operator of order $p$ with constant coefficients; $b_{\alpha }$ are constant $(m\times n)$-matrices. It is assumed that $m⩾n$ and that the symbol $b(\mathbf{\xi })$ has maximal rank for any $0\ne \mathbf{\xi }\in {\mathbb{C}}^d$. We find approximations for the resolvent ${\left(A_{N,\epsilon }-\zeta I\right)}^{-1}$ in the $L_2(\mathcal{O};{\mathbb{C}}^n)$-operator norm and in the norm of operators acting from $L_2(\mathcal{O};{\mathbb{C}}^n)$ to the Sobolev space $H^p(\mathcal{O};{\mathbb{C}}^n)$, with error estimates depending on $\epsilon$ and $\zeta$.

34 pages. arXiv admin note: text overlap with arXiv:1702.00550

Cited by in corpus (2)

• Uniform Boundary Estimates in Homogenization of Higher Order Elliptic Systems
• Convergence Rates and Interior Estimates in Homogenization of Higher Order Elliptic Systems