Homogenization of elliptic and parabolic equations with periodic coefficients in a bounded domain under the Neumann condition

Abstract

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 a selfadjoint matrix elliptic second-order differential operator $B_{N,\varepsilon}$, $0<\varepsilon\leqslant1$, with the Neumann boundary condition. The principal part of this operator is given in a factorized form. The operator involves first-order and zero-order terms. The coefficients of $B_{N,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. We study the generalized resolvent $(B_{N,\varepsilon}-\zeta Q_0( {\cdot} /\varepsilon))^{-1}$, where $Q_0$ is a periodic bounded and positive definite matrix-valued function, and $\zeta$ is a complex-valued parameter. We obtain approximations for the generalized resolvent in the operator norm on $L_2(\mathcal{O};\mathbb{C}^n)$ and 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 two-parametric error estimates (with respect to $\varepsilon$ and $\zeta$). The results are applied to study the behaviour of the solutions of the initial boundary value problem with the Neumann condition for the parabolic equation $Q_0(\mathbf{x}/\varepsilon) \partial_t \mathbf{u}_\varepsilon(\mathbf{x},t) = -(B_{N,\varepsilon} \mathbf{u}_\varepsilon)(\mathbf{x},t)$ in a cylinder $\mathcal{O} \times (0,T)$, where $0<T \le \infty$.