Homogenization of the stationary Maxwell system with periodic coefficients in a bounded domain

In a bounded domain $\mathcal{O}\subset\mathbb{R}^3$ of class $C^{1,1}$, we consider a stationary Maxwell system with the perfect conductivity boundary conditions. It is assumed that the dielectric permittivity and the magnetic permeability are given by $\eta({\mathbf x}/ \varepsilon )$ and $\mu({\mathbf x}/ \varepsilon )$, where $\eta( {\mathbf x})$ and $\mu({\mathbf x})$ are symmetric $(3 \times 3)$-matrix-valued functions; they are periodic with respect to some lattice, bounded and positive definite. Here $\varepsilon >0$ is the small parameter. We use the following notation for the solutions of the Maxwell system: ${\mathbf u}_\varepsilon$ is the electric field intensity, ${\mathbf v}_\varepsilon$ is the magnetic field intensity, ${\mathbf w}_\varepsilon$ is the electric displacement vector, and ${\mathbf z}_\varepsilon$ is the magnetic displacement vector. It is known that ${\mathbf u}_\varepsilon$, ${\mathbf v}_\varepsilon$, ${\mathbf w}_\varepsilon$, and ${\mathbf z}_\varepsilon$ weakly converge in $L_2({\mathcal O})$ to the corresponding homogenized fields ${\mathbf u}_0$, ${\mathbf v}_0$, ${\mathbf w}_0$, and ${\mathbf z}_0$ (the solutions of the homogenized Maxwell system with the effective coefficients), as $\varepsilon \to 0$. We improve the classical results and find approximations for ${\mathbf u}_\varepsilon$, ${\mathbf v}_\varepsilon$, ${\mathbf w}_\varepsilon$, and ${\mathbf z}_\varepsilon$ in the $L_2({\mathcal O})$-norm. The error terms do not exceed $C \sqrt{\varepsilon} (\| {\mathbf q}\|_{L_2}+\|{\mathbf r}\|_{L_2})$, where the divergence free vector-valued functions ${\mathbf q}$ and ${\mathbf r}$ are the right-hand sides of the Maxwell equations.Comment: 42 pages. arXiv admin note: text overlap with arXiv:1810.1132

Homogenization of a stationary periodic Maxwell system in a bounded domain in the case of constant magnetic permeability

In a bounded domain $\mathcal{O}\subset\mathbb{R}^3$ of class $C^{1,1}$, we consider a stationary Maxwell system with the boundary conditions of perfect conductivity. It is assumed that the magnetic permeability is given by a constant positive $(3\times 3)$-matrix $\mu_0$ and the dielectric permittivity is of the form $\eta({\mathbf x}/ \varepsilon)$, where $\eta({\mathbf x})$ is a $(3 \times 3)$-matrix-valued function with real entries, periodic with respect to some lattice, bounded and positive definite. Here $\varepsilon >0$ is the small parameter. Suppose that the equation involving the curl of the magnetic field intensity is homogeneous, and the right-hand side $\mathbf r$ of the second equation is a divergence-free vector-valued function of class $L_2$. It is known that, as $\varepsilon \to 0$, the solutions of the Maxwell system, namely, the electric field intensity ${\mathbf u}_\varepsilon$, the electric displacement vector ${\mathbf w}_\varepsilon$, the magnetic field intensity ${\mathbf v}_\varepsilon$, and the magnetic displacement vector ${\mathbf z}_\varepsilon$ weakly converge in $L_2$ to the corresponding homogenized fields ${\mathbf u}_0$, ${\mathbf w}_0$, ${\mathbf v}_0$, ${\mathbf z}_0$ (the solutions of the homogenized Maxwell system with effective coefficients). We improve the classical results. It is shown that ${\mathbf v}_\varepsilon$ and ${\mathbf z}_\varepsilon$ converge to ${\mathbf v}_0$ and ${\mathbf z}_0$, respectively, in the $L_2$-norm, the error terms do not exceed $C \varepsilon \| {\mathbf r}\|_{L_2}$. We also find approximations for ${\mathbf v}_\varepsilon$ and ${\mathbf z}_\varepsilon$ in the energy norm with error $C\sqrt{\varepsilon} \|{\mathbf r}\|_{L_2}$. For ${\mathbf u}_\varepsilon$ and ${\mathbf w}_\varepsilon$ we obtain approximations in the $L_2$-norm with error $C\sqrt{\varepsilon} \| {\mathbf r}\|_{L_2}$.Comment: 28 page

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

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 $A_{N,\varepsilon}$ of order $2p$ given by the expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon) b({\mathbf D})$, $\varepsilon >0$, 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 $m\geqslant n$ and that the symbol $b({\boldsymbol \xi})$ has maximal rank for any $0 \ne {\boldsymbol \xi}\in {\mathbb C}^d$. We find approximations for the resolvent $\left(A_{N,\varepsilon}-\zeta I \right)^{-1}$ 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 $\varepsilon$ and $\zeta$.Comment: 34 pages. arXiv admin note: text overlap with arXiv:1702.0055

Homogenization of hyperbolic equations with periodic coefficients

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$ we consider selfadjoint strongly elliptic second order differential operators ${\mathcal A}_\varepsilon$ with periodic coefficients depending on ${\mathbf x}/ \varepsilon$, $\varepsilon>0$. We study the behavior of the operator cosine $\cos( {\mathcal A}^{1/2}_\varepsilon \tau)$, $\tau \in \mathbb{R}$, for small $\varepsilon$. Approximations for this operator in the $(H^s\to L_2)$-operator norm with a suitable $s$ are obtained. The results are used to study the behavior of the solution ${\mathbf v}_\varepsilon$ of the Cauchy problem for the hyperbolic equation $\partial^2_\tau {\mathbf v}_\varepsilon = - \mathcal{A}_\varepsilon {\mathbf v}_\varepsilon +\mathbf{F}$. General results are applied to the acoustics equation and the system of elasticity theory.Comment: 88 pages. arXiv admin note: substantial text overlap with arXiv:1508.0764

Homogenization of high order elliptic operators with periodic coefficients

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, we study a selfadjoint strongly elliptic operator $A_\varepsilon$ of order $2p$ given by the expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon) b({\mathbf D})$, $\varepsilon >0$. Here $g({\mathbf x})$ is a bounded and positive definite $(m\times m)$-matrix-valued function in ${\mathbb R}^d$; it is assumed that $g({\mathbf x})$ is periodic with respect to some lattice. Next, $b({\mathbf D})=\sum_{|\alpha|=p}^d b_\alpha {\mathbf D}^\alpha$ is a differential operator of order $p$ with constant coefficients; $b_\alpha$ are constant $(m\times n)$-matrices. It is assumed that $m\ge n$ and that the symbol $b({\boldsymbol \xi})$ has maximal rank. For the resolvent $(A_\varepsilon - \zeta I)^{-1}$ with $\zeta \in {\mathbb C} \setminus [0,\infty)$, we obtain approximations in the norm of operators in $L_2({\mathbb R}^d;{\mathbb C}^n)$ and in the norm of operators acting from $L_2({\mathbb R}^d;{\mathbb C}^n)$ to the Sobolev space $H^p({\mathbb R}^d;{\mathbb C}^n)$, with error estimates depending on $\varepsilon$ and $\zeta$.Comment: 53 page

Homogenization of the Neumann problem for elliptic systems with periodic coefficients

Let ${\mathcal O} \subset {\mathbb R}^d$ be a bounded domain with the boundary of class $C^{1,1}$. In $L_2({\mathcal O};{\mathbb C}^n)$, a matrix elliptic second order differential operator ${\mathcal A}_{N,\varepsilon}$ with the Neumann boundary condition is considered. Here $\varepsilon>0$ is a small parameter, the coefficients of ${\mathcal A}_{N,\varepsilon}$ are periodic and depend on ${\mathbf x} /\varepsilon$. There are no regularity assumptions on the coefficients. It is shown that the resolvent $({\mathcal A}_{N,\varepsilon}+\lambda I)^{-1}$ converges in the $L_2({\mathcal O};{\mathbb C}^n)$-operator norm to the resolvent of the effective operator ${\mathcal A}_N^0$ with constant coefficients, as $\varepsilon \to 0$. A sharp order error estimate $|({\mathcal A}_{N,\varepsilon}+\lambda I)^{-1} - ({\mathcal A}_{N}^0 +\lambda I)^{-1}|_{L_2\to L_2} \le C\varepsilon$ is obtained. Approximation for the resolvent $({\mathcal A}_{N,\varepsilon}+\lambda I)^{-1}$ 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 an error $O(\sqrt{\varepsilon})$ is found. Approximation is given by the sum of the operator $({\mathcal A}^0_N +\lambda I)^{-1}$ and the first order corrector. In a strictly interior subdomain ${\mathcal O}'$ a similar approximation with an error $O(\varepsilon)$ is obtained.Comment: 54 pages. arXiv admin note: text overlap with arXiv:1201.214

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

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 $\zeta$.Comment: 75 pages. arXiv admin note: text overlap with arXiv:1212.114

Homogenization of the higher-order Schr\"odinger-type equations with periodic coefficients

In $L_2({\mathbb R}^d; {\mathbb C}^n)$, we consider a matrix strongly elliptic differential operator ${A}_\varepsilon$ of order $2p$, $p \geqslant 2$. The operator ${A}_\varepsilon$ is given by ${A}_\varepsilon = b(\mathbf{D})^* g(\mathbf{x}/\varepsilon) b(\mathbf{D})$, $\varepsilon >0$, where $g(\mathbf{x})$ is a periodic, bounded, and positive definite matrix-valued function, and $b(\mathbf{D})$ is a homogeneous differential operator of order $p$. We prove that, for fixed $\tau \in {\mathbb R}$ and $\varepsilon \to 0$, the operator exponential $e^{-i \tau {A}_\varepsilon}$ converges to $e^{-i \tau {A}^0}$ in the norm of operators acting from the Sobolev space $H^s({\mathbb R}^d; {\mathbb C}^n)$ (with a suitable $s$) into $L_2({\mathbb R}^d; {\mathbb C}^n)$. Here $A^0$ is the effective operator. Sharp-order error estimate is obtained. The results are applied to homogenization of the Cauchy problem for the Schr\"odinger-type equation $i \partial_\tau {\mathbf u}_\varepsilon = {A}_\varepsilon {\mathbf u}_\varepsilon + {\mathbf F}$, ${\mathbf u}_\varepsilon\vert_{\tau=0} = \boldsymbol{\phi}$.Comment: 22 page

Homogenization of nonstationary periodic Maxwell system in the case of constant permeability

In $L_2({\mathbb R}^3;{\mathbb C}^3)$, we consider a selfadjoint operator ${\mathcal L}_\varepsilon$, $\varepsilon >0$, given by the differential expression $\mu_0^{-1/2}\operatorname{curl} \eta(\mathbf{x}/\varepsilon)^{-1} \operatorname{curl} \mu_0^{-1/2} - \mu_0^{1/2}\nabla \nu(\mathbf{x}/\varepsilon) \operatorname{div} \mu_0^{1/2}$, where $\mu_0$ is a constant positive matrix, a matrix-valued function $\eta(\mathbf{x})$ and a real-valued function $\nu(\mathbf{x})$ are periodic with respect to some lattice, positive definite and bounded. We study the behavior of the operator-valued functions $\cos (\tau {\mathcal L}_\varepsilon^{1/2})$ and ${\mathcal L}_\varepsilon^{-1/2} \sin (\tau {\mathcal L}_\varepsilon^{1/2})$ for $\tau \in {\mathbb R}$ and small $\varepsilon$. It is shown that these operators converge to the corresponding operator-valued functions of the operator ${\mathcal L}^0$ in the norm of operators acting from the Sobolev space $H^s$ (with a suitable $s$) to $L_2$. Here ${\mathcal L}^0$ is the effective operator with constant coefficients. Also, an approximation with corrector in the $(H^s \to H^1)$-norm for the operator ${\mathcal L}_\varepsilon^{-1/2} \sin (\tau {\mathcal L}_\varepsilon^{1/2})$ is obtained. We prove error estimates and study the sharpness of the results regarding the type of the operator norm and regarding the dependence of the estimates on $\tau$. The results are applied to homogenization of the Cauchy problem for the nonstationary Maxwell system in the case where the magnetic permeability is equal to $\mu_0$, and the dielectric permittivity is given by the matrix $\eta(\mathbf{x}/\varepsilon)$.Comment: 29 page

Homogenization of hyperbolic equations with periodic coefficients in Rd{\mathbb R}^d: sharpness of the results

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, a selfadjoint strongly elliptic second order differential operator ${\mathcal A}_\varepsilon$ is considered. It is assumed that the coefficients of the operator ${\mathcal A}_\varepsilon$ are periodic and depend on ${\mathbf x}/\varepsilon$, where $\varepsilon >0$ is a small parameter. We find approximations for the operators $\cos ( {\mathcal A}_\varepsilon^{1/2}\tau)$ and ${\mathcal A}_\varepsilon^{-1/2}\sin ( {\mathcal A}_\varepsilon^{1/2}\tau)$ in the norm of operators acting from the Sobolev space $H^s({\mathbb R}^d)$ to $L_2({\mathbb R}^d)$ (with suitable $s$). We also find approximation with corrector for the operator ${\mathcal A}_\varepsilon^{-1/2}\sin ( {\mathcal A}_\varepsilon^{1/2}\tau)$ in the $(H^s \to H^1)$-norm. The question about the sharpness of the results with respect to the type of the operator norm and with respect to the dependence of estimates on $\tau$ is studied. The results are applied to study the behavior of the solutions of the Cauchy problem for the hyperbolic equation $\partial_\tau^2 {\mathbf u}_\varepsilon = - {\mathcal A}_\varepsilon {\mathbf u}_\varepsilon + {\mathbf F}$.Comment: 95 page