Abstract

An elliptic fourth-order differential operator \(A_\varepsilon\) on \(L_2(\mathbb{R}^d;\mathbb{C}^n)\) is studied. Here \(\varepsilon >0\) is a small parameter. It is assumed that the operator is given in the factorized form \(A_\varepsilon = b(\mathbf{D})^* g(\mathbf{x}/\varepsilon) b(\mathbf{D})\), where \(g(\mathbf{x})\) is a Hermitian matrix-valued function periodic with respect to some lattice and \(b(\mathbf{D})\) is a matrix second-order differential operator. We make assumptions ensuring that the operator \(A_\varepsilon\) is strongly elliptic. The following approximation for the resolvent \((A_\varepsilon + I)^{-1}\) in the operator norm of \(L_2(\mathbb{R}^d;\mathbb{C}^n)\) is obtained:

$$(A_{\varepsilon}+I)^{-1}=(A^{0}+I)^{-1}+\varepsilon K_{1}+\varepsilon^{2} K_{2}(\varepsilon)+O(\varepsilon^{3}).$$

Here \(A^0\) is the effective operator with constant coefficients and \(K_{1}\) and \(K_{2}(\varepsilon)\) are certain correctors.

中文翻译：

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考虑修正的具有周期系数的四阶椭圆算子的均质化

摘要

研究了\（L_2（\ mathbb {R} ^ d; \ mathbb {C} ^ n）\）上的椭圆四阶微分算子\（A_ \ varepsilon \）。这里\（\ varepsilon> 0 \）是一个小参数。假设以因子分解形式\（A_ \ varepsilon = b（\ mathbf {D}）^ * g（\ mathbf {x} / \ varepsilon）b（\ mathbf {D}）\）给出运算符，其中\（g（\ mathbf {x}）\）是关于某个晶格周期性的Hermitian矩阵值函数，而\（b（\ mathbf {D}）\）是矩阵二阶微分算子。我们做一些假设来确保运算符\（A_ \ varepsilon \）是强椭圆的。解析物\（（A_ \ varepsilon + I）^ {-1} \）的以下近似值在\（L_2（\ mathbb {R} ^ d; \ mathbb {C} ^ n）\）的算子范数中得到：

$$（A _ {\ varepsilon} + I）^ {-1} =（A ^ {0} + I）^ {-1} + \ varepsilon K_ {1} + \ varepsilon ^ {2} K_ {2}（ \ varepsilon）+ O（\ varepsilon ^ {3}）。$$

在这里，\（A ^ 0 \）是具有恒定系数的有效算子，并且\（K_ {1} \）和\（K_ {2}（\ varepsilon）\）是某些校正器。