We consider the spectrum of a class of positive, second-order elliptic systems of partial differential equations defined in the plane $ \mathbb{R}^2 $. The coefficients of the equation are assumed to have a special form, namely, they are doubly periodic and of high contrast. More precisely, the plane $ \mathbb{R}^2 $ is decomposed into an infinite union of the translates of the rectangular periodicity cell $ \Omega^0 $, and this in turn is divided into two components, on each of which the coefficients have different, constant values. Moreover, the second component of $ \Omega^0 $ consist of a neighborhood of the boundary of the cell of the width $ h $ and thus has an area comparable to $ h $, where $ h>0 $ is a small parameter.Using the methods of asymptotic analysis we study the position of the spectral bands as $ h \to 0 $ and in particular show that the spectrum has at least a given, arbitrarily large number of gaps, provided $ h $ is small enough.

中文翻译：

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砌体类型周期性介质中光谱的带隙结构

我们考虑在平面$ \ mathbb {R} ^ 2 $中定义的一类偏微分方程的正二阶椭圆系统的谱。假定方程的系数具有特殊形式，即，它们是双周期的并且具有高对比度。更准确地说，平面$ \ mathbb {R} ^ 2 $被分解为矩形周期性像元$ \ Omega ^ 0 $的平移的无限并集，而这又分为两个分量，每个分量系数具有不同的恒定值。此外，$ \ Omega ^ 0 $的第二个分量由宽度为$ h $的单元格边界的邻域组成，因此具有与$ h $相当的面积，其中$ h> 0 $是一个小参数。