振荡介质Maxwell方程透射特征值的下界估计

摘要

内部透射特征值问题在研究非均匀介质的逆散射问题中起着至关重要的作用，它源于入射波关于非均匀介质的散射问题。主要研究带有小周期振荡系数的Maxwell方程组的透射特征值问题，此模型在物理上对应于电磁波在周期微结构介质中的传播。给出了该类数学问题对应的电磁场散射的物理背景，明确了透射特征值实际上是一类特殊频率，使得该频率的入射波在给定的介质中不产生散射波和透射波。对此重要的应用问题，给出了正透射特征值的下界估计。%The interior transmission eigenvalue problems ( ITEP ) are of fundamental importance in inverse scattering problems for inhomogeneous medium. Since the corresponding differential operator is neither elliptic nor self-adjoint,such kinds of problems have been concerned widely. This paper concerns with the ITEP for Maxwell's equations of oscillatory refraction index with small period. The physical backgrounds for such kinds of nonstandard problems are clarified,showing that the positive eigenvalues are in fact the wave frequencies of incident plan waves for which the scattered waves vanish. Then we give the lower bounds for the transmission eigenvalues of Maxwell′s equations with oscillatory coefficient.