二维薄层介质中中频TM模式电场的严格渐近性
Rigorous Asymptotics For The Elecric Field In TM Mode At Mid-Frequency In A Bidimensional Medium With Thin Layer
摘要：
考虑环境介质和由薄膜包围的二维材料组成的异质实体。这些材料的电磁常数是不同的。通过与生物细胞类比，我们称这种实体为细胞。研究了薄膜厚度趋于零时横磁模中电场的渐近行为。膜的厚度为hf（s），其中s为曲线坐标。我们提供了一个严格的推导的前两项的渐近展开h趋于零。在膜中，这些项在局部坐标中根据边界数据和函数f明确给出，而在膜外部，它们是标量亥姆霍兹方程的解，其具有根据边界数据和上述函数f明确给出的适当边界和传输条件。我们证明了余项的阶为O（h<sup>3/2</sup>）.此外，如果薄膜中的复介电常数z_m比h更快地趋于零，我们给出了精确解与上述z_m = 0时的渐近解之间的差，其阶数为O（h^{3/2}+ |z_m|）.
英文摘要：
Consider an ambient medium and a heterogeneous entity composed of a bidimensional material surrounded by a thin membrane. The electromagnetic constants of these materials are different. By analogy with biological cells, we call this entity a cell. We study the asymptotic behavior of the electric field in the transverse magnetic (TM) mode, when the thickness of the membrane tends to zero. The membrane is of thickness of hf(s), with s a curvilinear coordinate. We provide a rigorous derivation of the first two terms of the asymptotic expansion for h tending to zero. In the membrane, these terms are given explicitly in local coordinates in terms of the boundary data and of the function f, while outside the membrane they are the solutions of a scalar Helmholtz equation with appropriate boundary and transmissions conditions given explicitly in terms of the boundary data and of the above function f. We prove that the remainder terms are of order O(h<sup>3/2</sup>). In addition, if the complex dielectric permittivity in the membrane, denoted by z_m, tends to zero faster than h, we give the difference between the exact solution and the above asymptotic with z_m = 0; it is of order O(h^{3/2}+|z_m|).
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