在航空领域, 涡轮机叶片作为航空发动机的重要组成部分, 往往处于极端高温高压的环境中. 这种环境也导致了涡轮叶片寿命短, 易损坏. 在二十世纪, 人们采用“热障涂层”来保护轮机叶片, 这一做法显著的延长了涡轮机叶片的寿命. 热障涂层的厚度以及其上的热传导系数都远远小于叶片, 人们也称其为“薄层”. 这时要想数值计算整体区域的热传导方程需要在薄层中打很细的网格, 这是十分耗时的过程, 也无法直接看出薄层所起的作用. 解决这些问题的一个办法是刻画薄层厚度趋于零时热传导方程的极限解所满足的方程以及边界条件 (实效边条件), 这就将薄层看成了无厚度的曲线或者曲面. 随后我们在涡轮机叶片区域就可方便地数值解热传导方程, 也可通过实效边条件看出薄层所起到的作用.
  在这种思想的指导下, 大量学者进行了不同情形下实效边条件的数学研究, 这些研究可以广泛应用于弹性力学、电磁学和声学等的薄层问题. Li 和 Rosencrans等人[13] 研究了在薄层厚度均匀时线性热传导方程的实效边条件. Li 和 Zhang[14]研究了周期振荡薄层情形下的 Poisson 方程对应的实效边条件. 本文的研究也是在此成果的基础上展开的.
  本学位论文主要研究带有周期振荡薄层区域上的热传导方程 Dirichlet 问题. 这是基于二维周期振荡薄层开展的研究, 是对薄层为固定厚度情形下热传导方程的实效边条件的推广. 此前对二维周期振荡薄层的研究停留在 Poisson 方程层面, 没有引入时间维度. 在本研究中, 我们构建了二维周期振荡薄层的数学模型, 研究了在此情况下区域的热传导方程. 通过构造试验函数的方法, 刻画了极限解满足的方程及边界条件.

Turbine blades, as an important part of aeroengines in the aviation sector, are often exposed to extreme high temperature and high pressure environment. This environment also leads to a short life of turbine blades that are easily damaged. In the twentieth century, people adopted ”thermal barrier coating”to protect turbine blades significantly extends turbine blade life. The thickness of the thermal barrier coating and the thermal conductivity coefficient on it are much smaller than the blade, which is also known as the ”thin layer”. At this time, to numerically calculate the heat equation of the overall region, we need to lay a fine mesh in the thin layer, which is a very time-consuming process, and it is impossible to directly see the role of the thin layer. One solution to these problems is to characterize the equation and boundary condition (effective boundary condition) satisfies the limit solution of the heat equation as the thickness of the thin layer tends to zero. Thin layer is seen as curve or surface without thickness. Then we can easily numerically depyretic the heat equation in the turbine blade area, and we can also see the role of thin layer through the effective boundary condition. 
Under the guidance of this idea, a large number of scholars have carried out mathematical studies of effective boundary condition in different situations, which can be widely used in thin layer problems such as elastic mechanics, electromagnetism and acoustics. Li and Rosencrans[6] studied the effective boundary condition of  linear heat conduction equation with uniform thickness of thin layers. Li and Zhang [7] studied the effective boundary condition corresponding to Poisson equation in the case of periodically oscillating thin layer. The research of this thesis is also carried out on the basis of this achievement. 
This thesis focuses on the Dirichlet problem for the heat equation on periodically oscillating thin layer. This is based on the two-dimensional periodically oscillating thin layer, which is a generalization of the effective boundary condition of the heat equation when the thin layer is fixed thickness. Previous studies of periodically oscillating thin layer have stayed at the level of the Poisson equation without introducing the temporal dimension. In this thesis, we construct a mathematical model of thin layer of two-dimensional periodic oscillation and study the heat equation of the region in this case. By constructing the test function, the equation and boundary condition satisfied by the limit solution are described.