We provide a comprehensive study on the asymptotic solutions of an interface problem corresponding to an elliptic partial differential equation with Dirichlet boundary condition and transmission condition, subject to the small geometric perturbation and/or the high contrast ratio of the conductivity. All asymptotic terms can be solved in the unperturbed reference domains, which significantly reduces computations in practice, especially for random perturbations. Our setting is quite general and allows two types of elliptic problems: the perturbation of the domain boundary where the Dirchlet condition is imposed and the perturbation of the interface where the transmission condition is imposed. As the perturbation size and the ratio of the conductivities tends to zero, the two-parameter asymptotic expansions on the reference domain are derived to any order after the single parameter expansions are solved beforehand. The results suggest the emergence of the Neumann or Robin boundary condition, depending on the relation of the two asymptotic parameters. Our method is the classic asymptotic analysis techniques but in a new unified approach to both problems.

中文翻译：

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高对比度介质中具有小扰动的椭圆方程组的渐近分析

我们提供了一个关于具有Dirichlet边界条件和透射条件的椭圆型偏微分方程的界面问题的渐近解的综合研究，该问题受较小的几何扰动和/或高电导率的影响。可以在不受干扰的参考域中求解所有渐近项，这实际上减少了计算量，尤其是对于随机扰动而言。我们的设置非常笼统，并允许两种椭圆问题：施加Dirchlet条件的域边界的扰动和施加传输条件的接口的扰动。由于扰动大小和电导率之比趋于零，事先解决了单参数展开后，参考域上的两参数渐近展开可以按任意顺序导出。结果表明，取决于两个渐近参数的关系，出现了诺伊曼或罗宾边界条件。我们的方法是经典的渐近分析技术，但是采用了新的统一方法来解决这两个问题。