四分之一平面的顶点格林函数。函数方程、加法交叉与Lamé函数之间的联系
Vertex Green's functions of a quarter-plane. Links between the functional equation, additive crossing and Lamé functions
摘要：
在我们以前的工作中（Assier \& Shanin，QJMAM，2019），我们给出了一个新的光谱公式，与四分之一平面的平面波衍射问题相关的两个复变数。特别地，我们证明了未知谱函数满足关于其分支集的加性交叉的条件。在本文中，我们研究了一类非常相似的谱问题，并展示了如何利用加法交叉来表达其解的Lamé函数。得到的解决方案可以被认为是定制的顶点格林函数，其行为在近场直接相关的拉普拉斯-贝尔特拉米算子的本征值。这是重要的，因为正确的近场行为的尖端的四分之一平面迄今为止从未获得通过多变量复分析的方法。
英文摘要：
In our previous work (Assier \& Shanin, QJMAM, 2019), we gave a new spectral formulation in two complex variables associated with the problem of plane-wave diffraction by a quarter-plane. In particular, we showed that the unknown spectral function satisfies a condition of additive crossing about its branch set. In this paper, we study a very similar class of spectral problem, and show how the additive crossing can be exploited in order to express its solution in terms of Lamé functions. The solutions obtained can be thought of as tailored vertex Green's functions whose behaviours in the near-field are directly related to the eigenvalues of the Laplace-Beltrami operator. This is important since the correct near-field behaviour at the tip of the quarter-plane had so far never been obtained via a multivariable complex analysis approach.
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