半圆域内的二维线性椭圆偏微分方程
TWO DIMENSIONAL LINEAR ELLIPTIC PDES IN A SEMI-DISK

本文研究了半圆域内的二维线性椭圆偏微分方程.利用Fokas提出的求解凸多边形区域内的线性椭圆偏微分方程的变换方法,我们改进了这个方法来研究半圆域内Laplace方程,修改Helmholtz方程和Helmholtz方程的解,并且导出了这些方程解的积分表达式,讨论了Helmholtz方程的广义Dirichlet到Neumann映射.
In this paper, two dimensional linear linear elliptic in a semi-disk are considered. By using the effective approach by Fokas to solve the linear elliptic PDEs in convex polygonal domain, we improve this method to study the boundary value problems for Laplace, Helmholtz and modified Helmholtz equations in a semi-disk domain. The integral representations for the solutions of these elliptic PDEs are derived. The generalized Dirichlet to Neumann map for the Helmholtz equation is investigated