中文摘要：

考虑边界含尖点的有界区域上Helmholtz方程即△u＋k^2u=0的Dirichlet问题解的存在及唯一性．对于边界是光滑的情况，利用位势理论将问题转化为第二类边界积分方程，由Fredholm选择性定理可得到其Dirichlet问题解的存在及唯一性．但对于边界含尖点的有界区域，由于尖点处法向导数不连续，上述方法会遇到困难．可以通过对位势跳跃条件作相应修正来克服这一困难．从而得到边界含尖点区域上Helmholtz方程Dirichlet问题解的存在及唯一性．

英文摘要：

This paper consideres the existence and uniqueness of solution for Dirichlet problem of the Helmholtz equation on the domain with corners. For smooth boundary, using potential theory to transform the problem into second kind boundary integral equation, and using Fredholm alternative theorem, it obtains the existence and uniqueness of Dirichlet problem. For bounded domain with corners, the normal derivative is not continuous, which brings some additional difficults. This paper uses modified potential jump relations to overcome the difficults. Thereby it obtains the existence and uniqueness for the Dirichlet problem of the Helmholtz equation on domain with corners.