一类Helmholtz方程解的存在性和唯一性

Existence and Uniqueness of a Solution to the Helmholtz Equation with Mixed Boundary Conditions

声波的散射问题中,散射体由两部分组成:里面是一个不可穿透的障碍物,外面是一条可穿透的裂缝.不可穿透障碍物由两部分组成,不同部分的边界条件不同.通过位势理论可以将此混合问题转化为边界积分方程.通过Fredholm算子理论可以得到这个边界积分方程解的存在性和唯一性,从而获得原问题解的存在性和唯一性.

In the acoustic scattering problems,a scattering problem will occur when an incident wave meets an impenetrable obstacle after crossing a penetrable crack first. If the impenetrable obstacle consists of two parts,this will lead to two different boundary conditions. We can reformulate this problem as a boundary integral system by using potential theory. We obtain the existence and uniqueness of the weak solution to the system by using the fredholm theory.