On Helmholtz Equation With Mixed Boundary Conditions

In the acoustic scattering problems, a scattering problem will happen when an incident wave meet an impenetrable obstacle after crossing a penetrable obstacle first. It can be converted to the following mixed boundary value problem of the Helmholtz equation.We consider it specific like this:HereΓis a crack,the boundary ofΩis consisted of S₁ and S₂,which isαΩ= S₁∪S₂,the n denotes the unit normal to the corresponding boundary,u is total wave.uⁱ is incident wave, d denotes the incident direction, besides, the scattering wave must meet Sommerfeld's radiation condition.In the some Sobolev space, we gave the definition of the weak solution of the (2). When k andλsatisfy some condition, the (2) has the unique weak solution by the Riesz theory and the Green formula if we bring in the operator of the Dirichlet to Neumann.Our proof can be divided into two parts. In the first part, we prove the uniqueness and the existence of the solution by the Riesz theory and the Green formula. In the second part, we prove the existence of the area derivative.