The Existence And Uniqueness Of The Solution For Mixed Boundary Value Problem Of Helmholtz Equation

We consider the direct and inverse scattering problems for partially coated obsta-cles.To this end,we first use the method of integral equations of the first kind together with variational methods to solve a scattering problem for the Helmholtz equation where the scattered field satisfies mixed Neumann-Robin boundary conditions. We consider it like this:Interior mixed boundary value problemObviously,if we can know the whole Dirichlet Cauchy data or Neumann Cauchy data of the solution on the whole boundaryΓ,the existence of the solution can be gotten from[1].For this,we use the following method which referred to [8]:By single- and double- layer potential theories and Green's formula,we first reformulate the interior mixed boundary problem(*)as a 2×2 system of boundary integral equation of the first kind which is equivalent to our original interior mixed boundary value problem in some senses.(see[6]).0nce the unknown Cauchy data are determined from the 2×2 system of boundary integral equation of the first kind, the representation of formula(8) determines the unique weak solution.Our proof can be divided into two parts. In the first part we use the boundary integral equation theory to prove the existence of the solution for problem (*). In the second part we introduce and prove the lemma which was used in the first part .