Existence And Uniqueness Of A Solution To The Helmholtz Equation With Mixed Boundary Conditions

We consider the direct and scattering problems for the obstacles with two different boundaries. One is a crack , the other is a closed and partially coated curve. The scattered field satisfies mixed Dirichlet-impedance boundary conditions on the smooth boundary.We consider this Exterior mixed boundaries value problem:We use the Rellich's lemma to solve the uniqueness of the exterior problem. For the existence , we referred to [3]. By the single- and double- layer potential theories and Green formula, we first reformulate the exterior mixed boundary problem(*)as a 3×3 system of boundary integral equation of the first kind which is equivalent to our original exterior mixed boundary value problem in some senses. (see[18]). Once the unknown Cauchy data are determined from the 3×3 system of boundary integral equation of the first kind, then the representation formula determines the only weak solution.Our proof can be divided into two parts. In the first part we use the Rellich's lemma to prove the uniqueness, In the second part we use the Fredholm theory of the boundary integral equation to prove the existence of the solution for problem (*).