The Dirichlet Problem for the Equation Δu−k2u=0 in the Exterior of Nonclosed Lipschitz Surfaces

Abstract

We study the Dirichlet problem for the equation Δu−k2u=0 in the exterior of nonclosed Lipschitz surfaces in R3. The Dirichlet problem for the Laplace equation is a particular case of our problem. Theorems on existence and uniqueness of a weak solution of the problem are proved. The integral representation for a solution is obtained in the form of single-layer potential. The density in the potential is defined as a solution of the operator (integral) equation, which is uniquely solvable.