The Neumann problem for the dissipative Helmholtz equation in the exterior of open Lipschitz surfaces

Abstract

We study the Neumann problem for the dissipative Helmholtz equation in the exterior of open Lipschitz surfaces in R ³. 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 a double layer potential. The density in the potential is defined as a solution of the operator (integral) equation, which is uniquely solvable.

Keywords:

• theory of potentials
• integral equation method
• boundary value problem
• cracks

AMS Subject Classification:

• 35J05
• 35J25
• 31A10
• 31A25