The Neumann problem for the equation Δ𝑢-𝑘²𝑢=0 in the exterior of non-closed Lipschitz surfaces

Abstract

We study the Neumann problem for the equation Δ u − k 2 u = 0 \Delta u - k^2u=0 in the exterior of non-closed Lipschitz surfaces in R 3 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.