ON THE NEUMANN BOUNDARY PROBLEM IN A DOMAIN WITH COMPLICATED BOUNDARY
E J Hruslov · Mathematics of the USSR-Sbornik · 1970
The second boundary value problem is studied for a Helmholtz equation in a domain G(n), which is the complement of a strongly disconnected set F(n), contained in a neighborhood of a fixed surface Γ. An approximate description of a solution u(n)(x) of this problem is based on the study of the sequence {u(n)(x), n = 1, 2,⋯} of solutions corresponding to a sequence {F(n)} such that for n→∞ the set F(n) becomes infinitely close to Γ and becomes increasingly disconnected. The sets F(n) are characterized by the notion of conductivity, introduced in this paper. Necessary and sufficient conditions are given (in terms of conductivity) for the existence of a function v(x) as a limit of the sequence {u(n)(x)} for n→∞ such that it satisfies the same conditions outside Γ, and on Γ the conjugacy conditions of the form where the limits of functions from different sides of Γ are indicated by the signs + and –; ν is the normal to Γ. Bibliography: 7 titles. One figure.