On~the Neumann boundary problem in a~domain with complicated boundary

E. Ya. Khruslov

Sbornik. Mathematics, Tome 12 (1970) no. 4, pp. 553-571

Résumé

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 $\Gamma$. An approximate description of a solution $u^{(n)}(x)$ of this problem is based on the study of the sequence $\{u^{(n)}(x),n=1,2,\dots\}$ of solutions corresponding to a sequence $\{F^{(n)}\}$ such that for $n\to\infty$ the set $F^{(n)}$ becomes infinitely close to $\Gamma$ 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 $n\to\infty$ such that it satisfies the same conditions outside $\Gamma$, and on $\Gamma$ the conjugacy conditions of the form $$ \biggl(\frac{\partial v}{\partial\nu}\biggr)_+=\biggl(\frac{\partial v}{\partial\nu}\biggr)_-=p(x)[v_+-v_-], $$ where the limits of functions from different sides of $\Gamma$ are indicated by the signs $+$ and $-$; $\nu$ is the normal to $\Gamma$. Figure: 1. Bibliography: 7 titles.