Asymptotics of the Solution of the Dirichlet Problem for the Laplace Equation in a Strip with Thin Branches

The paper studies the asymptotic behavior of the solution of the Dirichlet problem for the Laplace operator in a domain obtained from an infinite horizontal strip by attaching a vertical infinite half-strip of small width. Using the methods of potential theory, the problem is reduced to an integral equation on the boundary of the domain. The Schwarz alternating method is applied to the resulting equation in an appropriate Banach space. The solution is expressed in terms of “reflection operators.” The formula for one of such operators can be obtained only under additional restrictions on the right-hand side of the equation, which require certain weight norms to be finite.