INTEGRAL EQUATION METHOD FOR BOUNDARY VALUE PROBLEMS IN MULTIPLY CONNECTED DOMAINS FOR THE TWO-DIMENSIONAL LAPLACE EQUATION

We consider Dirichlet and Neumann boundary value problems for the two-dimensional Laplace equation in multiply-connected domain bounded by two smooth closed curves. The solutions of this problems we present as a sum of potentials of double layer with unknown densities. Existence and uniqueness of solutions of the posed problems in appropriate functional spaces is proved. Using integral representation of solutions of the initial boundary value problems we obtain some systems of boundary integral and singular integro-differential equations. Inasmuch the obtained systems have not unique solutions we consider some approach based on modified system of boundary equations which have unique solutions. As a result we get densities of integral representations of the solutions of the Dirichlet and Neumann boundary value problems which satisfies some additional integral conditions.

Automated summary

This paper discusses Dirichlet and Neumann boundary value problems for the two-dimensional Laplace equation in a multiply-connected domain enclosed by two smooth closed curves. The solutions are presented as a sum of potentials of double layer with unknown densities, and existence and uniqueness of solutions in appropriate functional spaces are proved. By using integral representation of solutions, systems of boundary integral and singular integro-differential equations are obtained, and a modified system of boundary equations with unique solutions is considered to determine densities of the solutions satisfying additional integral conditions.