An Integral Equation Method for a Problem with Mixed Boundary Conditions

Abstract

Consider the problem of finding a complex function $f = u + iv$, which is holomorphic in a given domain, with the real part u taking prescribed values on one part of the boundary, and the imaginary part v taking prescribed values on the remainder of the boundary. (This is essentially equivalent to solving Laplace’s equation subject to mixed Dirichlet and Neumann boundary conditions.) By virtue of the Cauchy integral formula, the unknown boundary values of u and v satisfy a $2 \times 2$ system of singular integral equations, which can be solved in a certain class of weighted $L_p $ spaces, by applying a result of I. Gohberg and N. Krupnik. The values of both u and v are then known over the whole of the boundary, and so f can be computed using the Cauchy integral formula.