On the integral equation method for the plane mixed boundary value problem of the Laplacian

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Abstract
Although the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ2and given Neumann data on the remaining part Γ2of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solutionuon Γ2and the also desired values of the normal derivatie on Γ1. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity ofubut, on the other hand, gradupossesses singularities at the collision points Γ1∩ Γ2even forC∞data,uis decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ2providing an improvement of regularity for the smooth part ofu. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ1and Γ2augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm ofuand also superconvergence as well as high orders in smoother norms if the given data are smooth (and not the solution).