The treatment of window problems by transform methods (Q1315054)

The authors consider boundary value problems for the Laplace equation in two- or threedimensional domains of the following type: The solution is zero on the boundary everywhere (including infinity, if present in the boundary) excluding a bounded interval (2D) or a flat bounded part of the surface (in 3D). This is the window, and here a second kind condition is given (perhaps with an oblique derivative, which however is excluded to be tangential). For continuous data, the full problem can now be reduced to a singular or weakly singular integral equation for a certain density on the window. Having this density the full solution can be obtained from a further integral equation (simple in 2D, fourfold in 3D). In case of a bounded 3D domain, a Teodorescu transform is used and a system of integral equations obtained which contain no singular integrals over the window. Similar representations of the solution are derived for the linear equations of elasticity. Finally the authors mention the possibility to obtain a numerical solution by collocation.