On the Geometric Properties of the Poisson Kernel for the Lamé Equation

Abstract

It is shown that the Poisson kernel for the Lamé equation in a disk can be interpreted as a bi-univalent mapping of the projection of an elliptic cone onto the projection of the surface of revolution of a hyperbola. The corresponding mapping \({{f}_{\sigma }}\) of these surfaces is bijective. Such an interpretation sheds light on the nature of the well-known special property of solutions of elliptic systems on a plane to map points to curves and vice versa. In particular, a singular point of the kernel under study can be considered as the projection of the cone element so that the mapping \({{f}_{\sigma }}\) is regular in the sense that this element is bijectively mapped into a curve.