Abstract
This survey paper deals with the unitary treatment of some classes of linear partial differential equations on Klein surfaces, which are the most general two-manifolds that support harmonic functions. We are mainly concerned with the study of harmonic functions with Dirichlet or Neumann boundary conditions. In such a way, the present paper extends several classical results to the abstract setting of dianalytic manifolds of complex dimension 1. We bring together systematically and concisely the Green and Neumann functions, the harmonic kernel function, and the harmonic measure on Klein surfaces. The technique to extend these concepts is to apply the classical methods to the complex double of a Klein surface, which is a Riemann surface endowed with an antianalytic involution. The existence of a harmonic function on a Riemann surface is given by the property that a harmonic function remains invariant under biholomorphic mappings. Similar problems on Riemann surfaces were developed by L.V. Ahlfors and L. Sario. The analysis developed in this paper offers perspectives to the qualitative analysis of other classes of linear or nonlinear elliptic equations on Klein surfaces.
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