Boundary value problems on planar graphs and flat surfaces with integer cone singularities, II: The mixed Dirichlet-Neumann problem

In this paper we continue the study started in Hersonsky (in press) [16]. We consider a planar, bounded, m-connected region Ω, and let ∂Ω be its boundary. Let T be a cellular decomposition of Ω∪∂Ω, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair (S,f) where S is a special type of a (possibly immersed) genus (m−1) singular flat surface, tiled by rectangles and f is an energy preserving mapping from T(1) onto S. In Hersonsky (in press) [16] the solution of a Dirichlet problem defined on T(0) was utilized, in this paper we employ the solution of a mixed Dirichlet-Neumann problem.