Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, II: The mixed Dirichlet-Neumann Problem

TL;DR

This paper extends the study of boundary value problems on planar graphs and flat surfaces with cone singularities by solving a mixed Dirichlet-Neumann problem, constructing a singular flat surface and an energy-preserving map.

Contribution

It introduces a method to construct singular flat surfaces and energy-preserving maps using solutions to mixed boundary value problems on planar graphs.

Findings

Constructed a genus (m-1) singular flat surface from cellular decompositions.

Developed a canonical pair (S,f) linking graphs to flat surfaces.

Extended previous work to include mixed Dirichlet-Neumann boundary conditions.

Abstract

In this paper we continue the study started in part I (posted). We consider a planar, bounded, $m$-connected region $\Omega$, and let $\bord\Omega$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $\Omega\cup\bord\Omega$, 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 ${\mathcal T}^{(1)}$ onto $S$. In part I the solution of a Dirichlet problem defined on ${\mathcal T}^{(0)}$ was utilized, in this paper we employ the solution of a mixed Dirichlet-Neumann problem.