Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, II: The mixed DirichletNeumann Problem
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 2cell 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 $(m1)$ 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 DirichletNeumann problem.
 Publication:

arXiv eprints
 Pub Date:
 May 2010
 arXiv:
 arXiv:1006.0026
 Bibcode:
 2010arXiv1006.0026H
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Geometric Topology;
 53C43;
 57M50
 EPrint:
 26 pages, 16 figures (color)