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arXiv · 1006.0026

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

Sa'ar Hersonsky
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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.

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Sa'ar Hersonsky. 2010-05-31. Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, II: The mixed Dirichlet-Neumann Problem. https://arxiv.org/abs/1006.0026

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