Boundary value problems on planar graphs and flat surfaces with integer cone singularities, I: The dirichlet problem

Sa'ar Hersonsky

Research output: Contribution to journalReview articlepeer-review

12 Scopus citations

Abstract

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 genus (m - 1) singular flat surface tiled by rectangles and f is an energy preserving mapping from T (1) onto S. By a singular flat surface, we will mean a surface which carries a metric structure locally modeled on the Euclidean plane, except at a finite number of points. These points have cone singularities, and the cone angle is allowed to take any positive value (see for instance [28] for an excellent survey). Our realization may be considered as a discrete uniformization of planar bounded regions.

Original languageEnglish
Pages (from-to)65-92
Number of pages28
JournalJournal fur die Reine und Angewandte Mathematik
Issue number670
DOIs
StatePublished - 1 Sep 2012
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics

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