Title: On the Klein-Gordon equation on the 3-torus Authors: Krausshar, Rolf SörenConstales, Denis # Issue Date: Jul-2009 Publisher: Bauhaus Universitaet Weimar Host Document: 18th International Conference on the Application of Computer Science and Mathematics in Architecure and Civil Engineering Conference: 18th International Conference on the Application of Computer Science and Mathematics in Architecure and Civil Engineering edition:18 location:Weimar date:7-9 July 2009 Article number: 106 Abstract: We consider the time independent Klein-Gordon equation $(\Delta-\alpha^2) u= 0$ ($\alpha \in \mathbb{R})$ on some conformally flat $3$-tori with given boundary data. We set up an explicit formula for the fundamental solution. We show that we can represent any solution to the homogeneous Klein-Gordon equation on the torus as finite sum over generalized $3$-fold periodic elliptic functions that are in the kernel of the Klein-Gordon operator. Furthermore we prove Cauchy and Green type integral formulas and set up a Teodorescu and Cauchy transform for the toroidal Klein-Gordon operator. These in turn are used to set up explicit formulas for the solution to the inhomogeneous Klein-Gordon equation $(\Delta-\alpha^2) u= f$ on the $3$-torus. Description: The proceedings will be published electronically on a CD and will be put online on http://euklid.bauing.uni-weimar.de/ikm2009-cd (Paper size: 10 pages) Publication status: submitted KU Leuven publication type: IC Appears in Collections: Analysis Section