Noncommutative variations on Laplace's equation

Details Noncommutative variations on Laplace's equation Noncommutative variations on Laplace's equation

2008-02-27

arxiv.org/abs/0802.4033v3

Anal. PDE 1 (2008), no. 1, 95-114

Mathematics

Operator Algebras

21 pages, with proof of Theorem 2.8 corrected; will appear in Analysis & PDE

Scientific paper

As a first step at developing a theory of noncommutative nonlinear elliptic partial differential equations, we analyze noncommutative analogues of Laplace's equation and its variants (some of the them nonlinear) over noncommutative tori. Along the way we prove noncommutative analogues of many results in classical analysis, such as Wiener's Theorem on functions with absolutely convergent Fourier series, and standard existence and non-existence theorems on elliptic functions. We show that many many classical methods, including the Maximum Principle, the direct method of the calculus of variations, the use of the Leray-Schauder Theorem, etc., have analogues in the noncommutative setting.

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