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2008 Noncommutative variations on Laplace's equation
Jonathan Rosenberg
Anal. PDE 1(1): 95-114 (2008). DOI: 10.2140/apde.2008.1.95

Abstract

As a first step toward developing a theory of noncommutative nonlinear elliptic partial differential equations, we analyze noncommutative analogues of Laplace’s equation and its variants (some of 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 nonexistence theorems on elliptic functions. We show that many classical methods, including the maximum principle, the direct method of the calculus of variations, and the use of the Leray–Schauder Theorem, have analogues in the noncommutative setting.

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Jonathan Rosenberg. "Noncommutative variations on Laplace's equation." Anal. PDE 1 (1) 95 - 114, 2008. https://doi.org/10.2140/apde.2008.1.95

Information

Received: 27 February 2008; Revised: 19 March 2008; Accepted: 14 July 2008; Published: 2008
First available in Project Euclid: 20 December 2017

zbMATH: 1159.58005
MathSciNet: MR2444094
Digital Object Identifier: 10.2140/apde.2008.1.95

Subjects:
Primary: 58B34
Secondary: 30D30 , 35J05 , 35J20 , 46L87 , 58J05

Keywords: calculus of variations , elliptic partial differential equations , Harmonic Maps , irrational rotation algebra , Leray–Schauder Theorem , maximum principle , meromorphic functions , noncommutative geometry

Rights: Copyright © 2008 Mathematical Sciences Publishers

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