Vol. 1, No. 1, 2008

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ISSN: 1948-206X (e-only)
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Noncommutative variations on Laplace's equation

Jonathan Rosenberg

Vol. 1 (2008), No. 1, 95–114

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.

noncommutative geometry, irrational rotation algebra, elliptic partial differential equations, maximum principle, calculus of variations, harmonic maps, Leray–Schauder Theorem, meromorphic functions
Mathematical Subject Classification 2000
Primary: 58B34
Secondary: 58J05, 35J05, 35J20, 30D30, 46L87
Received: 27 February 2008
Revised: 19 March 2008
Accepted: 14 July 2008
Published: 2 October 2008
Jonathan Rosenberg
Department of Mathematics
University of Maryland
College Park, MD 20742-4015
United States