Vol. 1, No. 1, 2008

Download this article
Download this article For screen
For printing
Recent Issues

Volume 14
Issue 6, 1671–1976
Issue 5, 1333–1669
Issue 4, 985–1332
Issue 3, 667–984
Issue 2, 323–666
Issue 1, 1–322

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
 
Other MSP Journals
Noncommutative variations on Laplace's equation

Jonathan Rosenberg

Vol. 1 (2008), No. 1, 95–114
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.

Keywords
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
Milestones
Received: 27 February 2008
Revised: 19 March 2008
Accepted: 14 July 2008
Published: 2 October 2008
Authors
Jonathan Rosenberg
Department of Mathematics
University of Maryland
College Park, MD 20742-4015
United States
http://www.math.umd.edu/~jmr