  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.
