Noncommutative variations on Laplace's equation

TL;DR

This paper develops a foundational theory for noncommutative nonlinear elliptic PDEs by analyzing noncommutative Laplace equations on tori, establishing analogues of classical analysis results and methods.

Contribution

It introduces noncommutative analogues of Laplace's equation and classical analysis theorems, extending key methods to the noncommutative setting.

Findings

Proved noncommutative versions of Wiener's Theorem.

Established existence and non-existence results for noncommutative elliptic equations.

Demonstrated that classical PDE methods have noncommutative analogues.

Abstract

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.