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
