Algebras which satisfy a second order linear partial differential equation

Herbert Stanley Bear, Jr. and Gerald Norman Hile

Vol. 75 (1978), No. 1, 21–36

DOI: 10.2140/pjm.1978.75.21

Abstract

Let A be an algebra of complex valued functions satisfying a second order linear partial differential equation in a plane domain. If the equation is hyperbolic or parabolic, the functions of A are locally functions of only one variable. If the equation is elliptic, there exists a unique complex function λ such that f_x = λf_y for each f in A, and after a change of variables each function in A is analytic. If an algebra of functions satisfies the maximum principle, and one nonconstant function and its square satisfy an elliptic equation, then every function in the algebra satisfies this equation.

Mathematical Subject Classification 2000

Primary: 46J10

Secondary: 35G05

Milestones

Received: 24 March 1977

Revised: 16 May 1977

Published: 1 March 1978