The Liouville Theorem for a Quasi-Linear Elliptic Partial Differential Equation

Authors

Elwood Bohn, Bowling Green State University
Lloyd K. Jackson, University of Nebraska-Lincoln

Date of this Version

1962

Citation

Published in Transactions of the American Mathematical Society, Vol. 104, No. 3. (Sep., 1962), pp. 392-397.

Comments

Copyright © 1962 American Mathematical Society. Used by permission.

Abstract

The classical Liouville Theorem of analytic function theory can be stated in either of two equivalent forms: The Liouville Theorem states: If f(w) is analytic and bounded throughout the finite w-plane, then f(w) is constant. If z(x, y) is a real valued function of the real variables x and y which is a solution of z_xx + z_yy = 0 and is bounded either above or below throughout the finite plane, then z(x, y) is a constant. Here we are concerned with the question of whether or not the second formulation of the above theorem is valid for solutions of more general elliptic partial differential equations.