The maximum principle and biharmonic functions

Abstract

This note concerns the maximum principle which applies to solutions of partial differential equations of elliptic type. This principle asserts that the maximum of a solution occurs on the boundary of a region. Consideration of the ratio of solutions of an elliptic equation shows that the ratio satisfies the same maximum principle. This result is then used to obtain a maximum principle relating to biharmonic functions. These maximum principles give inequalities which biharmonic functions must satisfy. The relations and concepts developed in this note have application in elasticity and in hydrodynamics.