The Payne’s Maximum Principles

This note, completely based on the elegant paper of L.E. Payne [here], deals primarily with maximum principles for solutions of second order and fourth order elliptic equations. However, some of the results hold for arbitrary sufficiently smooth functions.

Throughout we assume to be a bounded domain in with sufficiently smooth boundary so that when necessary the governing differential equation will be satisfied on the boundary. In some of the applications we will need to be a surface but in most cases this excessive differentiality can be dispensed with.

We shall adopt the summation convention in which repeated Latin indices are to be summed from to , and we shall use the comma to denote differentiation. The symbol will be used for the normal derivative directed outward from on .

Following is the results

1. Inequalities based on the geometry of

We start with the maximum value of the gradient of a function whose normal derivative vanishes on a portion of .

Theorem I. Let have vanishing normal derivative on a portion of . Then if the Gaussian curvature o is everywhere positive the maximum value of can occur on if and only if in .

We also have the following result

Theorem II. Let vanish on a portion of . Then if the average curvature is positive at every point of the maximum value of

cannot occur on if in .