# Ngô Quốc Anh

## January 15, 2011

### The Payne’s Maximum Principles

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 15:34

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 $D$ to be a bounded domain in $\mathbb R^n$ with sufficiently smooth boundary $\partial D$ so that when necessary the governing differential equation will be satisfied on the boundary. In some of the applications we will need $\partial D$ to be a $C^{4+\alpha}$ 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 $1$ to $n$, and we shall use the comma to denote differentiation. The symbol $\partial/\partial \nu$ will be used for the normal derivative directed outward from $D$ on $\partial D$.

Following is the results

1. Inequalities based on the geometry of $D$

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

Theorem I. Let $u \in C^2(\overline D)$ have vanishing normal derivative on a portion $\Gamma$ of $\partial D$. Then if the Gaussian curvature o$\Gamma$ is everywhere positive the maximum value of $|{\rm grad} u|^2$ can occur on $\Gamma$ if and only if $u\equiv {\rm constant}$ in $D$.

We also have the following result

Theorem II. Let $u \in C^2(\overline D)$ vanish on a portion $\Gamma_1$ of $\partial D$. Then if the average curvature $K$ is positive at every point of $\Gamma_1$ the maximum value of

$|{\rm grad} u|^2 -2u\Delta u$

cannot occur on $\Gamma_1$ if $u\not \equiv 0$ in $D$.