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.


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