The Payne’s Maximum Principles

January 15, 2011

The Payne’s Maximum Principles

Filed under: PDEs — Tags: Maximum Principle — 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 $D$ .

Besides, we have

Theorem III. Let $u \in C^2(\overline D)$ vanish with its Laplacian 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 $|{\rm grad} u|^2$ cannot occur on $\Gamma_1$ if $u\not \equiv 0$ in $D$ .

Analogous results hold for combinations of higher derivatives of a function $u\in C^4(D)$ . A typical theorem is

Theorem IV. Let $u \in C^4(\overline D)$ vanish together with its normal derivative on $\partial D$ . Then if the average curvature is positive at every point of $\Gamma_1$ , the quantity

$H(x)=u_{,ij}u_{,ij}-2\gamma u_{,i}\Delta u_{,i}+(\gamma-1)(\Delta u)^2$

with $\gamma>0$ but arbitrary, can take its maximum value on $\Gamma_1$ if and only if $u\equiv 0$ in $D$ .

2. Second order equations

Let $u(x)$ be a solution of

$-\Delta u=f(u)$ .

We seek a maximum principle for the function $F(x)$ defined by

$\displaystyle F(x) = g(u){u_{,i}}{u_{,i}} + 2\int_0^u {g(\eta )f(\eta )d\eta }$

where $g(u)$ is a function to be determined. We first have

Theorem V. Let g(u) satisfy

$g(u) \geqslant 0$ ;

$f(u)g'(u)<0$ for $u\ne 0$ ;

$\displaystyle g(u)g''(u) - \frac{{3n - 4}}{{2(n - 1)}}{(g'(u))^2} \geqslant 0$ ;

for solutions $u$ of the pde. Then the function $F(x)$ , defined as above will take its maximum value either on $\partial D$ or at a critical point of $u$ provided $u\in C^3(D)$ .

With somewhat more restrictive assumptions on $f(u)$ it is possible to obtain other types of bounds for $|{\rm grad}u|^2$ . For instance

Theorem VI. Let $u \in c^3(\overline D)$ satisfy the pde and suppose $f''(u) \leqslant O$ in $D$ then the function $G(x)$ defined by

$\displaystyle G(x) = {u_{,i}}{u_{,i}} + \frac{2}{n}\int_0^u {f(\eta )d\eta }$

takes its maximum value on $\partial D$ .

3. Auxiliary inequalities

In this section we wish to mention an extension of a well known inequality which will be used in the next section to obtain maximum principles for solutions of fourth order equations. We prove the following

Theorem VII. Let $u(x) \in C^3(\overline D)$ then

$\displaystyle {u_{,ijk}}{u_{,ijk}} - \frac{3}{{n + 2}}\Delta {u_{,i}}\Delta {u_{,i}} \geqslant 0$ .

4. Fourth order equations

Let $u\in C^5(D)$ be a solution of

$\Delta^2u=f(u)$ .

We establish in this section a number of maximum principles based on specific assumptions on the form of f. These inequalities are in some sense generalizations of the inequality of Miranda for biharmonic functions in the plane. We prove first the following

Theorem VIII. Suppose $f(u)>0$ in $D$ then the function $\Phi (x)$ defined by

$\displaystyle\Phi (x) = {u_{,ij}}{u_{,ij}} - {u_{,i}}\Delta {u_{,i}} + \int_0^u {f(\eta )d\eta } + \frac{{2(n - 1)}}{{n + 2}}{(\Delta u)_M}\Delta u + \frac{{n - 4}}{{2(n + 2)}}{(\Delta u)^2}$

takes its maximum value on $\partial D$ . Here

$\displaystyle {(\Delta u)_M} = \mathop {\max }\limits_{\partial D} \Delta u$ .

Alternatively with a different assumption on $f$ we may establish the following result

Theorem IX. Let $f'(u)<0$ in $D$ ; then the function

$\displaystyle\Psi (x) = {u_{,ij}}{u_{,ij}} - {u_{,i}}\Delta {u_{,i}} + \frac{{4 - n}}{{n + 2}}\int_0^u {f(\eta )d\eta } + \frac{{n - 4}}{{2(n + 2)}}{(\Delta u)^2}$

takes its maximum value on $\partial D$ .

In an analogous way it can be shown that if $f'(u)<0$ in $D$ then the quantity

$\displaystyle {(\Delta u)^2} - 2\int_0^u {f(\eta )d\eta }$

takes its maximum value on the boundary of $D$ . This says in particular that

Theorem X. If $f'(u)<0$ in $D$ and $u =0$ , $\Delta u =0$ on $\partial D$ then

$\displaystyle {(\Delta u)^2} \leqslant 2\int_0^u {f(\eta )d\eta }$

in $D$ .

For the special case in which $f(u) \equiv c$ (a constant) various other maximum principles are possible.

Theorem XI. If $f(u) \equiv c$ and $\widetilde \psi$ satisfies the following

$-\Delta \widetilde\psi=u$ in $D$ and $\widetilde \psi=0$ on $\partial D$

then for $n =2$ the function

$\displaystyle\chi (x) = {u_{,i}}{u_{,i}} - u\Delta u - c\widetilde \psi$

takes its maximum value on $\partial D$ .

A number of applications of the various theorems in this paper could easily be exhibited. We have listed only a few as an indication of the types of results that can be obtained.

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Comments (3)

3 Comments »

I feel lucky to see your post. Would you mind if you share this thesis to me? Here is my e-mail address: sunghan290@snu.ac.kr
I’m a senior in Seoul National University, Korea, and this “P-ftn method” is highly related to my senior thesis. Unfortunately, I couldn’t find this paper online without purchasing(it’s about $40, which I cannot afford).
Good luck.

Comment by Sunghan Kim — November 16, 2012 @ 22:05

Reply
- Dear Kim, thank you for your interest in my post. You can email to L. E. Payne to get a free copy of the paper.
  
  Comment by Ngô Quốc Anh — November 16, 2012 @ 23:31
  
  Reply
  - Dear Anh, oh, really? I didn’t know that. Thanks for the information.
    
    Comment by Sunghan Kim — November 19, 2012 @ 9:52

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January 15, 2011