I am going to summarize some known comparison principles. I will start with a weak comparison principle for -Laplacian with Neumann boundary condition. This is adapted from a recent paper by J. Giacomoni et al. published in *Differential Integral Equations* [here]. I will keep the numbering for the convenience.

**Lemma 3.1**. Let be non-negative functions satisfying

in and

on the boundary in the strong sense. Then in .

*Proof*. The trick is that if we want to prove in then we just show that

in . To this purpose, we use as a test function in the equation

and integrate to achive

and use the integration by parts to arrive at

Keep in mind that

We now use the following inequality

that proof can be found in this topic. To be precise, we get

and

Hence,

This proves the positivity of the first term. For the second term, it is clear to see that

Lastly, we know that the integral on is non-positive since , thus the left hand side is non-negative. This gives the fact that .

### Like this:

Like Loading...

*Related*

[…] Quốc Anh: Weak comparison principle: p-Laplacian with Neumann boundary condition, Several interesting limits from a paper by Chang-Qing-Yang, The (original) Picone […]

Pingback by Third Xamuel.com Linkfest — April 10, 2011 @ 23:57

Hi there,

thanks for the many interesting articles in your website. Nevertheless, I am slightly confused: in my terminology the negative part of a function, is a nonnegative function, What is the definition of $(u-v)^{-}$ in your context? I ask, since I have no access to the article itself.

Thanks in advance,

ZNS

Comment by ZNS — September 28, 2015 @ 23:07

Thanks for your interest in my post.

I prefer to the following definition: .

Comment by Ngô Quốc Anh — September 28, 2015 @ 23:13

Thanks for that. I thought about incorporating a minus sign in front of the usual definition, but wasn’t sure at all. In any case, thanks again, keep up the good work.

Comment by ZNS — September 28, 2015 @ 23:18