0
$\begingroup$

I was reading a paper of Lu-Wei named "On a Sobolev inequality with remainder terms" (link at AMS site).

and I have quite a few questions regarding that.

(1) In page 78 they got a system of two equations involving $\phi$ and $w_i$ and the claim is $\phi_i=0$ when $\mu\leq p$ and $i>N+2$. Then it's written WLOG assume that $\phi_i<0$ on $(0,r_1)$ and $\phi_i(r_1)=0$ ( why it has to be the case, can't it be negative everywhere, why it has to hit $0$?) Also in the very next line it's written that as $i>N+2$ ,$\phi_i(0)=w_i(0)=\phi_i'(0)=w_i'(0)=0$.I couldn't see how this follows ?

Any help is very much helpful for my better understanding of the material.

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ It would be easier for the readers if you can either provide the full citation information of the paper you are looking at, or, better yet, include a DOI link to the electronic version (or to an arXiv preprint). $\endgroup$
    – Willie Wong
    Commented Jan 3, 2023 at 16:23
  • $\begingroup$ Sure I am adding it. $\endgroup$
    – User1723
    Commented Jan 3, 2023 at 17:55

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .