ChapterZero

A Sobolev inequality

It turns out that if is in $W^{2,2}$ , $\|Df\| \leq C \|D^2f\|^\alpha \|f\|^{1-\alpha}$ , where $C,\alpha$ depend on the dimension. I have no idea how to prove this in general, and don’t really care.

BUT, it makes for a nice problem in the one-dimensional case. Show that if $f \in W^{2,2}(\R)$ (i.e. its second derivative exists and both it and its first and second derivatives are square integrable), then $\|f^\prime\| \leq C\|f^{\prime\prime}\|^\alpha \|f\|^{1-\alpha}$ , and find $C,\alpha$ .

I haven’t a clue how to proceed, but it looks like mighty fun.

Update: the proof is, in retrospect (always in retrospect, damn it), obvious.

This entry was posted on Friday, July 11th, 2008 at 6:09 pm and is filed under Mathematics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

One Response to “A Sobolev inequality”

Alex Says:
July 12th, 2008 at 5:01 pm
For the 1d case, I used integration by parts on $\|f^\prime\|^2$ , then cauchy-schwarz. Maybe you can do the same in higher dimensions.

A Sobolev inequality

One Response to “A Sobolev inequality”

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