Sobolev Embedding

**lvleph** · May 9th 2010, 06:54 AM

This question could also go in Differential Equations, but I felt it would be more likely answered here. I just need a small question answered about a norm.

What is the norm $\|\cdot\|_{C^m(\Omega)}$ in the following:
For $0\le m < k - \frac{n}{p}$
$<br /> W^{k,p}_0(\Omega) \subset C^m(\bar{\Omega}),<br />$
i.e., $\|u\|_{C^m(\Omega)} \le c\|u\|_{W^{k,p}_0}.$

**Jose27** · May 9th 2010, 12:32 PM

Originally Posted by lvleph

This question could also go in Differential Equations, but I felt it would be more likely answered here. I just need a small question answered about a norm.

What is the norm $\|\cdot\|_{C^m(\Omega)}$ in the following:
For $0\le m < k - \frac{n}{p}$
$<br /> W^{k,p}_0(\Omega) \subset C^m(\bar{\Omega}),<br />$
i.e., $\|u\|_{C^m(\Omega)} \le c\|u\|_{W^{k,p}_0}.$

It's the usual, ie. $\| u\| _{C^m(\Omega )} = \sum_{|\alpha | \leq m} \| D^{\alpha } u \| _{\infty } = \sum_{ |\alpha |\leq m } \sup_{\Omega } |D^{\alpha }u |$ .

**lvleph** · May 9th 2010, 02:43 PM

That is what I thought.

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