# Sobolev Embedding

• May 9th 2010, 06:54 AM
lvleph
Sobolev Embedding
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 $\displaystyle \|\cdot\|_{C^m(\Omega)}$ in the following:
For $\displaystyle 0\le m < k - \frac{n}{p}$
$\displaystyle W^{k,p}_0(\Omega) \subset C^m(\bar{\Omega}),$
i.e., $\displaystyle \|u\|_{C^m(\Omega)} \le c\|u\|_{W^{k,p}_0}.$
• May 9th 2010, 12:32 PM
Jose27
Quote:

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

It's the usual, ie. $\displaystyle \| u\| _{C^m(\Omega )} = \sum_{|\alpha | \leq m} \| D^{\alpha } u \| _{\infty } = \sum_{ |\alpha |\leq m } \sup_{\Omega } |D^{\alpha }u |$.
• May 9th 2010, 02:43 PM
lvleph
That is what I thought.