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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., \(\displaystyle \|u\|_{C^m(\Omega)} \le c\|u\|_{W^{k,p}_0}.\)
 

Jose27

MHF Hall of Honor
Apr 2009
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México
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. \(\displaystyle \| u\| _{C^m(\Omega )} = \sum_{|\alpha | \leq m} \| D^{\alpha } u \| _{\infty } = \sum_{ |\alpha |\leq m } \sup_{\Omega } |D^{\alpha }u |\).
 
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