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Math Help - Sobolev Embedding

  1. #1
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    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 \|\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}.
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  2. #2
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    Quote Originally Posted by lvleph View Post
    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 |.
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  3. #3
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    That is what I thought.
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