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Given a smooth domain $\Omega$, one can define the sobolev space

$$W_{k,p}^0 = \text{ closure of } C_c^{\infty} \text{ in } W_{k,p}(U)$$

One interpretation of this space is given in the book Partial Differential Equations by Evans as

$W^{k,p}_0$ comprises those functions in $W^{k,p}(U)$ such that

$$ ``D^{\alpha}u = 0" \text{ on } \partial \Omega \qquad \forall |\alpha| \leq k-1 $$

Why is the $|\alpha| = k$ not included in this interpretation?

tgtt
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  • I'm not sure what you're asking; can you add more details? Are you familiar with the proof, at least for the case $k=1$? This would answer "why" in a way. And considering the case $k=0$ yourself can also shed some light on the problem. – Michał Miśkiewicz Oct 25 '18 at 23:07
  • @MichałMiśkiewicz Doesn't the $W_{k,p}$ norm include the derivative of up to $k$th order? So intuitively the $k$-th derivative of $u$ in $W_{k,p}$ should have zero boundary. Shouldn't it? What is special about the highest order derivative? – tgtt Oct 26 '18 at 00:33
  • Well, it depends on your intuition. I really think it would help if you considered the case $k=0$ and also read the proof in Evans' book (and included your thoughts or doubts about it in the question). – Michał Miśkiewicz Oct 26 '18 at 14:39

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