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?