Characterization of Sobolev Space I have just started learning about Sobolev spaces. So this might be trivial. I am working through the book "Partial Differential Equations" by Lawrence Evans, it came highly recommended.
Taking $\Omega \subset \mathbb{R}^{n}$ to be some open set. He defines the space $$W^{k,p}_{o}(\Omega) := \lbrace u \in W^{1,p}(\Omega): u|_{\partial \Omega} = 0 \rbrace$$ for $1 \leq p < \infty$. 
He does not however define $W^{1,\infty}_{o}(\Omega)$. Does anyone know how this is generally defined? Is it simply $$W^{1,\infty}_{o}(\Omega) := \lbrace u \in W^{1,\infty}(\Omega): u|_{\partial \Omega} = 0 \rbrace$$ Why is it not dealt with in the same manner as for $1 \leq p < \infty$? I also checked Brezis book, he also does not deal with the case $p = \infty$.
Thanks. 
 A: The alternative definition is to define $W_0^{1,p}(\Omega)$ as the closure of $C_c^{\infty}(\Omega)$ in the sobolev norm. For $1\leq p < \infty$ these definitions are equivalent, but not for $  p = \infty $. He probably does not define it to avoid that ambiguity.
EDIT: Here is a more detailed answer. Let us define $W_0^{1,p}(\Omega) := \{f \in W^{1,p}(\Omega) \mid f|_{\partial \Omega} = 0\}$ and $W_c^{1,p}(\Omega) := \overline{C_c^{\infty}(\Omega)}$, where the closure is taken in the $W^{1,p}$ norm.
First note that Evans actually defines $W_0^{1,p}(\Omega)$  the way I have defined $W_c^{1,p}(\Omega)$ (see page 259 in my edition of the book).
Evans then shows (5.5 Theorem 2, page 273) that (in my notation) $W_0^{1,p} = W_c^{1,p}$ as long as $p< \infty$.
This is not the case for $p=\infty$, because $W_c^{1,\infty}(\Omega)$ is (by definition) the set of all limits $f = \lim_n f_n$, where $f_n \in C_c^{\infty}(\Omega)$ andthe limit is taken in $W^{1,\infty}$, which means $\Vert f - f_n \Vert_\infty \rightarrow 0$ and $\Vert \partial^{\alpha}f - \partial^{\alpha} f_n\Vert_\infty \rightarrow 0$. You can easily check that this implies $f \in C^1(\Omega)$ with $f|_{\partial \Omega} = 0$ and $\partial^{\alpha}f |_{\partial \Omega} = 0$.
But one can construct functions $f \in W_0^{1,\infty}(\Omega)$ (with my above definition), whose partial derivatives are discontinuous.
