Questions March 31

Note that $LaTeX: W^{0,p}(U)=L^p(U)$ $W^{0,p}(U)=L^p(U)$ for $LaTeX: 1\le p \le \infty$ $1\le p \le \infty$ . What is $LaTeX: W^{0,p}_0(U)$ $W^{0,p}_0(U)$ when $LaTeX: 1\le p<\infty$ $1\le p<\infty$ ?
For those of you who know some integration theory: do exercise 4 in Evans, ch. 5.10 (see the discussion at the bottom of p. 259).
In Theorems 1-3 in ch. 5.3, the case $LaTeX: p=\infty$ $p=\infty$ is avoided. Why?
Can you identify $LaTeX: W^{k,p}_0(\mathbb{R} ^n)$ $W^{k,p}_0(\mathbb{R} ^n)$ with another space when $LaTeX: 1\le p <\infty$ $1\le p <\infty$ ? What happens in the case $LaTeX: p=\infty$ $p=\infty$ ?
For those who are interested in the proof of the extension theorem in ch 5.4 of Evans: would the even extension operator work instead of (3)? Why/why not? How about the case $k=2$ ?
Theorem 3 in ch. 5.6 is stated for $LaTeX: q\in [1, p^*]$ $q\in [1, p^*]$ , whereas Theorem 2 is stated just for $p^*$ . Why?
What changes if you replace the bounded domain $U$ by $LaTeX: \mathbb{R} ^n$ $\mathbb{R} ^n$ in Theorem 3?