Brezis' exercise 8.7: prove that $\bar{u} \in W^{1, p}(\mathbb{R})$

Question

I'm trying to solve below exercise in Brezis' Functional Analysis, i.e.,

Exercise 8.7 Let $I=(0,1)$. Given a function $u$ defined on $I$, set $$ \bar{u}(x)= \begin{cases} u(x) & \text { if } x \in I, \\ 0 & \text { if } x \in \mathbb{R}, x \notin I. \end{cases} $$

1. Assume that $u \in W_0^{1, p}(I)$ with $1 \leq p<\infty$. Prove that $\bar{u} \in W^{1, p}(\mathbb{R})$.
2. Conversely, let $u \in L^p(I)$ with $1 \leq p<\infty$ be such that $\bar{u} \in W^{1, p}(\mathbb{R})$. Show that $u \in W_0^{1, p}(I)$.
3. Let $u \in L^p(I)$ with $1<p<\infty$. Show that $u \in W_0^{1, p}(I)$ IFF there exists a constant $C$ such that $$ \left|\int_{\mathbb{R}} \bar{u} \varphi^{\prime}\right| \leq C\|\varphi\|_{L^{p^*}(\mathbb{R})} \quad \forall \varphi \in C_c^1(\mathbb{R}), $$ where $p^*$ is the Hölder conjugate of $p$.

There are possibly subtle mistakes that I could not recognize in my below attempt. Could you please have a check on it?

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We need auxiliary results (in the same book), i.e.,

• Theorem 8.12 Let $u \in W^{1, p}(I)$ with $p \in [1, \infty)$. Then $u \in W_0^{1, p}(I)$ IFF $u=0$ on $\partial I$.
• Proposition 8.3 Let $u \in L^p (I)$ with $p \in (1, \infty]$. Then $u \in W^{1, p} (I)$ IFF there is a constant $C>0$ such that $$ \left|\int_I u \varphi' \right| \leq C\|\varphi\|_{L^{p^*}(I)} \quad \forall \varphi \in C_c^1(I) . $$

1. Let $v := \overline{u'}$, i.e., $v$ is the extension of $u'$ on $\mathbb R$. Clearly, $v \in L^p (\mathbb R)$. There is a sequence $(u_n) \subset C_c^1 (I)$ such that $|u_n -u|_{W^{1, p}(I)} \to 0$. For $\varphi \in C_c^1 (\mathbb R)$, we have $$ \begin{align*} \int_{\mathbb R} \bar{u} \varphi' = \int_I u \varphi' = \lim_n \int_I u_n \varphi' = \lim_n \int_I u_n' \varphi = \lim_n \int_I u' \varphi = \lim_n \int_{\mathbb R} v \varphi. \end{align*} $$ It follows that $\bar u \in W^{1, p}(\mathbb{R})$ with $\bar u' =v$.

2. It follows from $\bar{u} \in W^{1, p}(\mathbb{R})$ that $u \in W^{1, p} (I)$. By Theorem 8.12, it remains to prove that $u = 0$ on $\partial I$. We have $$ u(0) = \lim_{x \downarrow 0} u(x) = \lim_{x \downarrow 0} \bar u(x) = \lim_{x \uparrow 0} \bar u(x) =0. $$ Similarly, we get $u(1)=0$.

• $\implies$ This follows from $(1)$ and Proposition 8.3.
• $\impliedby$ This follows from Proposition 8.3 and $(2)$.