I'm trying to solve below exercise in Brezis' Functional Analysis, i.e.,
Exercise 8.6.1 Let $I=(0, 1)$ and $p \in (1, \infty]$.
- Check that $W^{2, p} (I) \subset C^1 (\bar I)$ with compact injection.
- Deduce that for $\varepsilon >0$ there is $C=C(\varepsilon, p)$ such that $$ \| u' \|_{L^\infty} + \| u \|_{L^\infty} \le \varepsilon \| u'' \|_{L^p} + C \| u \|_{L^1}, \quad u \in W^{2, p} (I). $$
- Let $q \in [1, \infty)$. Prove that for $\varepsilon >0$ there is $C=C(\varepsilon, q)$ such that $$ \| u' \|_{L^q} + \| u \|_{L^\infty} \le \varepsilon \| u'' \|_{L^1} + C \| u \|_{L^1}, \quad u \in W^{2, 1} (I). $$
More generally, let $m \ge 2$ be an integer.
- Prove that for $\varepsilon >0$ there is $C=C(\varepsilon, m, p)$ such that $$ \sum_{j=0}^{m-1} \| D^j u \|_{L^\infty} \le \varepsilon \| D^m u \|_{L^p} + C \| u \|_{L^1}, \quad u \in W^{m, p} (I). $$
There are possibly subtle mistakes that I could not recognize in my below attempt of $(4)$. Could you have a check on it?
- First, we prove that the injection $W^{m, p}(I) \subset C^{m-1}(\bar I)$ is compact.
We proceed by induction on $m$. The base case $m=2$ has been proved in $(1)$. Assume that the injection $W^{m, p}(I) \subset C^{m-1}(\bar I)$ is compact. We will prove that the statement holds for $m+1$. Let $(u_n)$ be a bounded sequence in $W^{m+1, p} (I)$. We have $$ \| u \|_{W^{m+1, p}} = \| u \|_{L^{p}} + \| u' \|_{W^{m, p}} = \| u \|_{W^{m, p}} + \| D^{m+1} u \|_{L^{p}}, $$ where $D^{m+1} u$ is the $(m+1)$-th derivative of $u$. Then $(u_n)$ and $(u'_n)$ are bounded sequences in $W^{m, p} (I)$. By induction hypothesis, there are $u, v \in C^{m-1}(\bar{I})$ and a subsequence $(n_k)$ such that $u_{n_k} \to u$ and $u'_{n_k} \to v$ in $C^{m-1}(\bar{I})$. In particular, $u_{n_k} \to u$ and $u'_{n_k} \to v$ uniformly. By this well-known result, $u'=v$. Then $u \in C^{m}(\bar{I})$ and $u_{n_k} \to u$ in $C^{m}(\bar{I})$.
- Next we prove our desired inequality in $(4)$. We need an auxiliary result, i.e.,
Exercise 6.12 Let $X,Y,Z$ be real Banach spaces with corresponding norms $|\cdot|_X, |\cdot|_Y, |\cdot|_Z$. Assume that $X \subset Y$ with compact injection and that $Y \subset Z$ with continuous injection. Prove that for every $\varepsilon>0$ there is $C_\varepsilon > 0$ such that $$ |u|_Y \le \varepsilon |u|_X + C_\varepsilon |u|_Z \quad \forall u \in X. $$
We apply above exercise with $X = W^{m, p} (I), Y=C^{m-1} (\bar I)$ and $Z= L^1 (I)$ and get $$ \sum_{j=0}^{m-1} \| D^j u \|_{L^\infty} \le \varepsilon \sum_{j=0}^{m} \| D^j u \|_{L^p} + C \| u \|_{L^1}, \quad u \in W^{2, p} (I). $$
We have $\| D^j u \|_{L^p} \le \| D^j u \|_{L^\infty}$. Then $$ \sum_{j=0}^{m-1} \| D^j u \|_{L^\infty} \le \frac{\varepsilon}{1-\varepsilon} \| D^m u \|_{L^p} + \frac{C}{1-\varepsilon} \| u \|_{L^1}, \quad u \in W^{m, p} (I). $$
The quantity $\frac{\varepsilon}{1-\varepsilon}$ can be arbitrarily small. The claim then follows.
