Followup question, does $\|u\|_{L^\infty(0, 1)} \le \epsilon\|u'\|_{L^p(0, 1)} + C\|u\|_{L^1(0, 1)}$ still hold when $p = 1$?

Question

This is a followup to this question.

Let $p > 1$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, p)$ such that$$\|u\|_{L^\infty(0, 1)} \le \epsilon\|u'\|_{L^p(0, 1)} + C\|u\|_{L^1(0, 1)}\tag*{$(1)$}$$for all $u \in W^{1, p}(0, 1)$?

Does $(1)$ still hold when $p = 1$?

Answer 1

Consider $u_n = \max \{ 1-nx, 0\}$. Then $\|u_n\|_\infty = 1$, $\|u'_n\|_{L^1} =1$ and $\|u_n\|_{L^1} \to 0$. Thus if

$$\|u\|_{L^\infty} \le \epsilon\|u'\|_{L^1} + C\|u\|_{L^1}$$

holds for some $C$, take $u = u_n$ and takes $n\to \infty$ gives $1\le \epsilon$, which is nonsense.

Answer 2

Other counterexample: $u_n(x)=x^n$.

As in the @JohnMa 's answer, $\|u_n\|_{L^\infty}=\|u_n'\|_{L^1}=1$ and $\|u_n\|_{L^1}=\frac{1}{n+1}\to0$.