Given $I=(0,1)$ and $u\in W^{2,p}(0,1)$ for $p>1$. I am trying to prove that for any $\epsilon>0$, the following hold:
$$ \|u\|_{L^\infty(I)}+\|u'\|_{L^\infty(I)}\leq \epsilon\|u''\|_{L^p(0,1)}+C_{\epsilon, p}\|u\|_{L^1(0,1)}$$
I have the following idea: I am taking the fact that for any Banach space $X$, $Y$, $Z$, if $X\subset\subset Y$, i.e., $X$ compact embedding in $Y$, and $Y$ is continuous embedding in $Z$, then I have $$\|u\|_Y\leq \epsilon \|u\|_X+C_\epsilon \|u\|_Z$$ for any $u\in X$. This is my old functional homework and I am sure it is correct. Thus, by taking $X=W^{2,p}(I)$, $Y=C^1([0,1])$, and $Z=L^1(I)$, I have the following result:
$$ \|u\|_{L^\infty(I)}+\|u'\|_{L^\infty(I)}\leq \epsilon\|u\|_{W^{2,p}(I)}+C_{\epsilon, p}\|u\|_{L^1(I)}=\epsilon(\|u''\|_{L^p}+\|u\|_{L^p}+\|u'\|_{L^p})+C_{\epsilon, p}\|u\|_{L^1(I)}$$
Then by using $L^p$ interpolating and Young's inequality, I am able to write $$\epsilon(\|u\|_{L^p}+\|u'\|_{L^p})\leq \frac{1}{2} (\|u\|_{L^\infty}+\|u'\|_{L^\infty})+ C_{\epsilon,p}(\|u\|_{L^1}+\|u'\|_{L^1}) $$ Combining the result above I have $$ \|u\|_{L^\infty(I)}+\|u'\|_{L^\infty(I)}\leq \epsilon\|u''\|_{L^p}+C_{\epsilon,p}\|u'\|_{L^1}+C_{\epsilon,p}\|u\|_{L^1}$$ Now looks like I am just one step away from what I want, however, I have no idea how to control the part $C_{\epsilon,p}\|u'\|_{L^1}$ by $C_{\epsilon,p}\|u\|_{L^1}$...Some help is really welcome!!