Is it true that $\|Du\|_{L^{2p}} \le C\|u\|_{L^\infty}^{1\over2} \|D^2u\|_{L^p}^{1\over 2}$? Is it true that$$\|Du\|_{L^{2p}} \le C\|u\|_{L^\infty}^{1\over2} \|D^2u\|_{L^p}^{1\over 2}$$for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$?

This is a particular case of the Gagliardo-Nirenberg inequality:
$$\|D^ju\|_{L^p} \le C\|u\|_{L^q}^{1-\alpha} \|D^m u\|_{L^r}^\alpha,$$
where
$$1\le r,q\le \infty, \quad \frac{1}{p} = \frac{j}{n} + \left(\frac{1}{r} - \frac{m}{n}\right)\alpha + \frac{1-\alpha}{q} \quad \text{and} \quad \frac{j}{m} \le \alpha \le 1.$$
There are two cases in which you need more restrictions, but they do not apply to yours. Take $j=1$, $m=2$, $\alpha=\frac{1}{2}$, $p=2p$, $q=\infty$ and $r=p$.
You can see the original result in the paper On elliptic partial differential equations, by L. Nirenberg. As a reference for this and related results, I particularly like the book Partial Differential Equations by Lawrence C. Evans.
