I'm trying to solve the following problem:
Integrate by parts to prove $$\|D u\|_{L^p}\le C \|u\|_{L^p}^\frac{1}{2}\|D^2u\|_{L^p}^\frac{1}{2}$$ for $2\le p <\infty$ and all $u\in C^\infty_c(U)$. (Hint: $\int_U |D u|^pdx=\sum_{i=1}^n \int_U u_{x_i}u_{x_i}|Du|^{p-2}dx$.)
Now, if one mindlessly computes using the hint, integrating by parts, and Hölder's inequality, it is not hard to arrive to the desired result. The problem lies in justifying the integration by parts, because, if $2< p\le 4$, then $$ |Du|^{p-2}=\left(\sum_{i=1}^n u_{x_i}^2\right)^\frac{p-2}{2}$$ is non-differentiable whenever $Du=0$, since $0<\frac{p-2}{2}\le 1$ (the case $p=2$ should be considered separately). We could try to integrate on $\{Du\neq 0\}$, but we have no information on the geometry of $\partial\{Du\neq 0\}$, i.e. the set need not have a $C^1$ boundary, so the integration by parts formula does not work.
Keep in mind, you are splitting the terms $u_{x_i}$ and $u_{x_i}|Du|^{p-2}$ apart, so differentiability of $u_{x_i}|Du|^{p-2}$ is all that is required. You can compute the derivative of $u_{x_i}|Du|^{p-2}$ in $x_j$ to be
$$(p-2)|Du|^{p-4} \sum_{i}^n u_{x_i}^2u_{x_ix_j} + u_{x_ix_j}|Du|^{p-2},$$
where $Du\neq 0$. Note this is continuous for $p>2$. At a point $x_0$ where $Du(x_0)=0$ we have $|Du(x_0)|\leq C|x-x_0|$ and so
$$|u_{x_i}(x_0)|Du(x_0)|^{p-2}|\leq C|x-x_0|^{p-1}.$$
Since $p-1>1$ it follows that $u_{x_i}|Du|^{p-2}$ is differentiable at $x_0$ and has zero gradient. This shows that $u_{x_i}|Du|^{p-2}$ is continuously differentiable, and integration by parts is justified.