Interpolation inequality Lef $u$ be at least a $C^2$ function on $\mathbb{R}^n$.   Let's denote the gradient by $D$. Also, (using the multiindex notation), define the seminorm $$||D^ku|| = \sup_{|\gamma|=k}{\sup_x{|D^{\gamma}u|}}$$
How can we prove the following : 
$$||Du|| \leq \epsilon||D^2u|| + C||u|| $$
where $C$ is some constant depending on $\epsilon$

By restricting the function to a line, i.e., considering the function, $t \mapsto u(a+tb)$ for some point $a\in \mathbb{R}^n$ and a unit vector $b\in\mathbb{R}^n$, you can reduce the problem to the case $n=1$. Now the problem is for a $\mathcal{C}^2$ function $f:\mathbb{R}\to\mathbb{R}$ to show $\|f'\| \le \epsilon \|f''\| + C\|f\|$. The idea is that if you have a point $x_0$ and a constant $M>0$ such that $f'(x_0)\ge M+1$ (or $-f'(x_0)\ge M+1$), and if you have a uniform bound $\|f''\|\le K$, then $f'\ge M$ (or $-f'\ge M$) whenever $|x-x_0| \le 1/K$, and then $|f(x_0+1/K)-f(x_0-1/K)| \ge 2M/K$, so $\| f\| \ge M/K$. This implies the desired inequality by juggling of constants.
