Let $B$ be the unit ball in the Euclidean space $\mathbb{R}^n$. Consider the set of functions
$$X=\{u\in C^2(\bar B)|\,\, u|_{\partial B}=0 \mbox{ and } ||\Delta u||_{L^2(B)}\leq 1\},$$
where $\Delta$ is the Laplacian.
>Is it true that the set of first derivatives $\{\frac{\partial u}{\partial x_1}|\, u\in X\}$ is pre-compact in $L^2(B)$?