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 $L^2(B)$?

Let $B$ be the unit ball in the Euclidean space $\mathbb{R}^n$. Consider the set of functions $$X=\{u\in C^2(\bar B) \mid u|_{\partial B}=0 \text{ and } \|\Delta u\|_{L^2(B)}\leq 1\},$$ 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 $L^2(B)$?