Estimate on first derivatives given $L^2$-norm of Laplacian

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)$ ?