# 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 $$L^2(B)$$?