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.



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> Is it true that the set of first derivatives $\{\frac{\partial u}{\partial x\_1}|\, u\in X\}$ is pre-compact in $L^2(B)$?
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Estimate on first derivatives given $L^2$-norm of Laplacian
Yes, this is true and the "proof" is as follows (I wouldn't really dare to call that a proof, though).
For any $u\in X$ write $f=-\Delta u$, and observe that trivially $u$ solves the elliptic PDE
$$
\left\{
\begin{array}{ll}
-\Delta u=f & \mbox{in }B\\
u=0 & \mbox{on }\partial B
\end{array}
\right..
$$
(This is a tautology, if you will.)
By standard elliptic regularity (see e.g. Evans' textbook on PDEs, section 6.3.2 and in particular remark (i) just after theorem 4) we get that
$$
\|u\|\_{H^2(B)}\leq C \|f\|\_{L^2(B)}=C \|\Delta u\|\_{L^2(B)}\leq C
$$
for some purely dimensional $C\equiv C(n)$.
In other words, your set $X$ is bounded in $H^2$. The precompactness then follows from the [Rellich-Kondrachov theorem](https://en.wikipedia.org/wiki/Rellich%E2%80%93Kondrachov_theorem) applied to $u\_i=\partial\_{x\_i}u$, with of course $\|\nabla u\_i\|\_{L^2}\leq \|u\|\_{H^2}$ (the full Hessian of $u$ controls any first derivative of $u\_i$).


For a more "serious" reference (by that I mean not a textbook) I can recommend Gilbarg-Trudinger.


