convergence of the derivatives I am trying to solve the question:
Let $u_n$ a sequence converging uniformly to $u$ where $u_n\in C^3(\Omega)$ for each $n$ and $\Omega$ is a subset limited of $\mathbb{R^n}$. Suppose $u_n=0$ on
$\partial\Omega$. To show that 
$$
\displaystyle\lim_{n\rightarrow0}\int_{\Omega}|\nabla u_n|^2=\int_{\Omega}|\nabla u|^2.
$$

You can suppose, without loss of generality, that $u \equiv 0$ (if not, take $v_n = u_n - u$, and prove the statement for $v_n$).
Then, you want to show that $\lim_{n\to\infty}\int_\Omega |\nabla u_n|^2 = 0$.
Use integration by parts:
$$\int_\Omega |\nabla u_n|^2  = \int_{\partial \Omega} u_n (\nabla u_n \cdot \hat n) - \int_\Omega u_n \Delta u_n$$
Now use the facts that $u_n = 0$ on $\partial\Omega$, that $u_n \to 0$ uniformly, and that $u_n \in C^3$.
