Lower semicontinuous energy functional on compact space of Lipschitz functions Let $\Omega \subset \mathbb{R}^{n}$ be a bounded open subset containing $0$ and let $L>0$ be some positive constant. Consider the space 
$A_{0}=\{f \in C^{\infty}(\overline{\Omega}) \mid f \text{ is L-Lipschitz}, f(0)=0\}$ 
 and let $A$ be the closure of $A_{0}$ with respect to the $C^{0}-$norm. By Arzela-Ascoli $A$ is compact space consisting of $L$-Lipschitz functions which vanish at $0$. 
Define the functional
$\mathcal{E} : A \rightarrow \mathbb{R}$ by $\mathcal{E}(f)=\int_{\Omega}|\nabla f|^{2}dx$. 
I have the following questions:


*

*How can I make sense of $\nabla f$ since for general $f\in A$, $f$ is not differentiable anymore? I thought using Rademacher theorem but I do not know exactly how. 

*Is $\mathcal{E}$ continuous or semicontinuous? If its semicontinuous then is it lower or upper semi-continous. My feeling tells me that it is lower semi-continous. But I dont know how to prove it.
Do you have any idea?

Rademacher tells you that $\nabla f$ exists a.e. and is bounded by $L$. Since $\Omega$ has finite measure we get that 
$$
\int_\Omega |\nabla f|^2 \leq |\Omega|\| \nabla f\|_\infty^2 \leq |\Omega| L^2.
$$
Therefore your functional is well defined.
As for the lower semicontinuity; if a sequence $f_k$ converges to $f$ in $A$, then it converges in $L^2(\Omega)$ and weakly in $W^{1,2}(\Omega)$ so that, by (weak) lower semicountinuity of the norm we get
$$
\int_\Omega |f|^2 + |\nabla f|^2 \leq \liminf_k \int_\Omega |f_k|^2 + |\nabla f_k|^2 .
$$
This is what you want. 
