The Legendre-Fenchel transform of $BV$ semi-norm I am reading a numerical paper in which it calls some "easy" facts from convex analysis but I can't justify it...
Let $\Omega\subset \mathbb R^2$ be open. Define for a function $u\in L^1(\Omega)$ and
$$
J[u]=\sup\left\{\int_\Omega u(x)\text{ div }\xi(x)\,dx:\,\xi\in C_c^1(\Omega;\,\mathbb R^2),\,|\xi(x)|\leq 1\right\}
$$
which is the standard $TV$ semi-norm.
Then it claims that the Legendre-Fenchel transform
$$
J^*(v)=\sup_u\left<u,v\right>_X-J[u]
$$
is the characteristic function of a closed convex set $K$ where $K$ is the closure of the set 
$$
\{\text{div }\xi:\xi\in C_c^1(\Omega;\mathbb R^2),|\xi(x)|\leq 1\text{ for all }x\in\Omega\}\tag 1
$$
and we can recover
$$
J[u]=\sup_{v\in K}\left<u,v\right>_X.\tag 2
$$
(I looked carefully over the paper, they never define what is $X$...So I guess it is the dual pair?)
The paper refers to book Ekeland and Temama. I know it is a classical book but it is hard to find where this statement is... Could anybody just explain to me how $(1)$ and $(2)$ be verified?
Thank you!
 A: This is not specific to total variation; this is how the Legendre-Fenchel transform works for norms. Here one should really think of the space of $C^1_c$ vector fields as the original space, $X$. It is given the supremum norm. The $L^1$ functions induce linear functionals on this space via 
$$
\langle u,\xi\rangle = \int_\Omega u(x)\text{ div }\xi(x)\,dx
$$
and the BV seminorm is just the dual norm (some functions end up being zero functionals). 
Let $X$ be any vector space with norm $\|\cdot\|$. For $v\in X^*$, the quantity 
$$J^*(v) = \sup_u (\left<u,v\right>-\|u\|)$$
can only attain the values  $0$ and $\infty$. Indeed, $J^*(v)\ge 0$ because $u=0$ is a choice. And if some value of $u$ gives a positive amount, then scaling it yields $J(v)=\infty$. 
More specifically, $J^*(v)=0$ iff $\left<u,v\right> \le 1$ for all vectors $u$ with $\|u\|\le 1$. This is exactly the definition of the unit ball for the dual norm. Conversely, 
$$\|u\|= \sup_v (\left<u,v\right>-J^*(v))$$
because the right hand side is simply the supremum taken over $v$ with dual norm at most $1$.
