Question about Minimization Let be $J$ a convex functional defined in Hilbert space H and with real values. What hypothesis I should assume to exist solution for the problem?:
$J(u) = \inf \left\{{J(v); v \in K}\right\} , u \in K$
For all  closed convex $K \subset H.$
I begin using the theorem 
A functional $J:E\rightarrow\mathbb{R}$ defined over a norm space $E$ is semi-continuous inferiorly if for all sequence $(u_n)_{n\in \mathbb{N}}$ converging to $u$ then:
$\lim_{n\rightarrow \infty}\inf J(u_n)\geq J(u)$.
But I don't know how make to only "=".

You get equality by taking $u_n$ such that $J(u_n)\to  \inf_K J$. Indeed, the weak limit is also an element of $K$ and therefore cannot have a smaller value of the functional than the infimum.
The term is "lower semicontinuous", by the way. What you need from $J$ is being bounded from below, and lower semicontinuous with respect to weak convergence of sequences. And if you allow unbounded $K$, it helps to have $J\to\infty $ at infinity, because this forces the sequence $u_n$ to be bounded. 
