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 "=".