My question is rather general and in advance I appologize for not being precise enough, which is very likely. It concerns the matter: what do people understand by optimality conditions?
Suppose we have a problem: Find $u^*\in F$ such that $$\inf_{u\in F} J(u) = u^*.$$ Also, let $u^*$ be the solution of this problem ($J$ is some energy functional assoiciated with some system of differential equations or differential inequality).
What do we have to do to obtain optimality conditions - what are the? To make myself clear, if we want to solve an equation we look for the solution - what do we have to do find optimality condidtions? Is there any definition (or intuition) of the term "optimality conditions"?
Edit. Propably $\frac{\partial J}{\partial u}(u^*)=0$ (Gateaux derivative) is not enough?
$$\frac{\partial J}{\partial u}(u^*)(u-u^*)\ge 0 $$ for all $u\in F$.