Let $\mathcal{E}(\phi,\alpha), \phi\in \mathcal{D}$ be a functional on some domain $\mathcal{D}$ that depends on a parameter $\alpha$.
In the expression
$$\frac{\partial}{\partial \alpha} \inf_{\phi \in D}\{\mathcal{E}(\phi,\alpha)\},$$
is there a way to justify exchanging the derivative and the infimum? What conditions does $\mathcal{E}$ and possibly the domain have to fulfill for that?