Semi-differentiability of generalised $L^\infty$ envelopes with applications to supremal functionals

J. Danskin proved in 1966 that the envelope of a family of continuous functions $\mathrm F : \mathbb R^n \times \mathrm K \longrightarrow \mathbb R$, parameterised by the points of a compact $\mathrm K \subseteq \mathbb R^m$, given by \[ f(u) := \max_{k \in \mathrm K} \mathrm F(u,k), \ \ \ \ f \, :