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

Abstract

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 \[ f(u) := \max_{k \in \mathrm K} \mathrm F(u,k), \ \ \ \ f \, : \, \mathbb R^n \longrightarrow \mathbb R, \] is semi-differentiable on $\mathbb R^n$, if $\mathrm F$ is sufficiently regular. This result is of utmost importance in numerous applications. Extensions have been proved towards almost every direction, but none permits to replace ``max over $\mathrm K$" with ``essential sup over $\mathrm K$". We establish the semi-differentiability of generalised $\mathrm L^\infty$ envelopes when $ \mathrm F$ is defined on the product of a Banach space with a measure space. As an application, we obtain a new foundational regularity result in the Calculus of Variations in $\mathrm L^\infty$, asserting that supremal functionals are semi-differentiable everywhere, with an explicit formula for their semi-derivative, despite generally being non-differentiable. Moreover, we establish a \emph{variational characterisation} of (absolute) minimisers of general level-convex $\mathrm L^\infty$ functionals via their semi-differentials, revealing the genuine $\mathrm L^\infty$ counterpart of the Euler-Lagrange equations. Conventional Aronsson equations and related divergence PDE involving measures, deducible from $\mathrm L^p$ approximations as $p\to \infty$, in general are not sufficient for minimality in $\mathrm L^\infty$. Our results dislodge the trademark necessity for extrinsic $\mathrm L^p$-approximations as $p\to \infty$ to derive and study PDEs, reinterpreting the existing $\mathrm L^\infty$ theory. Most crucially, they provide a new powerful intrinsic $\mathrm L^\infty$ framework, evocative of the straightforward variational methods for integral functionals.