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 $\mathrm{F}:{\mathbb{R}}^n\times \mathrm{K}⟶\mathbb{R}$, 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.

arXiv comment48 pages, 2 figures