Semi-differentiability of generalised L∞ envelopes with applications to supremal functionals
Abstract
J. Danskin proved in 1966 that the envelope of a family of continuous functions F : ℝn × K ⟶ ℝ, parameterised by the points of a compact K ⊆ ℝm, given by f(u) := maxk ∈ K F(u,k), f : ℝn ⟶ ℝ, is semi-differentiable on ℝn, if 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 K" with ``essential sup over K". We establish the semi-differentiability of generalised L∞ 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 L∞, 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 L∞ functionals via their semi-differentials, revealing the genuine L∞ counterpart of the Euler-Lagrange equations. Conventional Aronsson equations and related divergence PDE involving measures, deducible from Lp approximations as p→ ∞, in general are not sufficient for minimality in L∞. Our results dislodge the trademark necessity for extrinsic Lp-approximations as p→ ∞ to derive and study PDEs, reinterpreting the existing L∞ theory. Most crucially, they provide a new powerful intrinsic L∞ framework, evocative of the straightforward variational methods for integral functionals.
Related
- Efficiency Optimality without Pathwise Differentiability: A Variational Theory for Marginal-Integral Functionals
- Disjunctive Submodular Functions: Envelopes and Applications to Inventory and 0-1 Quadratic Optimization
- Microlocal defect functionals in VMO: Geometric localisation and applications to highly heterogeneous media
- Higher-order differentiability of Korevaar--Schoen energy forms and energy measures
- On differentiability in weakly o-minimal expansions of fields
BibTeX
@misc{katzourakis2026semidifferentiability,
title = {{Semi-differentiability of generalised $L^\infty$ envelopes with applications to supremal functionals}},
author = {Nikos Katzourakis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27918},
eprint = {2609.27918},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27918},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Semi-differentiability of generalised L^∞ envelopes with applications to supremal functionals
AU - Nikos Katzourakis
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.27918
UR - https://arxiv.org/abs/2609.27918
AB - J. Danskin proved in 1966 that the envelope of a family of continuous functions F : ℝ^n × K ⟶ ℝ, parameterised by the points of a compact K ⊆ ℝ^m, given by f(u) := max_k ∈ K F(u,k), f : ℝ^n ⟶ ℝ, is semi-differentiable on ℝ^n, if 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 K" with ``essential sup over K". We establish the semi-differentiability of generalised L^∞ 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 L^∞, 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 L^∞ functionals via their semi-differentials, revealing the genuine L^∞ counterpart of the Euler-Lagrange equations. Conventional Aronsson equations and related divergence PDE involving measures, deducible from L^p approximations as p→ ∞, in general are not sufficient for minimality in L^∞. Our results dislodge the trademark necessity for extrinsic L^p-approximations as p→ ∞ to derive and study PDEs, reinterpreting the existing L^∞ theory. Most crucially, they provide a new powerful intrinsic L^∞ framework, evocative of the straightforward variational methods for integral functionals.
ER -