广义 $L^\infty$ 包络的半可微性及其在极大泛函中的应用
Semi-differentiability of generalised $L^\infty$ envelopes with applications to supremal functionals
- University of Reading（雷丁大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明广义 $L^\infty$ 包络的半可微性，并应用于极大泛函，给出其半导数显式公式及极小元的变分刻画，建立了内在 $L^\infty$ 框架，无需外在 $L^p$ 近似。
AI中文摘要：
J. Danskin 于1966年证明了，若函数族 $\mathrm F : \mathbb R^n \times \mathrm K \longrightarrow \mathbb R$ 由紧集 $\mathrm K \subseteq \mathbb R^m$ 中的点参数化，且 $\mathrm F$ 足够正则，则由 $f(u) := \max_{k \in \mathrm K} \mathrm F(u,k)$（其中 $f : \mathbb R^n \longrightarrow \mathbb R$）给出的包络在 $\mathbb R^n$ 上是半可微的。该结果在众多应用中极为重要。已有研究几乎向所有方向推广了这一结果，但均未允许将“在 $\mathrm K$ 上取最大值”替换为“在 $\mathrm K$ 上取本质上确界”。我们建立了当 $\mathrm F$ 定义在 Banach 空间与测度空间的乘积上时广义 $\mathrm L^\infty$ 包络的半可微性。作为应用，我们在 $\mathrm L^\infty$ 变分学中获得了新的基础正则性结果，断言极大泛函处处半可微，并给出了其半导数的显式公式，尽管这些泛函通常不可微。此外，我们通过半微分建立了一般水平凸 $\mathrm L^\infty$ 泛函的（绝对）极小元的变分刻画，揭示了 Euler-Lagrange 方程在 $\mathrm L^\infty$ 中的真正对应物。传统的 Aronsson 方程及涉及测度的相关散度 PDE（可由 $\mathrm L^p$ 近似在 $p\to\infty$ 时推出）通常不足以刻画 $\mathrm L^\infty$ 中的极小性。我们的结果摒弃了通过 $p\to\infty$ 时外在 $\mathrm L^p$ 近似来推导和研究 PDE 的典型必要性，重新诠释了现有的 $\mathrm L^\infty$ 理论。最重要的是，它们提供了一个新的强有力的内在 $\mathrm L^\infty$ 框架，令人联想到积分泛函的直接变分方法。
英文摘要：
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.
