About the general chain rule for functions of bounded variation

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications Pub Date : 2024-02-19 DOI:10.1016/j.na.2024.113518

Camillo Brena , Nicola Gigli

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引用次数: 0

Abstract

We give an alternative proof of the general chain rule for functions of bounded variation (Ambrosio and Maso, 1990), which allows to compute the distributional differential of $φ \circ F$ , where $φ \in LIP (R^{m})$ and $F \in BV (R^{n}, R^{m})$ . In our argument we build on top of recently established links between “closability of certain differentiation operators” and “differentiability of Lipschitz functions in related directions” (Alberti et al., 2023): we couple this with the observation that “the map that takes $φ$ and returns the distributional differential of $φ \circ F$ is closable” to conclude.

Unlike previous results in this direction, our proof can directly be adapted to the non-smooth setting of finite dimensional RCD spaces.

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关于有界变化函数的一般链式法则

我们给出了有界变化函数一般链式法则的另一种证明（Ambrosio 和 Maso, 1990），它允许计算φ∘F 的分布微分，其中φ∈LIP(Rm) 和 F∈BV(Rn,Rm).在我们的论证中，我们建立在最近建立的 "某些微分算子的可闭性 "与 "相关方向上的 Lipschitz 函数的可微性 "之间的联系之上（Alberti 等人，2023 年）：我们将其与 "取 φ 并返回 φ∘F 的分布微分的映射是可闭的 "这一观察结合起来，得出结论。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Nonlinear Analysis-Theory Methods & Applications 数学-数学

CiteScore

3.30

自引率

0.00%

发文量

265

审稿时长

60 days

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