weak derivative

弱导数

Analysis, Measure & PDE · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For locally integrable u and v on an open Omega, v is the jth weak derivative of u if integral u partial_j phi=-integral v phi for every smooth compactly supported test phi. Higher-order derivatives use the sign (-1)^{|alpha|}.

开集 $\Omega$ 上的局部可积函数 $u,v$ 若对每个 $\varphi\in C_c^\infty(\Omega)$ 满足 $\int_\Omega u\partial_j\varphi\,dx=-\int_\Omega v\varphi\,dx$，则 $v$ 为 $u$ 的第 $j$ 个弱导数；高阶情形的分部积分符号为 $(-1)^{|\alpha|}$。

Example

For $u(x)=|x-1|$ on $(0,2)$, splitting at $1$ gives weak derivative $-1$ for $x<1$ and $1$ for $x>1$.

Usage and distinctions

Every distribution has distributional derivatives, but a weak derivative in this sense must have a locally integrable representative.

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Definition sources

1. John K. Hunter — Notes on Partial Differential Equations§3.1, Definition 3.1 · Checked 2026-10-14

Translation sources

1. 孙天阳 — 偏微分方程2§3.2 弱导数，定义3.2.1，printed p.47 · Checked 2026-10-14

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