Sobolev space

Sobolev空间

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Translation evidence: Literature usage

Meaning in context

For open Omega in R^n, an integer k>=0, and 1<=p<=infinity, W^{k,p}(Omega) consists of Lp equivalence classes whose distributional derivatives of every order at most k are represented by Lp functions.

对欧氏空间开集、非负整数 k 及 1≤p≤∞，Sobolev 空间 W^{k,p} 由本身及直到 k 阶的分布导数均由 Lp 函数表示的几乎处处相等类组成。

Example

On $(-2,2)$, $u(x)=|x|$ lies in $W^{1,2}$: its weak derivative is the sign function, with squared integral $4$.

Usage and distinctions

Classical differentiability everywhere is unnecessary; a jump indicator has a Dirac derivative and is not in W^{1,p} across the jump.

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Definition sources

1. John K. Hunter — Notes on Partial Differential Equations§3.5, Definition 3.23 · Checked 2026-10-14

Translation sources

1. 电子科技大学 — 泛函分析教学大纲第三章，3.2 Sobolev空间的定义及其基本性质 · Checked 2026-10-14

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