Sobolev trace

Sobolev函数的迹

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

Translation evidence: Literature usage

Meaning in context

For a bounded smooth domain Omega and 1 <= p < infinity, the first-order Sobolev trace is the bounded linear map from W^{1,p}(Omega) to L^p(partial Omega, dS) extending boundary restriction of smooth functions on the closure; its kernel is W^{1,p}_0(Omega).

对有界光滑区域 Ω 及 1≤p<∞，一阶 Sobolev 迹是把闭包上光滑函数的边界限制连续延拓所得的有界线性映射，从 W^{1,p}(Ω) 到边界表面测度下的 L^p(∂Ω)，其核为 W^{1,p}_0(Ω)。

Example

On $(0,1)$, the trace of $u(x)=x(1-x)$ is the endpoint pair $(0,0)$; the trace of the constant function $1$ is $(1,1)$.

Usage and distinctions

This record concerns function boundary values, not operator or matrix trace. Higher-order zero boundary conditions also require derivative traces. The Chinese title is a descriptive qualification of the inspected usage 迹 for Sobolev boundary values, not a claim of normative standardization.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. John K. Hunter — Notes on Partial Differential Equations§3.9, Theorem 3.44 and proof (first-order case), printed pp.72–73; §3.11, transfer to smooth bounded domains by boundary flattening · Checked 2026-10-15

Translation sources

1. 程代展、赵寅 — 系统与控制中的近代数学基础（第2版），清华大学出版社预览§3.9, 定理3.9.1（迹定理）, printed p.134: Sobolev函数在边界Γ上的迹及迹映射γ（取m=1即函数值迹） · Checked 2026-10-15

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.