weak solution of a PDE

弱解

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

Translation evidence: Literature usage

Meaning in context

In the zero-Dirichlet Poisson problem on bounded open Omega with f in L2, a weak solution is u in H_0^1(Omega) satisfying integral gradient u dot gradient phi=integral f phi for every phi in H_0^1(Omega); other PDEs require their own test identity and solution space.

对有界开集 $\Omega$ 上的零边界 Poisson 问题及 $f\in L^2(\Omega)$，弱解是 $u\in H_0^1(\Omega)$，对每个 $\varphi\in H_0^1(\Omega)$ 满足 $\int_\Omega\nabla u\cdot\nabla\varphi\,dx=\int_\Omega f\varphi\,dx$；其他方程须另定测试式与解空间。

Example

On $(0,1)$, $u=x(1-x)$ obeys $\int_0^1u'\varphi'=2\int_0^1\varphi$, so it weakly solves $-u''=2$ with zero boundary values.

Usage and distinctions

A distributional identity without the specified H_0^1 membership does not impose this boundary condition or finite energy.

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Definition sources

1. John K. Hunter — Notes on Partial Differential Equations§4.1, Definition 4.2 and equation (4.4), printed p.92 · Checked 2026-10-14

Translation sources

1. 孙天阳 — 偏微分方程2§4.2 散度型椭圆方程弱解的定义，printed p.73 · Checked 2026-10-14

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