Laplace equation

Laplace方程

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

Translation evidence: Literature usage

Meaning in context

The Laplace equation on an open set in R^n is the homogeneous second-order PDE Delta u=sum_{j=1}^n partial_j^2 u=0, classically for u in C^2 or with derivatives interpreted weakly.

Laplace 方程是各坐标二阶偏导数之和等于零的齐次偏微分方程，经典解要求二阶连续可微，也可在弱意义下解释。

Example

An affine function u(x,y)=2x−5y+3u(x,y)=2x-5y+3 solves Δu=0\Delta u=0 since all second derivatives vanish.

Usage and distinctions

Boundary data are a separate part of the problem; the equation alone does not determine a unique solution.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. John K. Hunter — Notes on Partial Differential EquationsChapter 2 opening, Laplace equation and coordinate Laplacian, printed p.19 · Checked 2026-10-14

Translation sources

  1. 北京大学数学科学学院 — 数学分析（实验班）课程介绍第三部分，级数理论：六、调和函数和Laplace方程初步 · Checked 2026-10-14
Edit source / propose a PRReport a correction

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