Harmonic Function / Laplace's equation

Yao Yao on June 5, 2018

harmonic 这个词的意思太多了,比如在 periodic signals 里翻译成 “谐波”。而 Harmonic Function 的翻译是 “调和函数

本篇 quote from V7. Laplace’s Equation and Harmonic Functions, M.I.T. 18.02 Notes, Exercises and Solutions by Jeremy Orloff

Laplace operator

The two-dimensional Laplace operator, or laplacian as it is often called, is denoted by $\nabla^2$ or $lap$, and defined by

\[\nabla^2 = \frac {\partial^{2}}{\partial x^{2}} + \frac {\partial^{2}}{\partial y^{2}}\]


\[\nabla^2 = \frac {\partial^{2}}{\partial x_1^{2}} + \frac {\partial^{2}}{\partial x_2^{2}} + \cdots + \frac {\partial^{2}}{\partial x_n^{2}}\]

注意 $\nabla^2$ 其实是一个参数为 $f$ 的函数,只是我们不写成 $\nabla^2(f)$ 而是直接用 $\nabla^2 f$ 表示:

\[\nabla^2 f = \frac {\partial^{2} f}{\partial x^{2}} + \frac {\partial^{2} f}{\partial y^{2}}\]

where $f(x, y)$ is a twice differentiable functions.

The notation $\nabla^2$ comes from thinking of the operator as a sort of symbolic scalar product:

\[\nabla^2 = \nabla \cdot \nabla = \left ( \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial x} \mathbf{j} \right ) \cdot \left ( \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial x} \mathbf{j} \right ) = \frac {\partial^{2}}{\partial x^{2}} + \frac {\partial^{2}}{\partial y^{2}}\]

Notice that the laplacian is a linear operator, that is it satisfies the two rules

  • $\nabla^2 (u + v) = \nabla^2 u + \nabla^2 v$
  • $\nabla^2 cu = c(\nabla^2 u)$

for any two twice differentiable functions $u(x, y)$ and $v(x, y)$ and any constant $c$.

Laplace equation


\[\nabla^2 f = 0\]

Harmonic Function

A function $\phi(x, y)$ which has continuous second partial derivatives and satisfies Laplace’s equation is called a harmonic function. I.e.

\[\phi \text{ is harmonic} \iff \nabla^2 \phi = 0\]

Considering laplacian is a linear operator, we have:

\[\phi \text{ and } \psi \text{ harmonic} \Rightarrow (\phi + \psi) \text{ and } c\phi \text{ are harmonic}\]

Examples of harmonic functions

仅列举 Harmonic homogeneous polynomials in two variables 的例子。更多请参考教程。

  • Degree $0$: all constants $c$ are harmonic.
  • Degree $1$: all linear polynomials $ax + by$ are harmonic.
  • Degree $2$: the quadratic polynomials $x^2 − y^2$ and $xy$ are harmonic; all other harmonic homogeneous quadratic polynomials are linear combinations of these, e.g.:
\[\phi(x, y) = a(x^2 − y^2) + bxy\]

where $a b$ are constants.

  • Degree $n$: the real and imaginary parts of the complex polynomial $(x + \mathrm{i} y)^n$ are harmonic.

blog comments powered by Disqus