Harmonic function

Definition

Let $U \subseteq \mathbb{R}^n$ be open. A function $u \colon U \to \mathbb{R}$ is harmonic on $U$ if $u$ is of class C^2 on $U$ and its Laplacian vanishes on $U$: $$\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \dots + \frac{\partial^2 u}{\partial x_n^2} = 0.$$

Remarks

Each affine function is harmonic. In the plane, $x^2 - y^2$, $x y$ and $e^x \sin y$ are harmonic, and so is $\ln(x^2 + y^2)$ on $\mathbb{R}^2 \setminus \{0\}$. On $\mathbb{R}^3 \setminus \{0\}$, $1 / \|x\|$ is harmonic. It is the potential of a unit point charge at the origin. The real and imaginary parts of a holomorphic function of $z = x + i y$ are harmonic functions of $(x, y)$. Mean value property: let the closed ball of centre $a$ and radius $r$ lie in $U$. Then $u(a)$ is the average of $u$ on the sphere of radius $r$ about $a$, and also on the ball.

Explanations

L1 human

A harmonic function has no local bumps or dips: its value at each point is the average of its values around the point. A stretched rubber sheet held at fixed heights on its rim has a harmonic height function inside.