Laplace's equation

Laplace's differential equation is the statement that a scalar-valued function's Laplacian is zero.

\nabla 2 𝑓 = 0

Properties of solutions

A solution $𝑓 : ℝ 𝑛 \to ℝ$ to Laplace's equation has the following properties in general

It has the shape of something under tension, e.g. a rubber band or rubber sheet under tension.
Let $⃗ 𝐱 \in ℝ 𝑛$ , and $𝜕 𝐵 𝑟 (⃗ 𝐱) \subseteq ℝ 𝑛$ be the set of all points with distance $𝑟$ from $⃗ 𝐱$ . Then the average of the image $𝑓 (𝜕 𝐵 𝑟 (⃗ 𝐱))$ is $𝑓 (⃗ 𝐱)$ .
The value of $𝑓$ within a region $𝑅 \subseteq ℝ 𝑛$ is completely and uniquely determined by the values of $𝑓$ on the boundary $𝜕 𝑅$ .
There are no local maxima or minima (this follows from the previous property)

Holomorphic extension

Any function $𝑢 : ℝ 2 \to ℝ$ satisfying Laplace's equation may be extended to a holomorphic function $𝑓 : ℂ \to ℂ$ with $𝑢 (𝑥, 𝑦) = ℜ (𝑥 + 𝑖 𝑦)$ unique up to an imaginary constant. This follows directly from the Cauchy-Riemann equations.

Laplace's equation

Properties of solutions

Holomorphic extension

See also