Laplace's equation

Laplace’s equation is the partial differential equation

${\nabla }^{2}u=0,$

where ${\nabla }^2={\partial }_{xx}+{\partial }_{yy}+{\partial }_{zz}$ is the Laplacian operator. Solutions are called harmonic functions.

Laplace’s equation is the most studied PDE in mathematical physics. It governs steady-state behavior of an enormous range of physical phenomena.

Where it shows up

Electrostatics. The electric potential $\varphi$ in a region of free space (no charge) satisfies ${\nabla }^2\varphi =0$. Comes from $\mathbf{E}=-\nabla \varphi$ and $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$ — set $\rho =0$.

Fluid dynamics. For an irrotational, incompressible flow, the velocity field is $\mathbf{v}=\nabla \varphi$ (irrotational means $\nabla \times \mathbf{v}=0$, which forces a potential on simply connected domains) and $\nabla \cdot \mathbf{v}=0$ (incompressible). Together: ${\nabla }^2\varphi =0$.

Steady-state heat conduction. The heat equation ${\partial }_{t}u=\alpha {\nabla }^{2}u$ in steady state (${\partial }_{t}u=0$) gives ${\nabla }^{2}u=0$.

Gravitational potential in vacuum regions (Newtonian gravity).

Equilibrium of an elastic membrane under tension.

Probability and Brownian motion. Solutions are harmonic; harmonic measure is a probability measure.

Boundary-value problems

A standard Laplace problem: given a region $\Omega$ and a prescription of $u$ (or $\partial u/\partial \hat{\mathbf{N}}$) on the boundary $\partial \Omega$, find $u$ inside.

• Dirichlet problem: $u$ specified on $\partial \Omega$.
• Neumann problem: $\partial u/\partial \hat{\mathbf{N}}$ specified on $\partial \Omega$.

Both have unique solutions (under standard regularity) — this is what makes Laplace’s equation so useful: the interior is fully determined by the boundary.

Solution techniques

• Separation of variables in Cartesian, polar, cylindrical, spherical coordinates: reduce to ODEs whose solutions are sinusoids, Bessel functions, Legendre polynomials, spherical harmonics, etc.
• Method of images for problems with simple boundary geometry.
• Conformal mapping in 2D: map the problem region to a disk or half-plane via an analytic function, solve there, pull back. Works because conformal maps preserve Laplace’s equation. See Conformality.
• Green’s functions.
• Numerical methods: finite difference, finite element.

Why complex analysis is so powerful for 2D Laplace problems

The real and imaginary parts of any analytic function are automatically harmonic. So you have an infinite supply of solutions to Laplace’s equation in 2D, parameterized by all the analytic functions $f(z)=u(x,y)+iv(x,y)$.

If the boundary geometry can be conformally mapped to a half-plane or disk, the analytic-function machinery solves the problem. This was a major theme of 19th-century mathematical physics. See Conformality and Möbius transformation for the simplest examples.

Properties of solutions

Uniqueness. Two solutions of the Dirichlet problem with the same boundary data agree everywhere.

Maximum principle. A non-constant harmonic function on a bounded domain attains its max and min on the boundary, never the interior. No “hills” in the interior.

Mean value property. The value of $u$ at the center of any disk inside the domain equals the average over the bounding circle:

$u(P)=\frac{1}{2\pi r}{\oint }_{|z-P|=r}uds.$

Smoothness. Solutions are automatically $C^{\infty }$.

All of these mirror properties of analytic functions, since (in 2D) harmonic functions are real / imaginary parts of analytic functions.

Generalization

The non-homogeneous version is Poisson’s equation, ${\nabla }^{2}u=f(\mathbf{r})$ — Laplace with a source term on the right. Newton’s potential of a continuous mass distribution satisfies Poisson’s equation with $f=-4\pi G\rho$.

In dynamics (non-steady-state), Laplace is replaced by the wave equation ${\partial }_t^{2}u=c^2{\nabla }^{2}u$ or the heat equation ${\partial }_{t}u=\alpha {\nabla }^{2}u$ — same operator ${\nabla }^2$ but with time evolution.