Laplace Equation

The equation

Pierre-Simon Laplace's equation is the simplest non-trivial second-order partial differential equation:

∇²φ = 0

equivalently:  ∂²φ/∂x² + ∂²φ/∂y² + ∂²φ/∂z² = 0   (in 3D)

The operator ∇² is the Laplacian — the divergence of the gradient. Functions satisfying ∇²φ = 0 are called harmonic. The equation says nothing about boundaries; on its own, it has infinitely many solutions, from the constant function to polynomials, logarithms, and reciprocals of distance. Picking a unique one requires specifying boundary data — that is the Dirichlet problem.

The Laplace equation is the second-order linear, elliptic, source-free workhorse of mathematical physics. Wherever you have a steady-state diffusion process with no internal sources, the equilibrium distribution satisfies ∇²φ = 0.

What harmonic means physically

The Laplacian ∇²φ at a point measures by how much φ at that point differs from the average of φ on a small sphere around it. Up to a positive constant:

∇²φ(x) ≈ (constant) · (φ̄_sphere(x, r) − φ(x))   for small r

If ∇²φ > 0, φ at the centre is less than the local average — neighbouring points are higher. If ∇²φ < 0, the centre is higher than its surroundings — a local hump. If ∇²φ = 0 everywhere, every point is the exact average of its neighbours. The function is locally balanced.

This local balance is what makes harmonic functions equilibrium states. Heat at a point with above-average neighbours will flow in until the point matches its surroundings (positive ∇²). Heat at a hot peak flows out (negative ∇²). The only steady states are exactly the harmonic functions.

The mean value property

The most beautiful theorem about harmonic functions is exact, not approximate: the value of a harmonic function at any point equals its average over any ball or sphere centred at that point (provided the ball is contained in the region of harmonicity):

φ(x) = (1 / surface area)  ∫_{∂B_r(x)} φ dA
     = (1 / volume)         ∫_{B_r(x)} φ dV

for any r such that B_r(x) ⊂ Ω

The "any r" is striking. The equality does not just hold in the limit r → 0 — it holds for arbitrarily large balls, all the way out to the boundary of where φ is defined.

Concrete check. In 3D, the mean value over a sphere of radius r centered at x: ∫_{∂B_r(x)} φ dA = φ(x) · 4πr². The integral of a harmonic function over any centered sphere scales precisely with the area of that sphere. The proof reduces to the divergence theorem applied to ∇φ.

The mean value property is a complete characterisation: a continuous function with the mean value property on every ball in its domain is automatically harmonic. There is no way to be "approximately" harmonic without being harmonic.

The maximum principle

The mean value property has a one-line consequence: a non-constant harmonic function cannot have an interior maximum. If it did — say φ(x₀) is the largest value, with x₀ in the interior — then the average over any small ball around x₀ is strictly less than φ(x₀), contradicting the mean value property.

Formally, on a bounded domain Ω:

max_{x ∈ closure(Ω)} φ(x) = max_{x ∈ ∂Ω} φ(x)
min_{x ∈ closure(Ω)} φ(x) = min_{x ∈ ∂Ω} φ(x)

This is the maximum principle: the extrema of φ live on the boundary, not in the interior, unless φ is constant.

One immediate consequence: uniqueness for the Dirichlet problem. If φ₁ and φ₂ are two harmonic functions on Ω with the same boundary values, then their difference φ₁ − φ₂ is harmonic with zero boundary values. By the maximum principle, its max and min over the closure equal zero, so φ₁ = φ₂ everywhere.

The Dirichlet problem

The Dirichlet problem is: given continuous boundary data g on ∂Ω, find φ such that

∇²φ = 0 in Ω,    φ = g on ∂Ω.

The problem is the basic well-posed setup of potential theory. For sufficiently regular domains (Lipschitz boundary, in particular smooth boundary), the Dirichlet problem has a unique solution for every continuous boundary data. The existence proof uses the Perron method (the supremum of all subharmonic functions ≤ g on the boundary), or constructive integral formulas where they exist.

The disk — Poisson integral formula

On the unit disk in 2D, the Dirichlet problem has an explicit closed-form solution. If φ = g(e^{iθ}) on the unit circle, then for any interior point r e^{iθ_0} (with r < 1):

φ(r e^{iθ_0}) = (1/2π) ∫_0^{2π} g(e^{iθ}) · (1 − r²) / (1 − 2r cos(θ − θ_0) + r²) dθ

This is the Poisson integral formula (1820). The kernel (1−r²)/(1−2r cos(θ−θ_0)+r²) is the Poisson kernel; it concentrates at θ = θ_0 as r → 1, recovering the boundary value, and spreads out into a smooth interior solution.

Worked example — temperature on a disk

Pick a disk of radius 1 with boundary temperature g(θ) = cos(θ). Find the steady-state interior temperature.

Try φ(r, θ) = r cos(θ) — equivalently φ(x, y) = x. Compute ∇²φ:

∇²x = ∂²x/∂x² + ∂²x/∂y² = 0 + 0 = 0  ✓

And on the boundary, r = 1, so φ = cos(θ) = g. Both conditions satisfied. The temperature inside is just φ(x, y) = x: a perfectly linear temperature gradient across the disk, hottest on the right (x = 1, where cos θ = 1) and coldest on the left.

The mean value property checks: at the centre (0, 0), the average of cos(θ) over the unit circle is (1/2π) ∫_0^{2π} cos θ dθ = 0 — exactly φ(0, 0). Pick any interior point, average φ = x around a small circle of radius ρ: ∫(x_0 + ρ cos α) dα/(2π) = x_0 = φ(x_0, y_0). Mean value holds everywhere, as it must.

Harmonic function gallery

Dimension	Harmonic functions	Where they appear
1D	Linear: φ(x) = ax + b	1D heat equilibrium, trivial
2D	x, y, x²−y², 2xy, log|x|, Re(z^n), Im(z^n)	Re/Im parts of holomorphic functions
2D	arg(z) = arctan(y/x)	Branch cut; angular potential
3D	1, x, y, z, x²−y², xy, …	Polynomial potentials
3D	1 / |x − x₀|	Coulomb / gravitational potential of a point
3D	Y_ℓ^m(θ, φ) · r^ℓ and r^{−ℓ−1}	Spherical harmonics (multipole expansion)
n ≥ 3	|x|^{2−n}	Fundamental solution of ∇² in n-D

Connection to complex analysis (2D)

In 2D, the Laplace equation has a remarkable bridge to complex analysis. A function f : ℂ → ℂ is holomorphic exactly when its real and imaginary parts u(x, y) and v(x, y) satisfy the Cauchy–Riemann equations:

∂u/∂x = ∂v/∂y,    ∂u/∂y = −∂v/∂x.

Differentiating the first equation with respect to x and the second with respect to y, then adding, the cross terms cancel and you get u_xx + u_yy = 0. Similarly v is harmonic. So the real (and imaginary) parts of every holomorphic function are harmonic. Conversely, every harmonic function on a simply connected 2D domain is the real part of some holomorphic function (its harmonic conjugate).

This bridge is the reason 2D potential theory is rich and tractable: conformal maps (holomorphic bijections) preserve harmonicity, so the Dirichlet problem on any simply connected domain can be transplanted to the unit disk via the Riemann mapping theorem, then solved with the Poisson integral formula.

Where Laplace's equation governs

• Electrostatic potential in vacuum. Maxwell's equation ∇·E = ρ/ε₀ becomes ∇²φ = −ρ/ε₀ for E = −∇φ. In vacuum (ρ = 0), Laplace's equation holds. The potential outside a charged conductor, in any cavity or region containing no charge, is harmonic.
• Steady-state heat distribution. The heat equation ∂u/∂t = α∇²u reaches steady state when ∂u/∂t = 0, i.e. ∇²u = 0. A metal plate at thermal equilibrium with prescribed edge temperatures has harmonic interior temperature.
• Gravity in empty space. ∇²φ_grav = 4πGρ becomes ∇²φ_grav = 0 outside masses. The gravitational potential outside a planet is harmonic.
• Incompressible, irrotational fluid flow. Velocity v = ∇φ (irrotational) with ∇·v = 0 (incompressible) means ∇²φ = 0. The velocity potential of inviscid potential flow is harmonic — the basic model for airfoil aerodynamics in classical theory.
• Membrane displacement. A taut elastic membrane (drumhead) under no transverse load satisfies ∇²u = 0 with u prescribed on the rim. The Dirichlet problem on the membrane shape.
• Markov processes — harmonic measures. The probability that a Brownian motion starting at x first exits a region through a given subset of the boundary is itself a harmonic function. Brownian-motion exit distributions are the probabilistic incarnation of harmonicity.
• Conformal field theory. 2D conformal field theory builds on the connection between harmonic functions and holomorphic functions; the central charge, primary operators, and partition functions all draw on potential-theoretic identities.

How Laplace relates to other PDEs

Equation	Form	Type	Reduces to Laplace when
Laplace	∇²φ = 0	Elliptic	— (it is)
Poisson	∇²φ = f	Elliptic	f = 0 (the homogeneous case)
Heat	∂u/∂t = α∇²u	Parabolic	Steady state ∂u/∂t = 0
Wave	∂²u/∂t² = c²∇²u	Hyperbolic	Static u (no time dep.)
Helmholtz	∇²ψ + k²ψ = 0	Elliptic	k = 0 (low-frequency limit)
Biharmonic	∇⁴φ = 0	Elliptic (higher order)	Solutions to ∇²φ = 0 are also biharmonic

Common mistakes

• Believing harmonic ⇒ smooth boundary behaviour. Harmonic functions are real-analytic in their interior, but boundary values can be just continuous. The interior is smooth no matter how rough the boundary data, but you cannot generally extend across the boundary smoothly.
• Confusing the Laplace equation with the Laplace transform. The Laplace equation is a PDE (∇²φ = 0); the Laplace transform is an integral transform F(s) = ∫ e^{−st} f(t) dt. Both named after Pierre-Simon Laplace, but they are different objects.
• Forgetting the Laplacian in spherical or cylindrical coordinates has metric factors. In spherical (r, θ, φ): ∇²φ = (1/r²) ∂(r² ∂φ/∂r)/∂r + (1/(r² sin θ)) ∂(sin θ ∂φ/∂θ)/∂θ + (1/(r² sin² θ)) ∂²φ/∂φ². Treating it as a sum of unweighted second partials gives wrong answers.
• Assuming the Dirichlet problem always has a solution. On domains with cusps or with non-Lipschitz boundary, the problem can be ill-posed: solutions can fail to attain the boundary data continuously even when the data is continuous. Wiener's criterion characterises exactly when each boundary point is "regular."
• Mistaking Liouville's theorem for the maximum principle. Liouville's theorem says bounded harmonic functions on all of ℝⁿ are constant. The maximum principle says interior maxima force constancy on bounded domains. They are related but distinct.
• Misapplying the mean value property near the boundary. The property requires the ball B_r(x) to lie entirely within the region of harmonicity. Approaching the boundary, you must shrink r — there is no mean value identity using a ball that crosses the boundary.
• Confusing ∇²φ = 0 with ∇φ = 0. ∇φ = 0 means φ is constant. ∇²φ = 0 admits a much richer class (every harmonic function, including non-constants like x²−y²). The Laplacian is a sum of second derivatives, not the first derivative.