The Laplace Equation

Laplace's equation is the second-order partial differential equation

\nabla^{2} u = 0

, stating that a function's value at each point equals the average of its surrounding values. Its solutions, called harmonic functions, model steady-state phenomena: equilibrium temperature, electrostatic potential in charge-free regions, incompressible irrotational fluid flow, and gravitational potential in vacuum. Named for Pierre-Simon Laplace (1749–1827), it is the canonical elliptic PDE and the foundation of Potential Theory.

Laplace's equation requires that the sum of the unmixed second partial derivatives of a function vanish everywhere in a domain. A function satisfying it is harmonic, and such functions enjoy extraordinary rigidity: they are infinitely differentiable (in fact real-analytic), they attain their maxima and minima only on the boundary, and they are uniquely determined by their boundary values. It governs every physical system that has reached equilibrium with no internal sources — the steady-state version of the heat, diffusion, and wave equations. The surprising fact is that harmonic functions are precisely the real and imaginary parts of complex-analytic functions in two dimensions, fusing PDE theory with Complex Analysis so tightly that conformal mapping becomes a tool for solving boundary-value problems.

The Core Insight

A harmonic function has no local extrema in the interior: at every point its value equals the average over any surrounding sphere. There are no "peaks" or "valleys" — heat has flowed until everything is as smooth and balanced as possible. This single averaging property explains uniqueness, the maximum principle, smoothness, and why boundary data alone determines the entire interior.

Formal Statement

Laplace's Equation and the Laplacian

For a scalar function $u$ defined on an open domain $Ω \subseteq R^{n}$ , Laplace's equation is

\nabla^{2} u \equiv Δ u = i = 1 \sum n \frac{\partial ^{2} u}{\partial x _{i}^{2}} = 0.

⟳The Laplacian

Δ

is the divergence of the gradient,

\nabla \cdot (\nabla u)

; expand in Cartesian coordinates to obtain the sum of pure second derivatives.

Here $u : Ω \to R$ is the unknown field, $x_{i}$ are Cartesian coordinates, and $Δ = \nabla \cdot \nabla$ is the Laplace operator. A function satisfying [1] on $Ω$ is called harmonic on $Ω$ . The inhomogeneous version $Δ u = f$ is Poisson's Equation.

In two and three dimensions the equation reads explicitly:

u_{xx} + u_{y y} = 0 (n = 2), u_{xx} + u_{y y} + u_{z z} = 0 (n = 3) .

⟳Specialize equation [1] to

n = 2

and

n = 3

, writing

u_{xx} = \partial^{2} u / \partial x^{2}

, etc.

In polar coordinates $(r, θ)$ the Laplacian becomes

Δ u = \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial u}{\partial r}) + \frac{1}{r ^{2}} \frac{\partial ^{2} u}{\partial θ ^{2}} = u_{r r} + \frac{1}{r} u_{r} + \frac{1}{r ^{2}} u_{θ θ} .

⟳Apply the chain rule to

x = r cos θ

y = r sin θ

, expressing

\partial_{x}, \partial_{y}

in terms of

\partial_{r}, \partial_{θ}

, and sum the second derivatives — the cross terms cancel.

Mean Value Property

If $u$ is harmonic on $Ω$ and the closed ball $\overline{B} (x_{0}, R) \subset Ω$ , then the value at the centre equals the average over the bounding sphere $\partial B$ and over the solid ball:

u (x_{0}) = \frac{1}{∣ \partial B ( x _{0} , R ) ∣} \int_{\partial B (x_{0}, R)} u d S = \frac{1}{∣ B ( x _{0} , R ) ∣} \int_{B (x_{0}, R)} u d V .

⟳Define

ϕ (r) = \fint_{\partial B (x_{0}, r)} u d S

; differentiate, apply the divergence theorem to get

ϕ^{'} (r) = \frac{r}{n} \fint_{B} Δ u d V = 0

, so

ϕ

is constant and equals its

r \to 0

limit

u (x_{0})

Remarkably, this property characterizes harmonic functions: a continuous function satisfying the mean value property everywhere is automatically harmonic and smooth.

Maximum Principle

A non-constant harmonic function on a connected domain $Ω$ attains neither a maximum nor a minimum at any interior point. If $Ω$ is bounded and $u$ is continuous on $\overline{Ω}$ ,

\overline{Ω} max u = \partial Ω max u, \overline{Ω} min u = \partial Ω min u .

⟳If an interior max were attained at

x_{0}

, the mean value property [4] would force

u \equiv u (x_{0})

on a small sphere, hence (by a connectedness argument) on all of

Ω

, contradicting non-constancy.

Why It Holds / Derivation

Uniqueness of the Dirichlet Problem via the Maximum Principle

We prove that the Dirichlet problem — solve $Δ u = 0$ in $Ω$ with $u = g$ on $\partial Ω$ — has at most one solution.

Step 1. Suppose

u_{1}

and

u_{2}

are both harmonic in

Ω

, continuous on

\overline{Ω}

, with

u_{1} = u_{2} = g

\partial Ω

. Step 2. Let

w = u_{1} - u_{2}

. Since the Laplacian is linear,

Δ w = Δ u_{1} - Δ u_{2} = 0

, so

w

is harmonic. Step 3. On the boundary,

w = g - g = 0

. Step 4. By the maximum principle [5],

max_{\overline{Ω}} w = max_{\partial Ω} w = 0

and

min_{\overline{Ω}} w = min_{\partial Ω} w = 0

. Step 5. Therefore

0 \leq w \leq 0

everywhere, i.e.

w \equiv 0

, so

u_{1} = u_{2}

■

Harmonicity of Real Parts of Analytic Functions

We show that if $f = u + i v$ is holomorphic, then $u$ and $v$ are harmonic.

Step 1. Holomorphy implies the Cauchy–Riemann equations:

u_{x} = v_{y}, u_{y} = - v_{x} .

⟳These follow from requiring the complex derivative

f^{'} (z)

to be independent of direction of approach.

Step 2. Differentiate the first equation with respect to

x

u_{xx} = v_{y x}

. Step 3. Differentiate the second with respect to

y

u_{y y} = - v_{x y}

. Step 4. Since

v

is smooth, mixed partials commute (

v_{y x} = v_{x y}

). Adding the two results:

u_{xx} + u_{y y} = v_{y x} - v_{x y} = 0.

⟳Add the results of Steps 2–3 and invoke Clairaut's theorem on equality of mixed partials.

An identical computation shows $v_{xx} + v_{y y} = 0$ . Thus both components are harmonic, and $v$ is the harmonic conjugate of $u$ . $■$

Pierre-Simon Laplace | 1749–1827

The French mathematician and astronomer who, in his monumental Mécanique céleste, used the operator now bearing his name to analyze the gravitational potential of planetary bodies. Laplace argued that the gravitational potential of a mass distribution satisfies $Δ V = 0$ in empty space, reducing the study of forces to a single scalar function. He famously claimed celestial mechanics had no need of the "hypothesis" of divine intervention — the equations sufficed. Though the equation appears earlier in Euler's and Lagrange's fluid work, Laplace's systematic exploitation of potential theory cemented his name to it.

Building Intuition

Imagine a thin metal plate whose edge temperature you fix arbitrarily, then wait for equilibrium. Heat flows from hot to cold until no point can change — meaning every interior point sits exactly at the average temperature of its neighbours. That balance condition is Laplace's equation. Now stretch a rubber membrane (a soap film) over a bent wire loop: the film minimizes its area, and for small slopes its height is harmonic. There are no bumps an ant could climb to find a local summit; the highest and lowest points are always on the wire. This is the maximum principle made tactile. Finally, in 2D, picture the flow lines of a frictionless incompressible fluid: the velocity potential is harmonic, and the streamlines are the level curves of its harmonic conjugate — the same magic that lets Conformal Mapping reshape one flow into another.

Steady-state temperature on a square plate: harmonic interior, fixed boundary, isotherms shown in gold

Worked Examples

Steady-State Temperature in a Disk — Poisson Integral

Given: A circular plate of radius

R = 2

m. The boundary temperature is

g (θ) = 50 + 30 cos θ

degrees Celsius. Find the interior temperature

u (r, θ)

, then evaluate at the center and at

(r, θ) = (1, 0)

. Step 1 — General solution. By separation of variables in [3], the bounded harmonic functions on a disk are

u (r, θ) = \frac{a _{0}}{2} + n = 1 \sum \infty (\frac{r}{R})^{n} (a_{n} cos n θ + b_{n} sin n θ) .

⟳Set

u = P (r) Θ (θ)

in [3]; the angular part gives

Θ^{''} = - n^{2} Θ

, and boundedness at

r = 0

selects the

r^{n}

radial solution over

r^{- n}

Step 2 — Match boundary data. On

r = R

u (R, θ) = \frac{a _{0}}{2} + \sum (a_{n} cos n θ + b_{n} sin n θ) = 50 + 30 cos θ

. Matching Fourier coefficients:

a_{0} /2 = 50

a_{1} = 30

, all others zero. Step 3 — Assemble. With

R = 2

u (r, θ) = 50 + 30 (\frac{r}{2}) cos θ = 50 + 15 r cos θ .

Step 4 — Evaluate. Center

(r = 0)

u = 50

°C — exactly the average of the boundary data, confirming the mean value property. At

(r, θ) = (1, 0)

u = 50 + 15 (1) (1) = 65

°C.

u (r, θ) = 50 + 15 r cos θ; u (0) = 5 0^{\circ} C, u (1, 0) = 6 5^{\circ} C

Electrostatic Potential Between Concentric Spheres

Given: Two concentric conducting spheres, inner radius

a = 0.10

m at potential

V_{a} = 100

V, outer radius

b = 0.30

m at

V_{b} = 0

V. The region between them is charge-free, so

V

is harmonic and depends only on

r

(spherical symmetry). Find

V (r)

and the field at

r = 0.20

m. Step 1 — Radial Laplacian. With spherical symmetry,

Δ V = \frac{1}{r ^{2}} \frac{d}{d r} (r^{2} \frac{d V}{d r}) = 0

, so

r^{2} \frac{d V}{d r} = C_{1} \Rightarrow V (r) = - \frac{C _{1}}{r} + C_{2} .

⟳Integrate

\frac{d}{d r} (r^{2} V^{'}) = 0

once to get

r^{2} V^{'} = C_{1}

, then integrate

V^{'} = C_{1} / r^{2}

again.

Step 2 — Apply boundary conditions.

V (a) = - C_{1} / a + C_{2} = 100

and

V (b) = - C_{1} / b + C_{2} = 0

. Subtracting:

- C_{1} (1/ a - 1/ b) = 100 \Rightarrow - C_{1} (10 - 3.333) = 100 \Rightarrow C_{1} = - 15.0

V·m. Then

C_{2} = C_{1} / b = - 15.0/0.30 = - 50.0

V. Step 3 — Potential.

V (r) = \frac{15.0}{r} - 50.0 (V, with r in metres) .

Check:

V (0.10) = 150 - 50 = 100

V ✓;

V (0.30) = 50 - 50 = 0

V ✓. Step 4 — Field at $r = 0.20$ m.

E = - \frac{d V}{d r} = \frac{15.0}{r ^{2}} = \frac{15.0}{0.04} = 375

V/m, directed radially outward.

V (r) = \frac{15.0}{r} - 50.0 V, E (0.20) = 375 V/m outward

Common Misconceptions

"Harmonic functions can have interior maxima." They cannot (unless constant). If your computed solution has a bump in the interior, you have made an error or your domain contains a hidden source — which means you are really solving Poisson's Equation, not Laplace's. Likewise, do not assume superposition of products gives every solution: separation of variables only works in specially shaped domains (rectangles, disks, spheres) where the boundary aligns with a coordinate surface. Boundary data must be self-consistent. For the Neumann problem (specifying the normal derivative

\partial u / \partial n = h

\partial Ω

), no solution exists unless the compatibility condition

\oint_{\partial Ω} h d S = 0

holds — physically, net heat flux must vanish at steady state. And Neumann solutions are unique only up to an additive constant. The mean value property fails for $Δ u \neq = 0$ . Consider

u (x, y) = x^{2} + y^{2}

on the unit disk. Then

Δ u = 2 + 2 = 4 \neq = 0

, so

u

is not harmonic. At the origin

u (0, 0) = 0

, but the average over the unit circle is

\fint_{0}^{2 π} (cos^{2} θ + sin^{2} θ) \frac{d θ}{2 π} = 1 \neq = 0

. The center value undershoots the boundary average — a hallmark of subharmonic functions (

Δ u \geq 0

), which lie below their boundary averages, the opposite of the harmonic balance.

Connections

Laplace's equation sits at the crossroads of several theories. It is the steady-state limit of the Heat Equation $u_{t} = Δ u$ (set $u_{t} = 0$ ) and of the diffusion equation. Adding a source term yields Poisson's Equation $Δ u = f$ , the workhorse of electrostatics. In two dimensions, harmonic functions are exactly the real/imaginary parts of holomorphic functions (Complex Analysis), so Conformal Mapping transplants solutions between domains. The mean value and maximum principles connect it to Brownian Motion: $u (x)$ equals the expected boundary value reached by a random walker started at $x$ . Finally, harmonic functions are the equilibrium states minimizing the Dirichlet energy $\int_{Ω} ∣\nabla u ∣^{2} d V$ , linking the topic to the Calculus of Variations and Euler–Lagrange theory.

Real-World Applications

Electrostatics (capacitors, transmission lines): the potential in any charge-free region between conductors is harmonic; solving $Δ V = 0$ gives the field, capacitance, and breakdown geometry.
Steady-state heat conduction: temperature distribution in machine parts, building walls, and heat sinks after thermal equilibrium is reached.
Incompressible irrotational flow (aerodynamics): the velocity potential $ϕ$ satisfies $Δ ϕ = 0$ ; classical airfoil theory and flow around obstacles use harmonic potentials and conformal maps.
Gravitational potential (geodesy, astrophysics): in vacuum, Newtonian gravitational potential obeys $Δ V = 0$ (Laplace's original application to planetary orbits and the shape of the Earth).
Image processing and computer graphics: Laplace/Poisson solvers perform seamless image blending ("Poisson image editing") and mesh smoothing by enforcing harmonic interpolation of pixel/vertex data.
Groundwater hydrology: steady seepage of water through soil (Darcy flow) gives a harmonic hydraulic head; used to design dams and predict flow nets.
Membrane and elasticity problems: the deflection of a taut, lightly loaded membrane and certain torsion problems in beams reduce to Laplace's (or Poisson's) equation.

Historical Development

1752 — Euler derives the equation for the velocity potential of incompressible fluid flow, the first appearance of $Δ ϕ = 0$ in physics.
1762 — Lagrange systematizes the velocity-potential approach, reinforcing the role of harmonic functions in fluid dynamics.
1782–1785 — Laplace introduces the operator in his memoirs on planetary attraction, proving the gravitational potential satisfies $Δ V = 0$ in empty space; the name becomes attached.
1813 — Poisson generalizes to $Δ V = - 4 π ρ$ inside matter, correcting Laplace's equation where mass density is nonzero.
1828 — George Green publishes his Essay, introducing Green's functions and Green's identities, the analytic backbone of boundary-value theory.
1840s — Gauss & Thomson (Kelvin) develop potential theory rigorously, including the Gauss mean value theorem and energy methods.
1850s — Dirichlet & Riemann formulate the Dirichlet principle, asserting existence of the energy-minimizing harmonic function — used freely until its rigor is questioned.
1870 — Weierstrass exhibits a counterexample casting doubt on Dirichlet's principle, sparking a search for rigorous existence proofs.
1900–1904 — Hilbert restores the Dirichlet principle on firm footing; Perron (1923) later gives a robust subharmonic-function method for existence under general boundary conditions.

Subtleties & Limits

Existence of solutions can fail at irregular boundary points: Lebesgue's spine (a sharp inward cusp) is a domain where the Dirichlet problem has no continuous solution attaining the prescribed boundary value at the cusp tip — motivating the theory of regular boundary points and Wiener's criterion. More broadly, the regularity theory for elliptic equations (how smooth solutions are, given rough coefficients or data) remains active: De Giorgi–Nash–Moser theory addresses divergence-form equations, but sharp regularity for general nonlinear and degenerate elliptic operators, and for free-boundary problems built on the Laplacian (e.g. the obstacle problem and optimal regularity of free boundaries), continues to be researched. The interplay between harmonic measure, potential theory, and the geometry of rough boundaries (rectifiability) is a vibrant modern frontier.

The Laplace Equation

The Core Insight

Formal Statement

Laplace's Equation and the Laplacian

Mean Value Property

Maximum Principle

Why It Holds / Derivation

Uniqueness of the Dirichlet Problem via the Maximum Principle

Harmonicity of Real Parts of Analytic Functions

Pierre-Simon Laplace | 1749–1827

Building Intuition

Steady-state temperature on a square plate: harmonic interior, fixed boundary, isotherms shown in gold

Worked Examples

Steady-State Temperature in a Disk — Poisson Integral

Electrostatic Potential Between Concentric Spheres

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also

Before this

Where this leads