The Laplacian and Harmonic Functions

Part 14, Chapter 14: Differential Equations

Learning objectives

Write the Laplacian $\nabla^2$ in Cartesian and polar (and spherical) coordinates
Recognize harmonic functions as solutions of $\nabla^2 u = 0$
State and use the maximum principle and the mean value property
Connect harmonic functions in the plane to real and imaginary parts of holomorphic functions

The Laplacian $\nabla^2$ is the most important differential operator in all of mathematics outside of $d/dx$ itself. It appears in Laplace's equation (steady-state heat, electrostatics, gravity in vacuum), the heat equation (time-dependent diffusion), the wave equation (sound, light, vibrations), and the Schrodinger equation (quantum mechanics). It also has a clean geometric meaning: $\nabla^2 u(p)$ measures the gap between $u$ at the point $p$ and the average of $u$ on a small sphere around $p$ . That single picture explains every theorem about harmonic functions in one image.

Definition

In Cartesian coordinates on $\mathbb{R}^n$ , the Laplacian is the sum of unmixed second partial derivatives:

$\nabla^2 u = \dfrac{\partial^2 u}{\partial x_1^2} + \dfrac{\partial^2 u}{\partial x_2^2} + \cdots + \dfrac{\partial^2 u}{\partial x_n^2}$ 12+dfracpartial2upartialx22+cdots+dfracpartial2upartialxn2

Some books write $\Delta u$ instead of $\nabla^2 u$ ; they mean the same thing. The notation $\nabla^2$ emphasises that it is the divergence of the gradient: $\nabla^2 = \nabla \cdot \nabla$ .

Laplace's equation is the homogeneous PDE $\nabla^2 u = 0$ . Solutions are called harmonic functions.

Polar form (2D)

In polar coordinates $(r, \theta)$ on $\mathbb{R}^2$ , the Laplacian becomes:

$\nabla^2 u = \dfrac{\partial^2 u}{\partial r^2} + \dfrac{1}{r} \dfrac{\partial u}{\partial r} + \dfrac{1}{r^2} \dfrac{\partial^2 u}{\partial \theta^2}$

The first two terms combine into the radial Laplacian $(1/r)(r u_r)_r$ . The polar form is essential for problems with circular symmetry (a heated disk, a vibrating drum, electrostatics around a cylinder).

The maximum principle

A fundamental property of harmonic functions: on a bounded domain $D$ , a non-constant harmonic function cannot attain its maximum (or minimum) in the interior of $D$ . The extrema occur on the boundary $\partial D$ .

Intuitively: if $u$ peaked at an interior point $p$ , then nearby values of $u$ would be on average smaller than $u(p)$ , but the mean-value property below says $u(p)$ equals the average over a small sphere centered at $p$ . Contradiction.

The maximum principle gives uniqueness for the Dirichlet problem: if $u, v$ are both harmonic on $D$ and $u = v$ on $\partial D$ , then $u = v$ throughout $D$ . (Apply the principle to $u - v$ , which is harmonic with zero boundary values, so its max and min on $\bar D$ are both zero.)

The mean value property

If $u$ is harmonic on a domain containing the closed ball $\bar B(p, r)$ , then

$u(p) = \dfrac{1}{|\partial B(p, r)|} \displaystyle\int_{\partial B(p, r)} u \, dS$ partialB(p,r)u,dS

The value of a harmonic function at the center of a sphere is the average over the sphere. This is equivalent to being harmonic (Gauss's theorem). It implies harmonic functions are infinitely differentiable, even when one might expect only the second derivatives in $\nabla^2 u = 0$ to exist.

Plot the harmonic function $u(x, y) = x^2 - y^2$ as a level-set / contour map. Notice the saddle structure at the origin: along the $x$ -axis $u$ has a minimum at $0$ , but along the $y$ -axis it has a maximum. That is precisely how harmonic functions avoid interior extrema, any apparent peak is balanced by a valley in a perpendicular direction.

Where this shows up

Electrostatics: The electric potential in a region with no charge satisfies $\nabla^2 V = 0$ . Boundary conditions are the voltages on conductors; the solution gives the potential everywhere else, this is the Dirichlet problem for Laplace.
Steady-state heat: The temperature distribution in a body that has reached thermal equilibrium satisfies Laplace's equation. The maximum principle says no hot spots can spontaneously appear in the interior, thermodynamically obvious, mathematically a theorem.
Complex analysis: If $f(z) = u(x, y) + i v(x, y)$ is holomorphic, then both $u$ and $v$ are harmonic. This is the Cauchy-Riemann engine: every 2D harmonic function is locally the real part of a complex-analytic function, which means harmonic-function theory in the plane is just complex analysis dressed up in real-variable notation.
Image processing: The "Poisson editing" technique used in Photoshop's healing brush solves a Laplace-style PDE on a small image patch with the surrounding pixels as boundary data, producing a seamless blend.

Pause and think: If a harmonic function is bounded on all of $\mathbb{R}^n$ , what can it be? (Hint: Liouville's theorem for harmonic functions says it must be constant.)

Try it

Verify directly that $u(x, y) = x^2 - y^2$ is harmonic on $\mathbb{R}^2$ .
Is $u(x, y) = e^x \sin y$ harmonic? If yes, identify the holomorphic function $f(z)$ whose imaginary part it is.
Compute $\nabla^2 u$ for $u(x, y, z) = x^2 + y^2 + z^2$ in 3D. Is it harmonic?
Use the maximum principle to argue that if $u$ is harmonic on the closed unit disk with $u = 3$ on the boundary circle, then $u \equiv 3$ .
Verify the mean value property by example: compute the average of $u(x, y) = x^2 - y^2$ over the unit circle and check that it equals $u(0, 0) = 0$ .

A trap to watch for

The Laplacian on the sphere (or in spherical coordinates in 3D) has extra terms from the curvature: $\nabla^2 u = (1/r^2) \partial_r(r^2 u_r) + (1/r^2) \nabla_S^2 u$ r(r2ur)+(1/r2)nablaS2u, where $\nabla_S^2$ S2 is the spherical Laplace-Beltrami operator. Do not just translate the 2D polar formula by adding a third coordinate, the right formula has the Jacobian $r^2$ built into the radial derivative. When in doubt, derive it from $\nabla^2 = \nabla \cdot \nabla$ using the curvilinear coordinate divergence formula.

What you now know

You can write the Laplacian in any coordinate system you need, verify whether a function is harmonic, and apply the maximum principle and mean value property to extract uniqueness and smoothness results for the Dirichlet problem. The next two sections put the Laplacian into time: the heat equation $u_t = \nabla^2 u$ t=nabla2u describes diffusion, and the wave equation $u_{tt} = c^2 \nabla^2 u$ tt=c2nabla2u describes propagation.

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References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 14.
Evans, L. C. (2010). Partial Differential Equations (2nd ed.). AMS, ch. 2 (Laplace and Poisson equations).
Strauss, W. A. (2007). Partial Differential Equations: An Introduction (2nd ed.). Wiley, ch. 6.
Axler, S., Bourdon, P., Ramey, W. (2001). Harmonic Function Theory (2nd ed.). Springer, ch. 1-2.
Ahlfors, L. V. (1979). Complex Analysis (3rd ed.). McGraw-Hill, ch. 4 (harmonic functions and Poisson integrals).