Harmonic Functions

Harmonic functions are solutions to Laplace's equation and arise naturally as the real and imaginary parts of holomorphic functions. They play a central role in potential theory, physics, and the study of conformal mappings.

Definition and Basic Properties

Definition8.1Harmonic function

A $C^2$ function $u: D \to \mathbb{R}$ on a domain $D \subseteq \mathbb{R}^2$ is harmonic if it satisfies Laplace's equation:

$\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0.$

Theorem8.1Real and imaginary parts of holomorphic functions

If $f = u + iv$ is holomorphic on a domain $D$ , then $u$ and $v$ are harmonic on $D$ . Conversely, if $u$ is harmonic on a simply connected domain $D$ , there exists a harmonic function $v$ (unique up to a constant) such that $f = u + iv$ is holomorphic. The function $v$ is called the harmonic conjugate of $u$ .

ExampleExamples of harmonic functions

$u(x,y) = x^2 - y^2$ is harmonic (it is $\text{Re}(z^2)$ ), with conjugate $v = 2xy$ .
$u(x,y) = \ln\sqrt{x^2+y^2} = \ln|z|$ is harmonic on $\mathbb{R}^2 \setminus \{0\}$ (it is $\text{Re}(\log z)$ ).
$u(x,y) = e^x\cos y$ is harmonic (it is $\text{Re}(e^z)$ ), with conjugate $v = e^x\sin y$ .
$u(x,y) = x/(x^2+y^2)$ is harmonic on $\mathbb{R}^2 \setminus \{0\}$ .

The Cauchy-Riemann Equations

RemarkCauchy-Riemann and harmonicity

The Cauchy-Riemann equations $u_x = v_y$ , $u_y = -v_x$ link harmonic conjugates. Differentiating:

$u_{xx} = v_{yx}, \quad u_{yy} = -v_{xy}.$

If $v$ is $C^2$ , then $v_{xy} = v_{yx}$ , so $u_{xx} + u_{yy} = 0$ . The Cauchy-Riemann equations thus imply harmonicity and provide a method to find the conjugate.

ExampleFinding a harmonic conjugate

Given $u(x,y) = x^3 - 3xy^2$ , find $v$ :

$v_y = u_x = 3x^2 - 3y^2 \implies v = 3x^2y - y^3 + \phi(x)$ .

$v_x = 6xy + \phi'(x) = -u_y = 6xy \implies \phi'(x) = 0 \implies \phi = C$ .

So $v = 3x^2y - y^3 + C$ and $f = (x^3 - 3xy^2) + i(3x^2y - y^3) = z^3$ .

Mean Value Property and Maximum Principle

Theorem8.2Mean value property

A continuous function $u$ on $D$ is harmonic if and only if it satisfies the mean value property: for every disk $\overline{D}(z_0, r) \subset D$ ,

$u(z_0) = \frac{1}{2\pi}\int_0^{2\pi} u(z_0 + re^{i\theta})\,d\theta.$

Theorem8.3Maximum principle for harmonic functions

If $u$ is harmonic and non-constant on a domain $D$ , then $u$ has no local maximum or minimum in $D$ . If $D$ is bounded and $u$ is continuous on $\overline{D}$ , then

$\min_{\partial D} u \leq u(z) \leq \max_{\partial D} u \quad \text{for all } z \in D.$

ExampleTemperature distribution

In steady-state heat conduction, the temperature $u$ is harmonic. The maximum principle says the hottest and coldest points must occur on the boundary — the temperature inside is always an average. This has immediate physical content: heat flows from hot to cold until equilibrium is reached.

Poisson Integral Formula

Theorem8.4Poisson integral formula

If $u$ is harmonic on $|z| < R$ and continuous on $|z| \leq R$ , then for $|z| < R$ :

$u(z) = \frac{1}{2\pi}\int_0^{2\pi} \frac{R^2 - |z|^2}{|Re^{i\theta} - z|^2}\,u(Re^{i\theta})\,d\theta.$

The kernel $P_R(z, \theta) = \frac{R^2 - |z|^2}{|Re^{i\theta} - z|^2}$ is the Poisson kernel.

RemarkSolving the Dirichlet problem on the disk

The Poisson formula solves the Dirichlet problem: given continuous boundary data $g$ on $|z| = R$ , the function $u(z) = \frac{1}{2\pi}\int_0^{2\pi} P_R(z,\theta)g(Re^{i\theta})\,d\theta$ is harmonic in $|z| < R$ and continuous on $|z| \leq R$ with $u = g$ on the boundary.