Harmonic Functions and Potential Theory

The real and imaginary parts of a holomorphic function solve Laplace's equation — they are harmonic — and much of complex analysis is the two-dimensional theory of such functions. This page develops potential theory on its own terms: the mean-value property and maximum principle, the Poisson integral solving the Dirichlet problem, Harnack's inequality, and subharmonic functions. It promotes the harmonic-conjugate remark of holomorphic.md to a full treatment, supplies the compactness-free half of conformal mapping, and feeds the Riemann mapping proof. It builds on complex-plane.md and the real analysis spine.

Harmonic functions and conjugates

A real function $u(x,y)$ on a domain is harmonic if it is $C^2$ and satisfies Laplace's equation $$\Delta u=u_{xx}+u_{yy}=0.$$ Differentiating the Cauchy–Riemann equations shows the real and imaginary parts of a holomorphic $f=u+iv$ are both harmonic, and $v$ is the harmonic conjugate of $u$ (unique up to a constant).

Theorem (local existence of a conjugate). On a simply connected domain every harmonic $u$ is the real part of a holomorphic function $f=u+iv$; the conjugate $v$ is recovered by integrating $-u_{y}dx+u_{x}dy$ (an exact form by $\Delta u=0$, cf. de Rham).

So locally "harmonic" and "real part of holomorphic" coincide; the obstruction on multiply connected domains is topological (e.g. $\log |z|$ is harmonic on $\mathbb{C}\setminus \{0\}$ but its conjugate $\arg z$ is multivalued). Harmonicity is preserved by holomorphic change of variables — this is why conformal mapping solves potential problems (conformal-mapping.md).

The mean-value property and the maximum principle

Cauchy's integral formula, taken on a circle, has a purely real shadow:

Mean-value property. A harmonic $u$ equals its average over every circle in its domain: $u(z_0)=\frac{1}{2\pi }{\int }_0^{2\pi }u(z_0+r{e}^{i\theta })>d\theta .$

From it follows the workhorse of the subject:

Maximum principle. A harmonic function on a domain attains no interior maximum or minimum unless it is constant; on a bounded domain its extrema lie on the boundary. Consequently a harmonic function is determined by its boundary values (uniqueness for the Dirichlet problem).

This is the harmonic parent of the holomorphic maximum modulus principle ($|f|=e^{\log |f|}$ with $\log |f|$ subharmonic).

The Poisson integral and the Dirichlet problem

The Dirichlet problem asks for a harmonic function on a domain with prescribed boundary values. On the unit disc it is solved explicitly by the Poisson kernel:

Poisson integral formula. For continuous boundary data $u(e^{i\theta })=g(\theta )$, the unique harmonic extension to the disc is $$u(r{e}^{i\phi })=\frac{1}{2\pi }{\int }_0^{2\pi }>\frac{1-r^2}{1-2r\cos (\phi -\theta )+r^2}g(\theta )d\theta .$$

The kernel $P_r(\phi -\theta )=\frac{1-r^2}{|1-r{e}^{i(\phi -\theta )}{|}^2}$ is positive with total mass $1$ and concentrates at the boundary as $r\to 1$ (an approximate identity), so $u\to g$ at the boundary. Via the Riemann mapping theorem the disc solution transports to any simply connected domain, making the Poisson integral the master formula of planar potential theory. It also exhibits the boundary-value ↔ interior-value duality that becomes the physicist's dispersion relation (analytic-continuation.md).

Harnack's inequality

Positivity of the Poisson kernel yields quantitative control of positive harmonic functions:

Harnack's inequality. If $u\ge 0$ is harmonic on $D(0,R)$, then for $|z|=r<R$ $$\frac{R-r}{R+r}u(0)\le u(z)\le \frac{R+r}{R-r}u(0).$$

Harnack's principle. An increasing sequence of harmonic functions either diverges to $+\infty$ everywhere or converges (locally uniformly) to a harmonic function.

Harnack's principle is the compactness input to the Perron method below and a harmonic analogue of the normal-family machinery.

Subharmonic functions and the Perron method

An upper-semicontinuous $u$ is subharmonic if it lies below its own circle averages (equivalently $\Delta u\ge 0$ when $C^2$); $\log |f|$ for holomorphic $f$ is the prototype. Subharmonic functions obey the maximum principle and are the flexible "test functions" of potential theory.

Perron's method. The Dirichlet problem on a general domain is solved by taking the supremum of all subharmonic functions dominated by the boundary data; the result is harmonic in the interior. Whether it attains the boundary data depends on a local barrier, giving a clean criterion (regularity of boundary points) for solvability.

Perron's construction generalizes the Poisson formula beyond the disc and connects to capacity and the classification of removable sets. It also supplies the existence half of the Riemann mapping theorem on domains without smooth boundary.

Physics: planar potentials

Any static, source-free 2D potential — electrostatic, ideal-fluid, or steady thermal — is harmonic, hence the real part of a holomorphic complex potential $\Phi =\varphi +i\psi$, with equipotentials $\varphi =\text{const}$ orthogonal to field lines $\psi =\text{const}$. This is the working content of conformal mapping; the physics forward-links are collected in physics-bridge.md.

References

• Ahlfors, Complex Analysis, Ch. 6 (harmonic functions, Poisson, Perron).
• Stein & Shakarchi, Complex Analysis, Ch. 3 §3 and Ch. 5.
• Axler, Bourdon & Ramey, Harmonic Function Theory.