Laplace's Equation - Key Properties

Harmonic functions possess remarkable properties that distinguish them from solutions of other PDEs and make them fundamental in complex analysis and potential theory.

DefinitionMean Value Property

If $u$ is harmonic in a ball $B_R(x_0) \subset \Omega$ , then: $u(x_0) = \frac{1}{|B_R|}\int_{B_R(x_0)} u(y)\,dy = \frac{1}{|\partial B_R|}\int_{\partial B_R(x_0)} u(y)\,dS_y$

That is, the value at the center equals both the volume average and the surface average over any ball centered at that point.

This property characterizes harmonic functions: a continuous function satisfying the mean value property on all balls is harmonic.

DefinitionMaximum Principle

Strong Maximum Principle: If $u$ is harmonic in a connected domain $\Omega$ and attains its maximum (or minimum) at an interior point, then $u$ is constant.

Weak Maximum Principle: $\max_{\overline{\Omega}} u = \max_{\partial\Omega} u$ and $\min_{\overline{\Omega}} u = \min_{\partial\Omega} u$ .

Remark

The maximum principle has immediate consequences:

Uniqueness: Two solutions with the same boundary data must be identical
Stability: Small changes in boundary data produce small changes in the solution
Comparison: If $u_1$ and $u_2$ satisfy $\nabla^2 u_1 \geq \nabla^2 u_2$ with $u_1 \geq u_2$ on $\partial\Omega$ , then $u_1 \geq u_2$ in $\Omega$

Infinite Differentiability: Every harmonic function is automatically $C^\infty$ and even real analytic. If $u$ is harmonic, all derivatives $\frac{\partial^k u}{\partial x_{i_1} \cdots \partial x_{i_k}}$ are also harmonic.

ExampleLiouville's Theorem

A bounded harmonic function on all of $\mathbb{R}^n$ must be constant. This follows from the mean value property: if $|u| \leq M$ everywhere, then averaging over larger and larger balls shows $u$ cannot vary.

More generally, a harmonic function with polynomial growth $|u(x)| \leq C(1 + |x|^k)$ must be a polynomial of degree at most $k$ .

DefinitionHarnack's Inequality

If $u$ is non-negative and harmonic in $B_R(x_0)$ , then for any $B_r(x_0)$ with $r < R$ : $\left(\frac{R-r}{R+r}\right)^{n-2}\frac{R^n}{(R+r)^{n-1}} \leq \frac{u(x)}{u(x_0)} \leq \frac{R^n}{(R-r)^{n-1}}$ for all $x \in B_r(x_0)$ .

These properties reflect the equilibrium nature of harmonic functions and their connection to probability theory (harmonic functions are expectations of Brownian motion).