Proof of the Maximum Principle for Harmonic Functions

The maximum principle for harmonic functions states that a non-constant harmonic function on a domain cannot attain its maximum or minimum in the interior.

Statement

Theorem8.15Strong maximum principle for harmonic functions

Let $u$ be harmonic on a connected domain $D \subseteq \mathbb{C}$ . If $u$ attains its maximum at some point $z_0 \in D$ , then $u$ is constant.

Equivalently, if $u$ is non-constant, then $u(z) < \sup_D u$ for all $z \in D$ .

Proof

Suppose $u(z_0) = M = \sup_D u$ for some $z_0 \in D$ .

Step 1: Mean value property forces constancy locally.

Choose $r > 0$ with $\overline{D}(z_0, r) \subset D$ . By the mean value property:

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

Since $u(z_0 + re^{i\theta}) \leq M$ for all $\theta$ , and the average equals $M$ , we must have $u(z_0 + re^{i\theta}) = M$ for all $\theta$ (if $u < M$ on a set of positive measure, the average would be strictly less than $M$ ).

This holds for all $r$ with $D(z_0, r) \subset D$ , so $u \equiv M$ on the largest disk centered at $z_0$ contained in $D$ .

Step 2: Connectedness argument.

Define $S = \{z \in D : u(z) = M\}$ . We show $S$ is both open and closed in $D$ .

$S$ is open: If $z_1 \in S$ , the argument above shows $u \equiv M$ on a disk around $z_1$ , so that disk is in $S$ .

$S$ is closed: $S = u^{-1}(M)$ is the preimage of a closed set under a continuous function.

Step 3: Conclusion.

Since $D$ is connected, $S$ is either empty or all of $D$ . Since $z_0 \in S$ , we have $S = D$ and $u \equiv M$ . $\blacksquare$

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Weak Maximum Principle

Theorem8.16Weak maximum principle

Let $D$ be a bounded domain and $u$ continuous on $\overline{D}$ and harmonic on $D$ . Then

$\max_{\overline{D}} u = \max_{\partial D} u.$

Similarly, $\min_{\overline{D}} u = \min_{\partial D} u$ .

Proof

Since $\overline{D}$ is compact and $u$ is continuous, $u$ attains its maximum at some point $z^*\in \overline{D}$ . If $z^* \in D$ , then by the strong maximum principle, $u$ is constant on $D$ and hence on $\overline{D}$ (by continuity), so the maximum is also attained on $\partial D$ . If $z^* \in \partial D$ , we are done. $\blacksquare$

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Applications

ExampleUniqueness of the Dirichlet problem

If $u_1, u_2$ are both harmonic on $D$ , continuous on $\overline{D}$ , and equal on $\partial D$ , then $w = u_1 - u_2$ is harmonic with $w = 0$ on $\partial D$ . By the maximum principle, $w \leq 0$ on $D$ ; by the minimum principle, $w \geq 0$ . Hence $w \equiv 0$ and $u_1 = u_2$ .

ExampleContinuous dependence on boundary data

If $|g_1(\zeta) - g_2(\zeta)| \leq \varepsilon$ for all $\zeta \in \partial D$ , and $u_1, u_2$ are the corresponding Dirichlet solutions, then $|u_1(z) - u_2(z)| \leq \varepsilon$ for all $z \in D$ . This follows from the maximum principle applied to $u_1 - u_2$ and $u_2 - u_1$ . The Dirichlet problem is stable: small changes in boundary data produce small changes in the solution.

RemarkComparison with subharmonic functions

The maximum principle extends to subharmonic functions: if $v$ is subharmonic on $D$ and $u$ is harmonic with $v \leq u$ on $\partial D$ , then $v \leq u$ on $D$ . This comparison principle is fundamental in potential theory and PDE.