§7.1 Basic Properties of Harmonic Functions

7.1 Basic Properties of Harmonic Functions

The first question we answer here is the cases when a harmonic function can be represented as the real part of a holomorphic function. We will see that this is always the case locally, and thus harmonic functions are locally the real parts of holomorphic functions. However, this is not always the case globally, and we will see examples of this later.

As a refresher, we restate Theorem 3.4.4 $Theorem 3.4.4 Theorem 3.4.4$ :

Theorem 7.1.1.

Let

𝐷 (𝑎, 𝑟)

be a disk in

ℂ

and

𝑢 : 𝐷 (𝑎, 𝑟) \to ℝ

be a harmonic function. Then there exists a holomorphic function

𝑓 : 𝐷 (𝑎, 𝑟) \to ℂ

such that

𝑢 \equiv ℜ𝔢 (𝑓)

Proof.

Letting

𝑓 = 𝑢 + i 𝑣

, where

𝑣

is given in (3.6.10)

(3.6.10) (3.6.10)

suffices. ⁠

□

Theorem 7.1.2 (Elliptic Regularity).

Let

𝑈 \subseteq ℂ

be a region. Then a harmonic function

𝑢 \in 𝐶^{2} (𝑈)

is infinitely differentiable.

Proof.

For each point

𝑎 \in 𝑈

, there is a disk

𝐷 (𝑎, 𝑟_{𝑎}) \subset 𝑈

and a holomorphic function

𝑓_{𝑎} : 𝐷 (𝑎, 𝑟_{𝑎}) \to ℂ

such that

𝑢 \equiv ℜ𝔢 (𝑓_{𝑎})

𝐷 (𝑎, 𝑟_{𝑎})

. Since

𝑓_{𝑎}

is holomorphic, it is infinitely differentiable, and thus

𝑢

is infinitely differentiable on

𝐷 (𝑎, 𝑟_{𝑎})

. Since

𝑎 \in 𝑈

was arbitrary,

𝑢

is infinitely differentiable on

𝑈

. ⁠

□

In Section 3.6.2 $Section 3.6.2 Section 3.6.2$ and Section 3.4 $Section 3.4 Section 3.4$ , we proved the equivalence between $\nabla^{2} 𝑢 \equiv 0$ and the global mean-value property, in which the identity holds on every disk compactly contained in the domain. However, to derive harmonicity it suffices to verify the local mean-value property: for each fixed point $𝑧_{0}$ , the identity need only hold for every $0 < 𝜀 < 𝜀^{'}$ for some $𝜀^{'} > 0$ . Once harmonicity follows, the global mean-value property follows as well. See Theorem 3.6.2.1 $Theorem 3.6.2.1 Theorem 3.6.2.1$ .

Moreover, the statement of the maximum principle (Theorem 3.4.5 $Theorem 3.4.5 Theorem 3.4.5$ ) can be extended to:

Theorem 7.1.3 (Minimum Principle for Harmonic Functions).

Let $𝑈 \subseteq ℂ$ be open and connected and let $𝑢 : 𝑈 \to ℝ$ be harmonic. Suppose that $\exists 𝑧_{0} \in 𝑈$ and a neighborhood $𝑉 \subseteq 𝑈$ of $𝑧_{0}$ such that either

𝑢 (𝑧_{0}) \geq 𝑢 (𝑧) \forall 𝑧 \in 𝑉 .

Then

𝑢

is constant on

𝑈

This can be obtained by applying Theorem 3.4.5 $Theorem 3.4.5 Theorem 3.4.5$ to $- 𝑢$ . Then it follows that

max_{𝑧 \in \overline{𝑈}} 𝑢 (𝑧) = max_{𝑧 \in 𝜕 𝑈} 𝑢 (𝑧) and min_{𝑧 \in \overline{𝑈}} 𝑢 (𝑧) = min_{𝑧 \in 𝜕 𝑈} 𝑢 (𝑧),

as otherwise, the maximum (or minimum) would be attained within $𝑈$ itself, which implies constancy.

We have seen that many holomorphic properties have harmonic analogs. This is also the case for the Schwarz Reflection Principle (Theorem 5.5.2 $Theorem 5.5.2 Theorem 5.5.2$ ), for which we now provide an approach that yields an alternative proof.

Theorem 7.1.4 (Schwarz Reflection Principle for Harmonic Functions).

Let $𝑈 \subseteq ℂ$ be a connected region on one side of the real axis such that there exists a non-degenerate curve $𝛾 \subseteq 𝜕 𝑈$ such that $𝛾 \subseteq ℝ$ . Let $𝑢 : 𝑈 \to ℝ$ be harmonic with continuity up to $𝑈 \cup 𝛾$ such that $𝑢$ vanishes on $𝛾$ , and let $\tilde{𝑈} = {\overline{𝑧} : 𝑧 \in 𝑈}$ be the reflection of $𝑈$ across the real axis. Then there exists a unique harmonic function

\tilde{𝑢} (𝑧) = {\begin{cases} 𝑢 (𝑧) & if 𝑧 \in 𝑈 \\ - 𝑢 (\overline{𝑧}) & if 𝑧 \in \tilde{𝑈} \\ 0 & if 𝑧 \in 𝛾 \end{cases} .

𝑈 \cup \tilde{𝑈} \cup 𝛾

Proof.

Because satisfying the Laplace equation is equivalent to satisfying the global mean-value property, it suffices to show that $\tilde{𝑢}$ satisfies the local mean-value property (Theorem 3.6.2.1 $Theorem 3.6.2.1 Theorem 3.6.2.1$ ).

For all $𝑧 \in 𝑈$ , there exists $𝜀^{'} > 0$ such that $\overline{𝐷 (𝑧, 𝜀^{'})} ⋐ 𝑈$ (without crossing $ℝ$ ). Then for all $𝜀 \in (0, 𝜀^{'})$ , $\tilde{𝑢}$ satisfies the local mean-value identity on $𝜕 𝐷 (𝑧, 𝜀)$ since $𝑢$ is harmonic.
For all $𝑧 \in \tilde{𝑈}$ , there exists $𝜀^{'} > 0$ such that $\overline{𝐷 (𝑧, 𝜀^{'})} ⋐ \tilde{𝑈}$ (without crossing $ℝ$ ). Then for all $𝜀 \in (0, 𝜀^{'})$ , we get that
$\begin{aligned} \tilde{𝑢} (𝑧) & = - 𝑢 (\overline{𝑧}) = - \frac{1}{2 π} \int_{0}^{2 π} 𝑢 (\overline{𝑧} + 𝜀 e^{i 𝜃}) d 𝜃 \\ = - \frac{1}{2 π} \int_{0}^{2 π} 𝑢 (\overline{𝑧} + 𝜀 e^{i 𝜃}) d 𝜃 \\ = \frac{1}{2 π} \int_{0}^{2 π} \tilde{𝑢} (𝑧 + 𝜀 e^{- i 𝜃}) d 𝜃 \\ = \frac{1}{2 π} \int_{0}^{2 π} \tilde{𝑢} (𝑧 + 𝜀 e^{i 𝜃}) d 𝜃 . \end{aligned}$
Therefore, for all $𝜀 \in (0, 𝜀^{'})$ , $\tilde{𝑢}$ satisfies the local mean-value identity on $𝜕 𝐷 (𝑧, 𝜀)$ .
For all $𝑧 \in 𝛾$ (so $𝑧 = \overline{𝑧}$ ), for any $𝜀 > 0$ such that
$\overline{𝐷 (𝑧, 𝜀)} ⋐ 𝑈 \cup \tilde{𝑈} \cup 𝛾,$
$\begin{aligned} \frac{1}{2 π} \int_{0}^{2 π} \tilde{𝑢} (𝑧 + 𝜀 e^{i 𝜃}) d 𝜃 = & \frac{1}{2 π} [\int_{0}^{π} 𝑢 (𝑧 + 𝜀 e^{i 𝜃}) d 𝜃 - \int_{π}^{2 π} 𝑢 (\overline{𝑧} + 𝜀 e^{- i 𝜃}) d 𝜃] \\ = \frac{1}{2 π} [\int_{0}^{π} 𝑢 (𝑧 + 𝜀 e^{i 𝜃}) d 𝜃 - \int_{π}^{2 π} 𝑢 (𝑧 + 𝜀 e^{- i 𝜃}) d 𝜃] \\ (by substituting 𝜃 \to 2 π - 𝜃) \\ = \frac{1}{2 π} [\int_{0}^{π} 𝑢 (𝑧 + 𝜀 e^{i 𝜃}) d 𝜃 + \int_{π}^{0} 𝑢 (𝑧 + 𝜀 e^{- i (2 π - 𝜃)}) d 𝜃] \\ = \frac{1}{2 π} [\int_{0}^{π} 𝑢 (𝑧 + 𝜀 e^{i 𝜃}) d 𝜃 - \int_{0}^{π} 𝑢 (𝑧 + 𝜀 e^{i 𝜃}) d 𝜃] \\ = 0 = \tilde{𝑢} (𝑧) . \end{aligned}$
Then $\tilde{𝑢}$ satisfies the local mean-value identity on $𝜕 𝐷 (𝑧, 𝜀)$ .

Then by Theorem 3.6.2.1 $Theorem 3.6.2.1 Theorem 3.6.2.1$ , $\tilde{𝑢}$ is harmonic. ⁠ $□$