Basic Theorems Regarding Harmonic Functions

Proposition 1: Let $A \subseteq \mathbb{R}^2$ be open and let $f, g : A \to \mathbb{R}$ be harmonic on $A$. Then: a) $f + g$ is harmonic on $A$. b) $kf$ is harmonic on $A$ for all $k \in \mathbb{R}$.

• Proof of a) Suppose that $f$ and $g$ are harmonic on $A$. Then the second partial derivatives of $f$ and $g$ exist and moreover we have that $\Delta (f) = 0$ and $\Delta (g) = 0$. Then on all of $A$ we have that:

(1)

\begin{align} \quad \frac{\partial^2 (f + g)}{\partial x^2} + \frac{\partial^2 (f + g)}{\partial y^2} &= \frac{\partial}{\partial x} \left [ \frac{\partial (f + g)}{\partial x} \right ] + \frac{\partial}{\partial y} \left [ \frac{\partial (f + g)}{\partial y} \right ] \\ &= \frac{\partial}{\partial x} \left [ \frac{\partial f}{\partial x} + \frac{\partial g}{\partial x} \right ] + \frac{\partial}{\partial y} \left [ \frac{\partial f}{\partial y} + \frac{\partial g}{\partial y} \right ] \\ &= \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 g}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 g}{\partial y^2} \\ &= \left ( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right ) + \left ( \frac{\partial^2 g}{\partial x^2} + \frac{\partial^2 g}{\partial y^2} \right ) \\ &= \Delta (f) + \Delta (g) \\ &= 0 \end{align}

• So $f + g$ is harmonic on $A$.

• Proof of b) Suppose that $f$ is harmonic on $A$ and $k \in \mathbb{R}$. Then:

(2)

\begin{align} \quad \frac{\partial^2 kf}{\partial x^2} + \frac{\partial^2 kf}{\partial y^2} = k \left [ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right ] = k \Delta (f) = 0 \end{align}

• So $kf$ is harmonic on $A$.