On the partial derivatives of a harmonic function

Question

Well, here is the thing. We know that the laplacian operator commutes with any partial derivative of a function, if the function is smooth. We also know that a harmonic function is infinitely differentiable, thus every partial derivative of a harmonic funtion is harmonic.

My problem is the following: $\frac 1 r$, where $r=(\sum_i x_i^2)^\frac 1 2$ is harmonic if $r\ne 0$. Is every partial derivative of this function going to be harmonic, even though the space where this function is harmonic is not compact nor simply connected?

Thank you for your help!

Answer 1

Clairaut's Theorem states that the order of taking mixed partial derivatives of a function $f(x_{1}, \ldots, x_{n})$ does not matter so long as $f$ is sufficiently smooth.

Also, note that $\frac{1}{r}$ is only harmonic in dimension $n = 3$. In general dimension, the function $f(x_{1}, \ldots, x_{n}) = r^{2-n}$ is harmonic, however.

Answer 2

The answer is yes. However, note that the domain of definition for $\frac{1}{r}$ is in fact connected.

EDIT:

To prove this let $u(x):D \subseteq \mathbb{R}^n \to \mathbb{R}$ be any harmonic function in some domain $D$, and let $\partial_i$ be any partial derivative operator. $u$ satisfies

$$\Delta u \equiv 0$$ for $x \in D$. Applying $\partial_i$ to both sides gives

$$\partial_i \Delta u \equiv \partial_i 0 \equiv 0 $$ for all $x \in D$. But since $\partial_i \Delta=\Delta \partial_i$, we have

$$\Delta(\partial_i u) \equiv 0 $$

for all $x \in D$, which means that $\frac{\partial u}{\partial x_i}$ is harmonic in the same domain.