Harmonic Functions and Mean Values in Rn — Examples

Let $n\ge 1$, $a\in \mathbb{R}$, and $b\in {\mathbb{R}}^n$. Every affine function $x\mapsto a+b\cdot x$ is harmonic on ${\mathbb{R}}^n$. If $n\ge 2$ and $i,j\in \{0,\dots ,n-1\}$ are distinct, then $x\mapsto x_i{x}_j$ is harmonic on ${\mathbb{R}}^n$.

For $z=x+iy$, the components $\mathrm{Re}{z}^2=x^2-y^2$ and $\mathrm{Im}{z}^2=2xy$, and likewise those of every $z^m$, are harmonic on ${\mathbb{R}}^2$.

Let $n\ge 1$ and $U\in C^2((0,\mathrm{\infty }))$. The radial function $u:{\mathbb{R}}^n\setminus \{0\}\to \mathbb{R}$ given by $u(x)=U(\mathrm{\mid }x\mathrm{\mid })$ is harmonic precisely when $U^{\mathrm{\prime }\mathrm{\prime }}(r)+\frac{n-1}{r}{U}^{\mathrm{\prime }}(r)=0(r>0)\mathrm{.}$ Thus the families are $U(r)=a+br$ for $n=1$, $U(r)=a+b\log r$ for $n=2$, and $U(r)=a+b{r}^{2-n}$ for $n\ge 3$.

A function harmonic on a punctured domain need not extend harmonically across the puncture.

One ball-mean identity at one centre forces harmonicity.

Let $n\ge 1$ and define $v:{\mathbb{R}}^n\to \mathbb{R}$ by $v(x)=\mathrm{\mid }{x}_1\mathrm{\mid }$. This apparent corner is not distributionally harmonic on ${\mathbb{R}}^n$; indeed $\mathrm{\Delta }v=2{\delta }_{\{x_1=0\}}$. Thus a distributionally harmonic locally integrable function has a unique smooth harmonic representative; in particular, the actual corner $\mathrm{\mid }{x}_1\mathrm{\mid }$ cannot be distributionally harmonic on any open set meeting the hyperplane $\{x_1=0\}$.