Given $y\in\mathbb{R}^N$, I am trying to show that the function $$V(x) = \frac{\lvert y\rvert^2-\lvert x\rvert^2}{\lvert y-x\rvert^N}$$ is harmonic in $\mathbb{R}^N\setminus\{y\}$. While it is always possible to try to compute the derivatives explicitly, I wonder whether there is a more elegant proof of this fact.

This example also motivates a more general question:

How can one show that a given $C^2$ function $f:\Omega\subset\mathbb{R}^N\to\mathbb R$ is harmonic?

I can give a partial answer to the question:

  1. If $f$ is radial, then $f$ is harmonic iff $$f(x) = a + b\lvert x\rvert^{2-N},\quad N \neq 2,$$ $$f(x) = a + b\log\lvert x\rvert,\quad N = 2,$$ for some $a,b\in\mathbb R$.

  2. If $f$ can be easily integrable on balls or spheres, one can check whether $f$ satisfies the mean property, i.e., $$f(x) = \frac{1}{R^{N-1}\omega_N}\int_{S_R(x)} f(y)d\sigma(y) = \frac{N}{R^N\omega_N}\int_{B_R(x)}f(z)dz.$$

  3. In the case where $N = 2$, one can try to find a holomorphic function $\zeta$ such that $f = \Re\zeta$.

For proofs of (1) and (2), see [Folland, Chap. 2].

Finally, at least to the extent of elementary PDEs textbooks, I find that there is a significant shortage of examples of harmonic functions in dimension $>2$. Since these references generally only treat properties (1) - (3), none of which are very adequate for $N > 2$ (other than (1)). A final question would then be:

Is there a large class of useful examples of harmonic functions for dimension $> 2$, such as the holomorphic for $N = 2$?


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