Find all possible functions, $F(r)$, harmonic in $2$ and $3$ dimensions,

Question

Sources: this is an old advanced calculus exam question, which I think is asking for harmonic functions.

The problem statement is:

Suppose $F(r)$ is a smooth function of $r$ for $r>0$ . Define $\Phi(x) = F(|x|)$ with $x = (x_1, ...,x_d) \in \mathbb{R}^d$ and $|x| = (\large \sum_i^dx_i^2)^{\frac{1}{2}}$.For $d=2$ and $d=3$, find all possible functions $F(r)$ so that $$\sum_{i=1}^d (\frac {\partial}{\partial x_i})^2\Phi(x) = 0.$$

Any hints or comments are welcome.

So, it appears that this problem statement is just a tricky way of asking to find all functions $\Phi$ such that

$$\Phi_{x_1x_1} + \Phi_{x_2x_2} = 0 $$

and

$$\Phi_{x_1x_1} + \Phi_{x_2x_2} + \Phi_{x_3x_3} = 0$$

I.e., we are looking for all functions that are harmonic. But what is weird about this question is that it doesn't really specify a domain in which we are to find these harmonic functions. Unless the domain is simply understood to be for all $| x| \ne 0$.

Thanks,

Answer 1

It's easy enough to carry out the calculation for all $d$. $$ \frac{\partial^2 }{\partial x_j^2}F(r)=\frac{\partial}{\partial x_j}\left(F'(r)\frac{x_j}{r}\right)=F''(r)\frac{x_j^2}{r^2}+F'(r)\frac{1}{r}-F'(r)\frac{x_j^2}{r^3} $$ Adding these together gives \begin{align} \nabla^2 F(r) &= F''(r)+F'(r)\frac{d}{r}-F'(r)\frac{1}{r} \\ &= F''(r)+F'(r)\frac{d-1}{r}. \end{align} In order to have a harmonic function $F(r)$ for $d \ge 2$, it is necessary and sufficient that, for $d > 1$, $$ rF''(r)+(d-1)F'(r)=0 \\ (r^{d-1}F'(r))'=0,\\ F'(r)=\frac{C}{r^{d-1}}. $$ For $d=2$, the solutions are $$ F(r)=A\ln r + B $$ For $d\ge 3$, the solutions are $$ F(r)= A\frac{1}{r^{d-2}}+B. $$

Answer 2

Added later: Let me finish the argument here, because it's quite simple. Suppose $u$ is harmonic and radial on $U = \mathbb C \setminus \{0\}.$ Consider the radial harmonic function

$$v(z) = u(1) + \frac{u(2)-u(1)}{\ln 2}\ln|z|, z \in U.$$

Then $u=v$ on the boundary of the the annulus $A=\{1<|z|<2\}.$ By the maximum/minimum principle for harmonic functions, $u=v$ in all of $A.$ The harmonic function $u-v$ is then harmonic on $U$ and vanishes on an open subset of $U.$ Again by the maximum principle, $u-v\equiv 0$ in $U.$

Thus $u$ has the form $a + b\ln |z|,$ for constants $a,b.$ Conversely, $a + b\ln |z|$ is radial and harmonic on $U$ for any constants $a,b.$

In $\mathbb R^n,n>2,$ we can replace $\ln |z|$ by the radial harmonic function $|x|^{2-n};$ the argument then will be exactly the same.

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Previous hint: In each case you have seen examples of radial harmonic functions. There are the constants for example. But there are others (not many though). Use these functions to match the values of $\Phi$ on the boundary of any annular region centered at the origin.