Let $X$ be the set of all harmonic functions external to the unit sphere on $\mathbb R^3$ which vanish at infinity, so if $V \in X$, then $\nabla^2 V(\mathbf{r}) = 0$ on $\mathbb R^3 – S(2)$ and $\lim_{r \rightarrow \infty} V(r) = 0$. Now consider a function $f: X \rightarrow \mathbb R$, defined by
$$
f(V)(\mathbf{r}) = || \nabla V(\mathbf{r}) ||^2
$$
For some given $V \in X$, I am looking for all functions $W \in X$ which satisfy
$$
f(V) = f(W)
$$
Certainly $W = \pm V$ will satisfy the condition. Can anyone find nontrivial solutions for $W$?
My approach so far:
The condition on $V$ and $W$ is
$$
\nabla V \cdot \nabla V = \nabla W \cdot \nabla W
$$
By defining $\phi = V + W$ and $\psi = V – W$, this is equivalent to
$$
\nabla \phi \cdot \nabla \psi = 0
$$
I then tried expanding $\nabla \phi$ and $\nabla \psi$ in a basis of vector spherical harmonics and plugging into the above formula. This step makes use of the fact $\nabla^2 \phi = \nabla^2 \psi = 0$ and leads to the following condition on the expansion coefficients:
$$
\nabla \phi \cdot \nabla \psi = \sum_{nm,n'm'} \phi_{nm} \psi_{n'm'} \left( \frac{1}{r} \right)^{n+n'+4} \left( (n+1)(n'+1) Y_{nm} Y_{n'm'} + \partial_{\theta} Y_{nm} \partial_{\theta} Y_{n'm'} + \frac{1}{\sin^2{\theta}} \partial_{\phi} Y_{nm} \partial_{\phi} Y_{n'm'} \right)
$$
Its not clear to me how to proceed from here, or whether this is even the correct approach to take. I could get rid of the sum over $n',m'$ by integrating both sides over a unit sphere and using the orthogonality relations for the spherical harmonics. Doing this gives:
$$
\sum_{nm} (n+1)(2n+1) \phi_{nm} \psi_{nm} = 0
$$
though I'm not sure that yields any additional insight. I would appreciate any ideas.
Best Answer
This problem has an important background in geomagnetism. When planning the MAGSAT satellite mission (1979/80) to determine the spherical harmonic coefficients of the Earth's magnetic field from space, Backus (JGR, 1970) showed that a measurement of the total field intensity $||\nabla V||$ on a spherical shell is in general not sufficient to uniquely determine $V$ (not regarding trivial non-uniqueness due to gauge and sign). He did this by explicitly constructing some counterexample by means of similar arguments as used in the question and comments here. As a consequence, MAGSAT became the first mission carrying a vector magnetometer instead of the much simpler absolute field sensors. In relation to the problem as posed here, Backus (Quart. Journ. Mech. and Applied Math., 21, 195-221 , 1968) proved the following theorem:
THEOREM 5: Suppose $\phi$ and $\phi'$ are harmonic outside some open bounded set $W$ in $\mathbf{R}^n$ and vanish at infinity. Suppose that $| \nabla \phi| = | \nabla \phi'|$ outside some sphere which contains $W$. If $n \geq 3$ then one of the two functions $\phi -\phi'$ and $\phi +\phi'$ vanishes identically outside $W$.