Because I'm doing research in the area of harmonic function theory I would like to know are there any conjectures in the theory of harmonic functions in $\mathbb{R}^{n}$ still open. I know that there are many conjectures related to manifolds, but I don't want them. I want conjectures that are just related to harmonic functions on open sets in $\mathbb{R}^{n}$. Are there some really famous ones and long open?



Conjectures in classical harmonic function theory
There are many conjectures about harmonic functions in $R^n$ which are open. Not surprisingly, most of them are very hard. Even about harmonic polynomials in $R^3$ there
are many unsolved problems.


For example:


1. A question of of Nadirashvili.
Let $u$ be a harmonic function in $R^3$, and consider the set $\{ x:u(x)=0\}$. Can this set
have finite area?
2. To the best of my knowledge, for $n>2$, local topological classification
of zero sets is not known.
Not speaking of the global classification, even for harmonic polynomials.
3. If harmonic functions in subregions of $R^n$ are permitted, there are any more problems, for example on the zero sets of gradients of potentials of discrete masses.


One outstanding problem goes back to
Maxwell: let $u$ be a potential of $m$ positive (or even unit) charges in $R^3$. How many
zeros can the gradient of $u$ have? One conjecture is that always finitely many. But suppose we know that. How many? Maxwell said at most $(m-1)^2$. 


Believe me or not, but this is unproved even for $m=3$.


Here is another old problem: is it true that the gravity force created by an infinite discrete
set of positive masses in $R^3$ equals to zero at some point? Even in $R^2$ this problem
is not completely solved.


You may look at <http://www.math.purdue.edu/~eremenko/uns1.html> for discussion of the last two problems. 


Update. Since I wrote this, 1 has been solved:
Logunov, Alexander, Nodal sets of Laplace eigenfunctions: proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture. Ann. of Math. (2) 187 (2018), no. 1, 241–262.


