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?
1 Answer
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:
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?
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.
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.
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$\begingroup$ Thank you very much. I think that someone should write an article about this topic because Google doesn't tell me much. What about the ones related with $L^{p}$ norms? Are there any like that? This question of Nadirashvili looks like Riemann hypothesis. It is so beautiful. $\endgroup$– AlemFeb 1, 2014 at 18:25