TITLE: Harmonic level sets and boundary data
QUESTION [8 upvotes]: This is probably a classic problem, so a good reference book or paper to get me started on this type of question would be great:
Let $\mathbb{D} \subset \mathbb{C}$ be the unit disk with boundary $\mathbb{T}$. Take a function $f : \mathbb{T} \to \mathbb{R}$ and let $u_f$ be the harmonic extension to $\mathbb{D}$ using the Poisson kernel, i.e. the Poisson integral of $f$.

What can be said about the levels sets of $u_f$ in relation to the data $f$?

For example, If the data $f$ has certain behaviour such as, say, oscillatory taking positive and negative values.

REPLY [2 votes]: I know that's been a while now that the question has been asked, but as I'm looking more or less into this topic, I think I should share some of my discoveries in the literature. I'm somewhat amazed at the really small number of occurrences of this theme.

I. De Carli & S. Hudson, Geometric remarks on the level curves of harmonic functions (2010), MR2586969
L. Flatto, D. Newman & H. Shapiro, The level curves of harmonic functions (1966), MR197755
W. Boothby, The topology of the level curves of harmonic functions with critical points (1951), MR43456