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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.

alt text

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The thing to concentrate on is that in the interior, connected components of level sets of a harmonic function are either nice arcs, as your graphic shows, or like the zero set of the imaginary part of $$ ( x + i y)^n $$ for integer $ n \geq 2.$ If you picked exactly the right level in your graphic you could see such a point where multiple arcs intersect. – Will Jagy Nov 12 '10 at 3:00
That's really interesting, is there a theorem that explains this? Is there a way to choose the boundary data in such a way to make the level sets not arcs? maybe to make the boundaries of the level sets to be only a Lipschitz curve? – Dale Roberts Nov 12 '10 at 12:50
Let $\theta$ be the angle around the boundary, from $0$ to $2 \pi$ as usual. Let your boundary function be $f(\theta) = \sin ( 3 \theta)$ and make a new graphic, and make sure that one of the level sets depicted is $ u_f = 0.$ – Will Jagy Nov 12 '10 at 22:36

(This really should be a comment, but is a bit long.)

Consider elliptic regularity and maximum principle. Interior regularity of harmonic function means that away from the boundary, your function $u_f$ is real analytic, and hence its level sets will also be analytic arcs away from where $\nabla u_f = 0$. Near the boundary the behaviour may degenerate, and the regularity will depend on the regularity of $f$.

Where $\nabla u_f = 0$, by the maximum principle for harmonic functions, the Hessian $\nabla^2 u_f$, if non-zero, must be indefinite (or you can see that just by noting it is trace free). So the non-degenerate critical points of $u_f$ are saddle points, and have the classical structure with two level-lines intersecting there.

In general, near a critical point, taking the Taylor expansion of the function $u_f$, you must have

$$ u_f(x) = u_f(x_0) + \sum_{|\alpha| \geq m} a_\alpha (x-x_0)^\alpha $$

where $\alpha$ are multi-indices, $a_\alpha$ are coefficients, and $m \geq 1$ is the highest number of derivatives to which $u_f$ vanish. The maximum principle states that

$$ \sum_{|\alpha| = m+1} a_\alpha x^\alpha $$

cannot be a signed function. So the critical point there correspond to an intersection of at least 2 and up to $m+1$ level curves.

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Also, if multiple arcs intersect, it will be at equal angles, as locally we are looking at the imaginary part of $z^n.$ – Will Jagy Nov 19 '10 at 23:43

Here are some images to complement the nice comments made by Will and Willie:

alt text

I should have been more precise. I actually meant:

What can be said about the geometry of the level regions of $u_f$ in relation to the data $f$?

That is, for $a, b \in \mathbb{R}$ with $a<b$, we define a level region as the set

$$ \{z:a \le u_f(z) \le b\}. $$

In particular, I am interested what can be said about the level regions for "bad" boundary data. Let say I have some boundary data that causes $u_f$ to blow-up as it approaches the boundary $\mathbb{T}$. For example, here is an image where I "cycle" colors for the level regions, so fast color cycling as one approaches a point $z \in \mathbb{T}$ radially is equivalent to fast blow-up.

alt text

It seems like the level regions have very particular shapes. What can be said about the geometry of these regions?

Maybe a good place to start is to assume that we have the estimate $$ \sup_{z \in \mathbb{D}} | u_f'(z)|(1-|z|^2) < \infty, $$ so that $u_f$ is a Bloch function?

I am just a novice to these types of questions. References to books, papers, and theorems to get me started would be much appreciated. I plan to start reading Garnett and Marshall Harmonic measure, would that be a good place to start in regards to obtaining these types of results?

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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.

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