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

  • 1
    $\begingroup$ 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. $\endgroup$ – Will Jagy Nov 12 '10 at 3:00
  • $\begingroup$ 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? $\endgroup$ – Dale Roberts Nov 12 '10 at 12:50
  • 2
    $\begingroup$ 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.$ $\endgroup$ – 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.

  • 2
    $\begingroup$ Also, if multiple arcs intersect, it will be at equal angles, as locally we are looking at the imaginary part of $z^n.$ $\endgroup$ – 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?


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
  • $\begingroup$ Could you please post the titles of the articles as well? The direct links to MathSciNet cannot always be accessed. $\endgroup$ – Beni Bogosel Feb 24 '16 at 14:52
  • $\begingroup$ Yeah, right. I do that. $\endgroup$ – Loïc Teyssier Feb 24 '16 at 17:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.