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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 http://dl.dropbox.com/s/ldluyhymy23asq4/levelsets0.png

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.

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

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

Update:

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

alt text http://dl.dropbox.com/s/fl3fr8vdoeyuaxv/levelsets1.png

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 http://dl.dropbox.com/s/ar9idpj8zf0x7k6/levelsets2.png

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

Maybe a good example is where one has 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 as it seems like something might be related to this question there?

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 http://dl.dropbox.com/s/ldluyhymy23asq4/levelsets0.png

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

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

Update:

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

alt text http://dl.dropbox.com/s/fl3fr8vdoeyuaxv/levelsets1.png

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 http://dl.dropbox.com/s/ar9idpj8zf0x7k6/levelsets2.png

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

Maybe a good example is where one has 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 as it seems like something might be related to this question there?

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

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

Update:

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

alt text http://dl.dropbox.com/s/fl3fr8vdoeyuaxv/levelsets1.png

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 http://dl.dropbox.com/s/ar9idpj8zf0x7k6/levelsets2.png

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

Maybe a good example is where one has 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 as it seems like something might be related to this question there?