# Notes - Complex Analysis MT23, Dirichlet problem

### Flashcards

What is the Dirichlet problem in the case of $v : \mathbb R^2 \to \mathbb R$?

Suppose:

- $v : \mathbb R^2 \to \mathbb R$
- $v$ harmonic, i.e. $\partial^2 _ x v + \partial^2 _ y v = 0$
- $D \subset \mathbb R^2$
- We know the values of $v$ on $\partial D$

Then the Dirichlet problem is about solving for the values of $v$ on all of $D$.

Suppose that $U \subset \mathbb C$ is a simply-connected, proper open subset of $\mathbb C$ and $v : U \to \mathbb R$ is a harmonic function. How can you link such a $v$ to a holomorphic function?

There is a holomorphic function $f : U \to \mathbb C$ such that $\text{Re}(f) = v$.

The Dirichlet problem states the following. Suppose:

- $v : \mathbb R^2 \to \mathbb R$
- $v$ harmonic, i.e. $\partial^2 _ x v + \partial^2 _ y v = 0$
- $D \subset \mathbb R^2$
- We know the values of $v$ on $\partial D$

Then we wish to determine the values of $v$ on all of $D$. There is a result that says such a $v$ is actually the real part of a holomorphic function. How can we then use properties of holomorphic functions and conformal maps to make studying such problems easier?

- By the identity theorem, holomorphic functions with values defined on the boundary will be completely determined on the rest of $D$, and we can calculate the values inside $D$ using Cauchy’s integral formula
- Composition with conformal maps means a function will still be harmonic
- This means we can transport solutions around