# MATH3401

Lecture 25

### Harmonic functions

Remark: $$v$$ is harmonic conjugate of $$u$$ $$\nRightarrow$$ $$u$$ is a harmonic conjugate of $$v$$.

Example: $$u(x,y) = x^2-y^2$$,  $$v(x,y) = 2xy$$. Since $$u+iv = z^2$$ is entire $$\implies$$ $$v$$ is a harmonic conjugate of $$u$$.

But if $$u$$ were a harmonic conjugate of $$v$$, then $$g = v+iu$$ would be analytic: Check via C/R, nowhere analytic.

Remark: Suppose $$u$$ is harmonic on a simple connected domain $$\Omega$$. Then $$u$$ has a harmonic conjugate on $$\Omega$$.

### Physical problems

"Physical" configurations often modelled by solutions of partial differential equations (PDE).

Generally, interested in solving a PDE subject to associated initial/boundary conditions.

### Dirichlet problem

$$\text{(D) } \left\{ \begin{array}{cc} \Delta u = 0 & \text{in } \Omega\\ u|_{\partial \Omega} = \varphi & (*) \end{array} \right.$$

$$(*)$$ Says $$u(\mathbf x)= \varphi(\mathbf x)$$ for all $$\mathbf x \in \partial \Omega$$

($$\varphi: \partial \Omega \ra \R$$).

$$\Omega, \varphi$$ are known/given, and $$u$$ is the unknown.

$$\text{(D)}$$ is called the Dirichlet problem for Laplace's equation, a. k. a. boundary problem of the first kind.

### Dirichlet problem

One way to "solve" $$\text{(D)}$$ is to find $$u$$ that minimises $\int_{\Omega} \left|\nabla u\right|^2 d\mathbf x$ subject to $$u|_{\partial \Omega} = \varphi$$.

### Neumann problem

Also important: Boundary conditions of the second kind, called Neumann boundary conditions

$$\text{(N) } \left\{ \begin{array}{cc} \Delta u = 0 & \\ \ds\frac{\partial u}{\partial \mathbf \nu} = \Psi & \text{on } \partial \Omega \end{array} \right.$$

In practice often have homogeneous Neumann boundary conditions, i. e., $$\Psi = 0$$.

#### Transformation of harmonic functions

Theorem: If $$f$$ is conformal and $$h$$ is harmonic in $$\Lambda$$, then $$H$$ is harmonic in $$\Omega$$, where $H(x,y) = h\left(u(x,y), v(x,y)\right).$

#### Transformation of harmonic functions

Proof: Mesy in general, but straightforward if $$\Lambda$$ is simply connected.

Example: $$h(x,y) = e^{-v}\sin u$$ is harmonic in UHP (upper half plane).

Define $$w=x^2$$ on $$\Omega =$$ 1st quadrant.

#### Transformation of harmonic functions

Example (cont): So $$u = x^2 -y^2$$ and $$v = 2xy$$.

The previous Theorem implies $H(x,y) = e^{-2xy}\sin \left(x^2-y^2\right).$ is harmonic on $$\Omega$$.