Abstract:

Let $\Omega$ be an open and connected subset of the complex plane. A real valued function $u: \omega \rightarrow \mathbb{r}$ is said to be harmonic if it has continuous first and second partial derivatives and satisfies Laplace's equation $\Delta u = \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} = 0$. We begin by investigating basic properties of harmonic functions and defining the harmonic conjugate gradient and harmonic conjugate functions. From this we prove the mean value property and maximum and minimum modulus principle for harmonic functions.

Suppose we have a two dimensional plate of homogeneous material whose boundaries are heated to a constant temperature. We investigate solving the Dirichlet problem, which involves Laplace's equation and specific boundary values. Along with examples, we show that we can solve the Dirichlet problem on a disk using the Poisson integral along with Fourier series.

We generalize the Dirichlet problem to any region by investigating conformal mappings. We find a solution to the Dirichlet problem on a general region in the plane by transforming it to a problem in the upper half plane and using the Poisson integral.