Green functions and conformal mapping

1 Introduction

The Green function for the Laplacian operator in two dimensions  is closely related to conformal mappings   to the unit disk. Given the Green function for a simply-connected region with Dirichlet boundary conditions, one can construct the mapping by exponentiating the sum of the Green function and its conjugate harmonic function. In practise, this can be used to construct mapping functions  for various regions for which it is possible to solve the Dirichlet problem. In principle, it can be used to prove results about conformal mappings using the theory of differential equations. For instance, one can prove the Riemann mapping theorem  as a consequence of the existence of a solution to the Dirichlet problem.

2 Definition

Let $D$ be a simply connected subset of the complex plane with boundary $\partial D$ and let $a$ be a point in the interior of $D$. The Green’s function is a function $g\colon D\to\mathbf{R}$ such that

1. 1.

$g=0$ on $\partial D$.

2. 2.

$\nabla^{2}g=0$ on the interior of $D$.

3. 3.

$g(z)-\log|z-a|$ is bounded as $z$ approaches $a$.

3 Construction of the mapping function

Let $h$ be the conjugate harmonic function of to $g$. It can be shown that $h(z)-\arg(z-a)$ is bounded as $z\to a$ and, consequently, that $h$ is a multiple-valued function with branch point  at $a$ which increases by $2\pi$ every time one encircles $a$.

Now consider the function $f$ defined as $e^{-(g+ih)}$. This function is single valued because, when one circles about $a$, the argument of the exponential increases by $2\pi i$, but adding $2\pi i$ to an exponential does not change its value. Since $h$ is the conjugate harmonic function of $g$, it follows that $g+ih$ is holomorphic and, hence $f$ is also holomorphic.

Therefore, $f$ maps $D\setminus\{a\}$ to $\mathbf{C}$. Various things can be said about this mapping.

Because of the maximum princliple, $g(z)>0$ for all $z$ in the interior of $D$. Hence, $f$ maps the interior of $D$ into the interior of the unit disk and maps $\partial D$ to the unit circle.

Furthermore, it can be shown that the function $f$ is invertible   so, in fact, it is a conformal diffeomorphism between $D$ and the unit disk.

Title Green functions and conformal mapping GreenFunctionsAndConformalMapping 2013-03-22 15:57:06 2013-03-22 15:57:06 rspuzio (6075) rspuzio (6075) 5 rspuzio (6075) Topic msc 30A99