Riemann Mapping theorem
Every simply connected region $\Omega \subset \mathbb C$ is
conformally equivalent to the open unit disk (except $\Omega = \mathbb C$)
What are application of this theorem? I mean an example of a problem that can be transformed to a different space and solved there, and then the solutions carried back. If it's possible something elementary, I am fairly new at complex analysis after all.
On the book there is written that if you have a problem on $H(\Omega_2)$ (all holomorphic function on $\Omega_2$) you can transport it back to $H(\Omega_1)$ if you have a biholomorphic function $\varphi: \varphi(\Omega_1) = \Omega_2$ using the ring isomorphism $f\mapsto f \circ \varphi$ which maps $H(\Omega_2)$ onto $H(\Omega_1)$.
If it's possible it would be nice to have an application along these lines! Thank you very much 🙂
Best Answer
In the plane, solving the Laplace equation with Dirichlet conditions is to find a function $u:\overline\Omega\subset\Bbb R^2\to\Bbb R$ such that: $$\begin{cases}\partial_{xx}u +\partial_{yy}u = 0, &(x,y)\in\Omega \\ u(x,y) = d(x,y), &(x,y)\in\partial\Omega \end{cases}$$ Verbally, we want a harmonic function $u$ defined in the domain $\Omega$, whose values coincide with those of some $\mathcal C^0 $ function $d$ on the border $\partial\Omega$. An easy version of the problem is when $\Omega = B_1(0)$ (the unit ball). Indeed, we are able to procure an analytic (in the sense that it's given by an explicit formula) solution, whose expression isn't important here, we just care that it exists.
Now say we wish to examine existence of solutions to the Laplace equation, where $\Omega$ is any ($\mathcal C^0$) simply connected region in $\Bbb R^2$ (except $\Bbb R^2$). You can probably see where this is going. Let $$\varphi:\Omega\to B_1(0)$$ be holomorphic (in the sense that the component functions $f,g$ make $f+ig$ holomorphic) with holomorhpic inverse (idem). Also, I'll require that $\varphi$ extend to $\partial\Omega$ continuously, and such that $\varphi(\partial\Omega) = \partial B_1(0)$. This is assured by a stronger version of the Riemann mapping theorem. Now consider the Dirichlet problem $$\begin{cases}\partial_{xx}u +\partial_{yy}u = 0, &(x,y)\in B_1(0) \\ u(x,y) = (d\circ \varphi^{-1})(x,y), &(x,y)\in\partial B_1(0) \end{cases}$$
This is exactly of the type that I first mention, so it has a solution $v$. I now claim that $v\circ \varphi$ is a solution to the Dirichlet problem $$\begin{cases}\partial_{xx}u +\partial_{yy}u = 0, &(x,y)\in \Omega \\ u(x,y) = d(x,y), &(x,y)\in\partial\Omega \end{cases}$$ The boundary condition is immediate, since on $\partial B_1(0)$, $v=d\circ \varphi^{-1}$. Therefore on $\partial\Omega$, $v\circ\varphi = d\circ\varphi^{-1}\circ\varphi = d$.
For the PDE, we use that since $v$ is harmonic (i.e. it satisfies the Laplace equation), there exists a harmonic conjugate $w$ such that $v+iw$ is holomorphic. Therefore $(v+iw)\circ\varphi$ (now interpreting $\varphi$ as complex) is holomorphic, and therefore its real part $v\circ\varphi$ is harmonic*, i.e. it satisfies $(\partial_{xx}+\partial_{yy})(v\circ\varphi) = 0$.
By the Riemann mapping theorem, we were able to solve the Laplace equation with Dirichlet conditions over a much wider range of domains, using only that we know a solution in the unit ball. Of course there are much more potent ways of approaching this particular equation, that give even more general results, but I think it's a nice example of an application regardless.
*It's easy to check that if $f+ig$ is holomorphic, then $f,g$ are harmonic, by using the Cauchy-Riemann equations