We can solve laplace equation $\Delta u=0$ with Dirichlet boundary condition $u(x,y)=f\in C(\partial D)$ for the unit disk $D$ $\subset$ $R^2$ .
If $f$ is a continuous function on the boundary $\partial D$ of the open unit disk $D$, then the solution to the Dirichlet problem is $u(z)$ given by
$$u(z) = \begin{cases} \frac{1}{2\pi}\int_0^{2\pi} f(e^{i\psi}) \frac {1-\vert z \vert ^2}{\vert 1-ze^{-i\psi}\vert ^2} d \psi & \mbox{if }z \in D \\ f(z) & \mbox{if }z \in \partial D \end{cases}$$ by Riemann mapping theorem any non-empty simply connected open subset of the complex number plane $\mathbb{C}$ which is not all of $\mathbb{C}$ like $U$ can be mapped by a biholomorphic (bijective and holomorphic) $f$ , onto the open unit disk $D = \{z\in \mathbf{C} : |z| < 1\}$. so we can solve laplace equation $\Delta u=0$ with Dirichlet boundary condition $u(x,y)=f$ for any non-empty simply connected open subset of the complex number plane $\mathbb{C}$ which is not all of $\mathbb{C}$ like $U$.
Now If we could have a way to prove that laplace equation $\Delta u=0$ with Dirichlet boundary condition $u(x,y)=f\in C(\partial U)$ for any non-empty simply connected open subset of the complex number plane $\mathbb{C}$ which is not all of $\mathbb{C}$ like $U$ have solution . I think we could proof Riemann mapping theorem as below:
Given $U$ and $z_0$, we want to construct a function $f$ which maps $U$ to the unit disk and $z_0$ to $0$. we will assume that $U$ is bounded and simply connected. Write $f(z) = (z - z_0)e^{g(z)}$ where $g = u + iv$ is some (to be determined) holomorphic function with real part $u$ and imaginary part $v$. It is then clear that $z_0$ is the only zero of $f$. We require $|f(z)| = 1$ for $z ∈ ∂U$, so we need $u(z) = -\log|z - z_0|$ on the boundary. Since $u$ is the real part of a holomorphic function, we know that $u$ is necessarily a harmonic function; i.e., it satisfies Laplace's equation.
The question then becomes : does a real-valued harmonic function $u$ exist that is defined on all of $U$ and has the given boundary condition? Once the existence of $u$ has been established, the Cauchy-Riemann equations for the holomorphic function $g$ allow us to find $v$ (this argument depends on the assumption that $U$ be simply connected). Once $u$ and $v$ have been constructed, one has to check that the resulting function $f$ does indeed have all the required properties.
Is this right way to prove Riemann mapping theorem?