Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner characterization is known. Has any progress been made since then? And what simpler characterization is known today, if one is known?

##Here is the problem definition:##

An open connected set $G\subseteq \mathbb{C}$ is called a Dirichlet Region if for each continuous function $f:\partial_\infty G\rightarrow \mathbb{R}$ there is a continuous function $u:G^- \rightarrow \mathbb{R}$ such that $u$ is harmonic in $G$ and $u(z)=f(z)$ for all $z$ in $\partial_\infty G$.

(The notation $\partial_\infty G$ refers to the boundary of $G$ in $\mathbb{C}\cup\{\infty\}$, and $G^-$ denotes the closure of $G$ in $\mathbb{C}\cup\{\infty\}$.)

##The characterization given in the book is:##

Given $a \in \partial_\infty G$, a barrier for $G$ at $a$ is a family $\{\psi_r: r>0\}$ of functions such that:

1. $\psi_r$ is well-defined and superharmonic on $B(a;r) \cap G$ with $0\leq \psi_r(z) \leq 1$
   
2. $\lim_{z\rightarrow a}\psi_r(z) = 0$, and
   
3. $\lim_{z\rightarrow w} \psi_r(z) = 1$ for $w$ in $G \cap \{w:|w-a|=r\}$.

An open connected set $G$ is a Dirichlet Region iff there is a barrier for $G$ at each point of $\partial_\infty G$.