Let $D \subset \Bbb R^2$ be a domain, and suppose that there is a subharmonic barrier at some point $\zeta \in \partial D$, i.e. a function $w$ negative on $D$, tending to $0$ on $\zeta$ and with negative partial limits at all other points of $\partial D$.
Does that force $D$ to be included in some domain with analytic boundary?
I read a book by Tsuji that seems to rely on this fact in a corollary, but this is not clear to me. It is enough if one proves that $D^c$ has a nonempty interior, as it implies the above claim. The main problem is that I lack necessary properties for the existence of a barrier; I only know sufficient conditions. (such as the existence of a continuum in the complement starting at the point of the barrier.)
replacing analytic by piecewise-smooth is also enough for the aim of Tsuji, though he seems to indicate it even for the analytic case.