Logarithmic exterior-sphere barrier in two dimensions
= Logarithmic exterior-sphere barrier in two dimensions
If $B(y,R)$ is an exterior ball tangent to a planar domain at $x_0$, then
$$
w(x)=\log\frac{|x-y|}{R}
$$
is a <harmonic function> in the domain, vanishes at $x_0$, and is positive at every other point of its closure. It is therefore a <barrier for the Dirichlet problem> and proves that $x_0$ is a <regular boundary point>.