 Solution

Let the exterior ball be $B(y,R)$, tangent at $x_0$. Then $|x-y|\ge R$ on $\overline{\Omega }$, with equality only at $x_0$. Choose $\delta =R^{-1}$ and defineIn two dimensions, $\log |x-y|$ is a harmonic function away from $y$, so $\Delta w=0\le 0$ in $\Omega$. Moreover $w(x_0)=0$ and $w>0$ on $\overline{\Omega }\setminus \{x_0\}$. Thus $w$ is a barrier for the Dirichlet problem, and $\boxed{x_0\text{ is a regular boundary point}}$.