Consider the Dirichlet problem for Laplace's equation in a bounded domain $\Omega \subset \mathbb R^n$: $$ -\Delta u = 0, \quad x \in \Omega, $$ with $u = \phi$ on $\partial\Omega$, and $\phi$ is continuous on $\partial\Omega$. For a boundary point $y$, if a barrier $w_y$ exists, i.e. a superharmonic function that is continuous on $\bar \Omega$ with $w_y(y) = 0$ and $w_y(x) > 0$ for $x\in \partial\Omega \setminus \{y\}$, then we know $y$ is a regular point: $\lim_{x\to y, x\in \Omega} u(x) = \phi(y)$. From this, we conclude some quantitative thickness of the exterior of $\Omega$ near $y$, in that the Wiener series diverges: $$ \sum_{i=1}^\infty 2^{i(n-2)} \mbox{Cap}((B_{2^{-i}}(y) \setminus B_{2^{-i-1}}(y)) \setminus \Omega) = \infty, $$ where $\mbox{Cap}$ is the harmonic capacity.
My question: If we assume in addition that $w_y$ is Lipschitz continuous, can we say anything more about the regularity of $\partial \Omega$? For example, can we conclude stronger quantitative thickness for $\mathbb R^n \setminus \Omega$ at $y$, such as a lower bound on the Lebesgue density?
