I am studying in PDE and I have next definition :

Definition. Let $\Omega\subset\mathbb{R}^n$ open, connected. Then $\xi\in\partial\Omega$ isregularif there exists a superharmonic function $p$ in $\Omega$ such that $p>0$ in $\overline{\Omega}\backslash\{\xi\}$ and $p(\xi)=0$.

And with this regularity of the boundary, we could show some useful results such as well-posedness of Dirichlet problem for the laplace equation on any regular (every points are regular) domain (moreover, if a harmonic solution exists then $\Omega$ is regular).

However, I couldn't find any proper example of non-regular boundary points. I also feel like this definition is 'given by God'. So the questions are :

- What is the example of non-regular boundary point?
- What was the motivation of this definition?

Thank you in advance!

sub-harmonic, is it not? $\endgroup$ – Mateusz Kwaśnicki May 16 at 22:27