I am studying in PDE and I have next definition :



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



What is the example of non-regular boundary point?
Example 1. In dimension 2, all isolated boundary points (punctures) are irregular.


Example 2. (Generalization) In dimension $n$ if you remove from a region $D$ a smooth
$n-2$ dimensional surface $S$, which does not separate $D$ then all points of this surface $S$ are irregular for
$D\backslash S$.


Example 3. (Further generalization) if you remove from a region $D$ any compact $E$ of
zero capacity (logarithmic capacity for $n=2$, Newtonian capacity for $n>2$), then
all points of $E$ will be irregular for $D\backslash E$.


Example 4. A spike. If $n\geq 3$, and you remove from a region $D$ containing the origin a very sharp spike
$$S=\{(x\_1,\ldots,x\_n):x\_1\geq 0, x\_2^2+\ldots+x\_n^2\leq\phi(x\_1)\},$$
where $\phi(x)>0, \;\phi(0)=0$ tends to zero sufficiently fast as $x\to 0$, then
the point $0\in D\backslash S$ is of zero capacity. This is called Lebesgue's spike.
In Lebesgue's original example $\phi(x)=e^{-1/x^2},x>0$.


There is a quantitative criterion (a necessary and sufficient condition) due to Norbert Wiener which says that
if the complement of the region is very small in a neighborhood of a boundary point
then this boundary point is irregular. Smallness is described in terms of capacity.


All this is stated for the Laplace operator (classical potential theory),
but there are analogous results for other elliptic and parabolic operators.


