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I am studying in PDE and I have next definition :

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 $p(\xi)=0$.

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!

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    $\begingroup$ The motivation is to solve the Dirichlet problem (it exists for every continuous boundary data if and only if every point is regular). Example: unit ball with a single point removed (in dimension $2$ or above). Another example: unit ball with its diameter removed (in dimension $3$ or above). This has a very nice probabilistic interpretation: if the Brownian motion started from $\xi$ hits the boundary of $\Omega$ immediately with probability one, then $\xi$ is regular; otherwise — it is not. $\endgroup$ – Mateusz Kwaśnicki May 16 at 22:10
  • $\begingroup$ @MateuszKwaśnicki I don't understand the example you gave. Let's say $\Omega=B_1(0)\backslash\{0\}$. Then $|x|$ is the superharmonic function in $\Omega$, $p>0$ in $\overline{\Omega}\backslash\{0\}=\overline{B_1(0)}\backslash\{0\}$, and $p(0)=0$, isn't it? $\endgroup$ – Jingeon An May 16 at 22:24
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    $\begingroup$ I do not think so: $|x|$ is sub-harmonic, is it not? $\endgroup$ – Mateusz Kwaśnicki May 16 at 22:27
  • $\begingroup$ @MateuszKwaśnicki Yes, you are right. I was confused. Now it's clear for me. Thank you so much. $\endgroup$ – Jingeon An May 16 at 22:38
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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.

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    $\begingroup$ Thank you so much for your effort to give me those concrete examples. This helps me a lot. I highly appreciate it. $\endgroup$ – Jingeon An May 17 at 1:19

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