Timeline for Superharmonicity of the distance function
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 12, 2021 at 20:10 | comment | added | Mateusz Kwaśnicki | No, this is not what I intended to say. I just wanted to point out that if $V$ is not the complement of a compact set, then there is a sequence $x_n$ such that $|x_n| \to \infty$, but $x_n$ is close enough to the boundary of $V$ so that $u(x_n) \to 0$. | |
| Apr 12, 2021 at 19:45 | comment | added | M. Rahmat | Thanks. If I am not mistaken, you are saying that if $V=\mathbb{R}^m\setminus K$ with $K$ compact, then the Green potential of $V$ goes to infinity at infinity (correct?). But it seems to me that the Green function of $V$ goes to zero at infinity; so how come the Green potential goes to infinity? | |
| Apr 12, 2021 at 7:10 | comment | added | Mateusz Kwaśnicki | If you replace "at each point of the boundary" by "at each regular point of the boundary", then yes, of course: just take the Green potential of an arbitrary finite, positive measure. It cannot go to infinity in the usual sense, though, unless the complement of $V$ is compact (in particular, if $V$ is an unbounded convex set). | |
| Apr 12, 2021 at 6:18 | history | asked | M. Rahmat | CC BY-SA 4.0 |