Timeline for If the first Dirichlet eigenfunction on a set $D$ is regular up to the boundary, is $D$ regular?
Current License: CC BY-SA 3.0
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| Aug 10, 2017 at 6:25 | comment | added | Mateusz Kwaśnicki | A barrier at a boundary point $x_0$ is a positive superharmonic function which converges to zero at $x_0$. Barriers are used to prove boundary continuity of harmonic functions: a barrier at $x_0$ exists if and only if $x_0$ is regular for the Dirichlet problem, that is, any harmonic function with continuous boundary data is continuous at $x_0$. I learned this from Wermer's book, but any textbook on (classical) potential theory will do. | |
| Aug 10, 2017 at 1:20 | comment | added | Neal | Would you be willing to elaborate what you mean by "indeed: the first eigenfunction is a barrier at any boundary point"? | |
| Aug 9, 2017 at 20:02 | history | answered | Mateusz Kwaśnicki | CC BY-SA 3.0 |