Timeline for Characterize where the Dirichlet Problem for the Laplacian is always solvable
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Aug 23, 2011 at 4:42 | comment | added | timur | Wiener criterion is probably also in $\Pi^1_2$, since it involves capacity. | |
| May 25, 2011 at 21:23 | comment | added | Will Jagy | I don't know that he is in town this instant, but L. C. Evans would have an informed opinion on your question, office 1033. Or one of his students. | |
| May 25, 2011 at 20:47 | answer | added | Yakov Shlapentokh-Rothman | timeline score: 3 | |
| May 25, 2011 at 20:46 | history | edited | Linda Brown Westrick |
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| May 25, 2011 at 20:45 | comment | added | Linda Brown Westrick | Thanks to those who have offered elegant sufficient conditions. But as Will suggested, I am indeed interested in exactly classifying the "always" regions. My interest is from mathematical logic. When I come across an unwieldy characterization which has resisted simplification for a long time, I often wonder if that unwieldiness is unavoidable. In logical terms, what I mean by unwieldiness can be made precise: the definition of a Dirichlet Region is syntactically $\Pi^1_2$, and on its face the barrier characterization is also syntactically $\Pi^1_2$; i.e. just as complicated. | |
| May 25, 2011 at 10:43 | history | edited | Charles Matthews | CC BY-SA 3.0 |
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| May 25, 2011 at 9:09 | comment | added | Spencer | This isn't an actual characterization but after Theorem 2.14 in Gilbarg and Trudinger, they also go on to remark that the boundary value problem is solvable (in 2 dimensions still, where we can use complex analysis) in any bounded domain where each connected component of the complement consits of of more than a single point. | |
| May 25, 2011 at 4:14 | answer | added | Nilima Nigam | timeline score: 3 | |
| May 24, 2011 at 23:55 | comment | added | GH from MO | I like the fact that there is always a unique solution when $\partial_\infty G$ is a simple closed curve on $\mathbb{C}\cup\{\infty\}$ as follows from Theorem 2.24 in Markushevich: Theory of Functions of a Complex Variable Vol. 3. | |
| May 24, 2011 at 23:38 | comment | added | Will Jagy | If the boundary is $C^{1,\alpha}$ there is always a solution. | |
| May 24, 2011 at 22:57 | comment | added | Will Jagy | I'm not sure why you are not satisfied. I assume there are large classes of regions, simple to describe, for which the Dirichlet problem can be satisfied with any continuous boundary data. I also assume there are some classes, simply described, for which some continuous boundary data causes failure (although 0 boundary data would seem to always work). Exactly classifying the "always" regions strikes me as subtle. Note that your result seems to be identical with Theorem 2.14 in Gilbarg and Trudinger, in the section on Perron's method. | |
| May 24, 2011 at 22:28 | history | asked | Linda Brown Westrick | CC BY-SA 3.0 |