Timeline for Asymptotic behaviour near the boundary in the Dirichlet problem for the Laplacian.
Current License: CC BY-SA 2.5
14 events
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| Jun 28, 2010 at 20:03 | vote | accept | Kaminoite | ||
| Jun 28, 2010 at 19:06 | answer | added | Scott Armstrong | timeline score: 1 | |
| Jun 28, 2010 at 15:53 | comment | added | Willie Wong | books.google.com/… ... this Gilbarg and Trudinger. | |
| Jun 28, 2010 at 15:33 | comment | added | Kaminoite | @ Willie Wong: I corrected the typo about the dependence of $C$ on $z$. About the literature, can you tell me exactly which are Gilbarg and Trudinger? | |
| Jun 28, 2010 at 15:31 | history | edited | Kaminoite | CC BY-SA 2.5 |
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| Jun 28, 2010 at 14:19 | comment | added | Willie Wong | It also looks like you may want to look at the literature for boundary regularity of elliptic equations. See Gilbarg and Trudinger for example. | |
| Jun 28, 2010 at 14:18 | comment | added | Willie Wong | Another point: your question is still a bit fishy. The constant $C(s)$ in your last formula MUST depend on $z$. Ideally instead of the limit being equal some $C(s)$, it should be some inequality (presumably less-than-or-equal-to). I think this may be an interesting problem, but as it stands, the phrasing of the question is not clear. I am not even sure what the question is! | |
| Jun 28, 2010 at 13:12 | comment | added | Andrey Rekalo | @Kaminoite: Thank you for the comment. I thought $u_s$ was supposed to be harmonic everywhere in $B$. | |
| Jun 28, 2010 at 12:57 | comment | added | Kaminoite | @Andrey Rekalo: The condition ${u_s}_{|K}(x) = {u_0}_{|K}(x)+sg(x)$ is not redundant. In fact, you can think that $K$ is part of the boundary for a new $\bar{\Omega} = \Omega-K$. | |
| Jun 28, 2010 at 12:55 | comment | added | Kaminoite | The solution must be harmonic in $B-K$ for all $s$. | |
| Jun 28, 2010 at 12:39 | comment | added | Willie Wong | @Kaminoite: for the perturbed problem, do you actually want $\triangle u_s = 0$ only on $B\setminus K$? If $u_s$ is not differentiable near $\partial K$ (as indicated by the bit after "MY QUESTION IS"), it can hardly be a harmonic function in $B$. If this is the case, aren't you just looking at the Dirichlet problem on $B\setminus K$ with $u | \partial B = 0$ and $u | \partial K = s g$? Then you are just comparing arbitrary extensions of $g$ into $K$ against harmonic extensions of $g$ into $B\setminus K$... | |
| Jun 28, 2010 at 12:14 | comment | added | Andrey Rekalo | I don't understand the question. The condition ${u_s}_{|K}(x) = {u_0}_{|K}(x)+sg(x)$ is redundant since a harmonic function on $B$ is uniquely determined by its trace on $\partial B$. | |
| Jun 28, 2010 at 12:05 | history | edited | Kaminoite | CC BY-SA 2.5 |
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| Jun 28, 2010 at 11:09 | history | asked | Kaminoite | CC BY-SA 2.5 |