Timeline for Reference request: Boundary behavior and quantitative lower bound for the principal eigenfunction of an elliptic PDE in a ball $B(r)$
Current License: CC BY-SA 3.0
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| Feb 7, 2014 at 9:27 | history | edited | Juhana Siljander | CC BY-SA 3.0 |
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| Feb 7, 2014 at 9:15 | comment | added | Juhana Siljander | One more remark: If we know that the principal eigenfunction attains its maximum in a point with a quantified distance to the boundary, then it is possible to show the estimate (2) by using the Harnack inequality - at least for $\sigma \ge \sigma_0$ with a quantitatively determinable $\sigma_0 <1$. Then we can probably adjust the constants to make it true for any $\sigma$. Moreover, I think this kind of result should be true as the structure of the matrix $A$ should give an upper bound for the speed of diffusion which determines where the maximum is attained. | |
| Feb 6, 2014 at 17:26 | history | edited | Juhana Siljander | CC BY-SA 3.0 |
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| Feb 6, 2014 at 12:59 | comment | added | Juhana Siljander | That's what people usually do, I guess. The problem with this scaling argument is, however, that then it is not immediate anymore what is the scaling behavior of the estimates and, in particular, whether they are scale invariant or not. So I usually prefer keeping the r in the estimates so that it is easier to see how they transform under scalings. | |
| Feb 6, 2014 at 12:54 | history | edited | Juhana Siljander |
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| Feb 5, 2014 at 13:10 | comment | added | Juhana Siljander | Oh, sorry about that. I forgot the eigenvalue in the problem definition. It's now corrected. :P | |
| Feb 5, 2014 at 13:09 | history | edited | Juhana Siljander | CC BY-SA 3.0 |
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| Feb 5, 2014 at 10:54 | answer | added | username | timeline score: 2 | |
| Feb 4, 2014 at 8:18 | comment | added | Juhana Siljander | Thanks for the paper. There seems to be some interesting references there. However, these papers seem overly complicated for my purposes. Shouldn't this kind of results for eigenfunctions be almost classical? The equation is linear after all. Yet, it seems difficult to find a reference for the results. | |
| Feb 4, 2014 at 8:15 | history | edited | Juhana Siljander | CC BY-SA 3.0 |
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| Feb 1, 2014 at 21:57 | comment | added | username | The keyword to look for is symmetrization, I think, as in this paper. | |
| Feb 1, 2014 at 18:02 | review | First posts | |||
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| Feb 1, 2014 at 17:43 | history | asked | Juhana Siljander | CC BY-SA 3.0 |