| Title:
|
Sets of determination for solutions of the Helmholtz equation (English) |
| Author:
|
Ranošová, Jarmila |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
38 |
| Issue:
|
2 |
| Year:
|
1997 |
| Pages:
|
309-328 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\alpha > 0$, $\lambda = (2\alpha)^{-1/2}$, $S^{n-1}$ be the $(n-1)$-dimensional unit sphere, $\sigma$ be the surface measure on $S^{n-1}$ and $h(x) = \int_{S^{n-1}} e^{\lambda\langle x,y\rangle }\,d\sigma(y)$. We characterize all subsets $M$ of $\Bbb R^n $ such that $$ \inf\limits_{x\in \Bbb R^n}{u(x)\over h(x)} = \inf\limits_{x\in M}{u(x)\over h(x)} $$ for every positive solution $u$ of the Helmholtz equation on $\Bbb R^n$. A closely related problem of representing functions of $L_1(S^{n-1})$ as sums of blocks of the form $ e^{\lambda\langle x_k,.\rangle }/h(x_k)$ corresponding to points of $M$ is also considered. The results provide a counterpart to results for classical harmonic functions in a ball, and for parabolic functions on a slab, see References. (English) |
| Keyword:
|
Helmholtz equation |
| Keyword:
|
set of determination |
| Keyword:
|
decomposition of $L^1$ |
| MSC:
|
31B10 |
| MSC:
|
35J05 |
| idZBL:
|
Zbl 0887.35035 |
| idMR:
|
MR1455498 |
| . |
| Date available:
|
2009-01-08T18:31:06Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118929 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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|
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| Reference:
|
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| . |
