| Title:
|
Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant (English) |
| Author:
|
Kawohl, B. |
| Author:
|
Fridman, V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
44 |
| Issue:
|
4 |
| Year:
|
2003 |
| Pages:
|
659-667 |
| . |
| Category:
|
math |
| . |
| Summary:
|
First we recall a Faber-Krahn type inequality and an estimate for $\lambda_p(\Omega)$ in terms of the so-called Cheeger constant. Then we prove that the eigenvalue $\lambda_p(\Omega)$ converges to the Cheeger constant $h(\Omega)$ as $p\to 1$. The associated eigenfunction $u_p$ converges to the characteristic function of the Cheeger set, i.e. a subset of $\Omega$ which minimizes the ratio $|\partial D|/|D|$ among all simply connected $D\subset\subset\Omega$. As a byproduct we prove that for convex $\Omega$ the Cheeger set $\omega$ is also convex. (English) |
| Keyword:
|
isoperimetric estimates |
| Keyword:
|
eigenvalue |
| Keyword:
|
Cheeger constant |
| Keyword:
|
$p$-Laplace operator |
| Keyword:
|
$1$-Laplace operator |
| MSC:
|
35J20 |
| MSC:
|
35J70 |
| MSC:
|
49Q20 |
| MSC:
|
49R05 |
| MSC:
|
49R50 |
| MSC:
|
52A38 |
| idZBL:
|
Zbl 1105.35029 |
| idMR:
|
MR2062882 |
| . |
| Date available:
|
2009-01-08T19:31:59Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119420 |
| . |
| Reference:
|
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