| Title:
|
Estimates of the principal eigenvalue of the $p$-Laplacian and the $p$-biharmonic operator (English) |
| Author:
|
Benedikt, Jiří |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
140 |
| Issue:
|
2 |
| Year:
|
2015 |
| Pages:
|
215-222 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet $p$-Laplacian and the Navier $p$-biharmonic operator on a ball of radius $R$ in $\mathbb R^N$ and its asymptotics for $p$ approaching $1$ and $\infty $. Let $p$ tend to $\infty $. There is a critical radius $R_C$ of the ball such that the principal eigenvalue goes to $\infty $ for $0<R\leq R_C$ and to $0$ for $R>R_C$. The critical radius is $R_C=1$ for any $N\in \mathbb N$ for the $p$-Laplacian and $R_C=\sqrt {2N}$ in the case of the $p$-biharmonic operator. When $p$ approaches $1$, the principal eigenvalue of the Dirichlet $p$-Laplacian is $NR^{-1}\*(1-(p-1)\log R(p-1))+o(p-1)$ while the asymptotics for the principal eigenvalue of the Navier $p$-biharmonic operator reads $2N/R^2+O(-(p-1)\log (p-1))$. (English) |
| Keyword:
|
eigenvalue problem for $p$-Laplacian |
| Keyword:
|
eigenvalue problem for $p$-biharmonic operator |
| Keyword:
|
estimates of principal eigenvalue |
| Keyword:
|
asymptotic analysis |
| MSC:
|
35J20 |
| MSC:
|
35J25 |
| MSC:
|
35J66 |
| MSC:
|
35J92 |
| MSC:
|
35P15 |
| MSC:
|
35P30 |
| idZBL:
|
Zbl 06486935 |
| idMR:
|
MR3368495 |
| DOI:
|
10.21136/MB.2015.144327 |
| . |
| Date available:
|
2015-06-30T12:20:52Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144327 |
| . |
| Reference:
|
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| Reference:
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| . |
