Estimates of the principal eigenvalue of the p -Laplacian and the p -biharmonic operator

Jiří Benedikt

Mathematica Bohemica (2015)

  • Volume: 140, Issue: 2, page 215-222
  • ISSN: 0862-7959

Abstract

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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 N and its asymptotics for p approaching 1 and . Let p tend to . There is a critical radius R C of the ball such that the principal eigenvalue goes to for 0 < R R C and to 0 for R > R C . The critical radius is R C = 1 for any N for the p -Laplacian and R C = 2 N in the case of the p -biharmonic operator. When p approaches 1 , the principal eigenvalue of the Dirichlet p -Laplacian is N R - 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 2 N / R 2 + O ( - ( p - 1 ) log ( p - 1 ) ) .

How to cite

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Benedikt, Jiří. "Estimates of the principal eigenvalue of the $p$-Laplacian and the $p$-biharmonic operator." Mathematica Bohemica 140.2 (2015): 215-222. <http://eudml.org/doc/271616>.

@article{Benedikt2015,
abstract = {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 $0<R\le R_C$ 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 $2N/R^2+O(-(p-1)\log (p-1))$.},
author = {Benedikt, Jiří},
journal = {Mathematica Bohemica},
keywords = {eigenvalue problem for $p$-Laplacian; eigenvalue problem for $p$-biharmonic operator; estimates of principal eigenvalue; asymptotic analysis},
language = {eng},
number = {2},
pages = {215-222},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Estimates of the principal eigenvalue of the $p$-Laplacian and the $p$-biharmonic operator},
url = {http://eudml.org/doc/271616},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Benedikt, Jiří
TI - Estimates of the principal eigenvalue of the $p$-Laplacian and the $p$-biharmonic operator
JO - Mathematica Bohemica
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 140
IS - 2
SP - 215
EP - 222
AB - 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 $0<R\le R_C$ 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 $2N/R^2+O(-(p-1)\log (p-1))$.
LA - eng
KW - eigenvalue problem for $p$-Laplacian; eigenvalue problem for $p$-biharmonic operator; estimates of principal eigenvalue; asymptotic analysis
UR - http://eudml.org/doc/271616
ER -

References

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