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

Benedikt, Jiří

Mathematica Bohemica, Tome 140 (2015) no. 2, pp. 215-222.

Résumé

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 $\mathrm{\infty }$. Let $p$ tend to $\mathrm{\infty }$. There is a critical radius $R_C$ of the ball such that the principal eigenvalue goes to $\mathrm{\infty }$ for $0$ 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 $N{R}^{-1}\text{*}(1-(p-1)\log R(p-1))+o(p-1)$ while the asymptotics for the principal eigenvalue of the Navier $p$-biharmonic operator reads $2N/R^2+O(-(p-1)\log (p-1))$.

Détail

DOI : 10.21136/MB.2015.144327

Classification : 35J20, 35J25, 35J66, 35J92, 35P15, 35P30
Mots-clés : eigenvalue problem for $p$-Laplacian; eigenvalue problem for $p$-biharmonic operator; estimates of principal eigenvalue; asymptotic analysis