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Keywords:
eigenvalue problem for $p$-Laplacian; eigenvalue problem for $p$-biharmonic operator; estimates of principal eigenvalue; asymptotic analysis
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 $2N/R^2+O(-(p-1)\log (p-1))$.
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