preprint

Inserted: 29 sep 2022

Year: 2022

Abstract:

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: \begin{equation} \lambda₁(\beta,\Omega)=\min_{\psi\in W^{1,p}(\Omega)\setminus\{0\} } \frac{\displaystyle\int_\Omega F(\nabla \psi)^p dx +\beta \displaystyle\int_{{\partial\Omega}
\psi}
^pF(\nu_{\Omega}) d\mathcal H^{N-1} }{\displaystyle\int_\Omega
\psi
^p dx}, \end{equation} where $p\in]1,+\infty[$, $\Omega$ is a bounded, mean convex domain in $\mathcal R^{N}$, $\nu_{\Omega}$ is its Euclidean outward normal, $\beta$ is a real number, and $F$ is a sufficiently smooth norm on $\mathcal R^{N}$. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $\beta$ and on geometrical quantities associated to $\Omega$. More precisely, we prove a lower bound of $\lambda_{1}$ in the case $\beta>0$, and a upper bound in the case $\beta<0$. As a consequence, we prove, for $\beta>0$, a lower bound for $\lambda_{1}(\beta,\Omega)$ in terms of the anisotropic inradius of $\Omega$ and, for $\beta<0$, an upper bound of $\lambda_{1}(\beta,\Omega)$ in terms of $\beta$.