preprint

Inserted: 18 oct 2026

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} λ₁(β,Ω)=\min_{ψ\in W^{1,p}(Ω)\setminus\{0\} } \frac{\displaystyle\int_ΩF(\nabla ψ)^p dx +β\displaystyle\int_{{\partialΩ}
ψ}
^pF(ν_Ω) d\mathcal H^{N-1} }{\displaystyle\int_Ω
ψ
^p dx}, \end{equation} where $p\in]1,+\infty[$, $Ω$ is a bounded, mean convex domain in $\mathbb R^{N}$, $ν_Ω$ is its Euclidean outward normal, $β$ is a real number, and $F$ is a sufficiently smooth norm on $\mathbb R^{N}$. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $β$ and on geometrical quantities associated to $Ω$. More precisely, we prove a lower bound of $λ_{1}$ in the case $β>0$, and a upper bound in the case $β<0$. As a consequence, we prove, for $β>0$, a lower bound for $λ_{1}(β,Ω)$ in terms of the anisotropic inradius of $Ω$ and, for $β<0$, an upper bound of $λ_{1}(β,Ω)$ in terms of $β$.