Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators

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*} λ_1(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 $p\in ]1,+\infty [$, $\Omega$ is a bounded, mean convex domain in ${\mathbb{R}}^N$, ${\nu }_{\Omega }$ is its Euclidean outward normal, $\beta$ 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 $\beta$ 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 $\beta$.

arXiv comment24 pages

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Cited by in corpus (1)

• On the first Robin eigenvalue of the Finsler $p$-Laplace operator as $p\to 1$