Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic $p$-Laplace operator, namely: egin{equation*} lambda_1(eta,Omega)= min_{psiin W^{1,p}(Omega)setminus{0} } rac{displaystyleint_Omega F( abla psi)^p dx +eta dsint_{deOmega}|psi|^pF( u_{Omega}) dcH^{N-1} }{displaystyleint_Omega|psi|^p dx}, end{equation*} where $pin]1,+infty[$, $Omega$ is a bounded, mean convex domain in $R^{N}$, $ u_{Omega}$ is its Euclidean outward normal, $eta$ is a real number, and $F$ is a sufficiently smooth norm on $R^{N}$. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on $eta$ and on geometrical quantities associated to $Omega$. More precisely, we prove a lower bound of $lambda_{1}$ in the case $eta>0$, and a upper bound in the case $eta<0$. As a consequence, we prove, for $eta>0$, a lower bound for $lambda_{1}(eta,Omega)$ in terms of the anisotropic inradius of $Omega$ and, for $eta<0$, an upper bound of $lambda_{1}(eta,Omega)$ in terms of $eta$.