Calculus of Variations and Geometric Measure Theory

F. Della Pietra - G. Piscitelli

Sharp estimates for the first Robin eigenvalue of nonlinear elliptic operators

created by piscitelli on 29 Sep 2022



Inserted: 29 sep 2022

Year: 2022

ArXiv: 2204.01814 PDF


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}
pF(\nu{\Omega}) d\mathcal H{N-1} }{\displaystyle\int\Omega
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 $\beta$.