Calculus of Variations and Geometric Measure Theory

F. Della Pietra - N. Gavitone - G. Piscitelli

On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

created by piscitelli on 31 Aug 2022
modified on 11 Sep 2022



Inserted: 31 aug 2022
Last Updated: 11 sep 2022

Year: 2017

ArXiv: 1704.00508 PDF


Let $\Omega$ be a bounded open set of $\mathbb R^{n}$, $n\ge 2$. In this paper we mainly study some properties of the second Dirichlet eigenvalue $\lambda_{2}(p,\Omega)$ of the anisotropic $p$-Laplacian \[ -\mathcal Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_\xi (\nabla u)\right), \] where $F$ is a suitable smooth norm of $\mathbb R^{n}$ and $p\in]1,+\infty[$. We provide a lower bound of $\lambda_{2}(p,\Omega)$ among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as $p\to+\infty$.