On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

Let $\mathrm{\Omega }$ 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 ${\lambda }_2(p,\mathrm{\Omega })$ of the anisotropic $p$-Laplacian $$-{\mathcal{Q}}_{p}u:=-\text{div}(F^{p-1}(\mathrm{\nabla }u)F_{\xi }(\mathrm{\nabla }u)),$$ where $F$ is a suitable smooth norm of ${\mathbb{R}}^n$ and $p\in ]1,+\mathrm{\infty }[$. We provide a lower bound of ${\lambda }_2(p,\mathrm{\Omega })$ among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as $p\to +\mathrm{\infty }$.