On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

Abstract

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 \[ -\mathcal Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_ξ(\nabla u)\right), \] 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 $p\to +\infty$.

References in corpus (2)

• Sharp estimates for the anisotropic torsional rigidity and the principal frequency
• A sharp weighted anisotropic Poincaré inequality for convex domains

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