On the second anisotropic Cheeger constant and related questions

Abstract

In this paper we study the behavior of the second eigenfunction of the anisotropic $p$-Laplace operator \[ - Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_ξ(\nabla u)\right), \] as $p\to 1^{+}$, where $F$ is a suitable smooth norm of ${\mathbb{R}}^n$. Moreover, for any regular set $\Omega$, we define the second anisotropic Cheeger constant as \begin{equation*} h_{2,F}(Ω):=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\},\; E_{1},E_{2}\subset Ω, E_{1}\cap E_{2}=\emptyset\right\}, \end{equation*} where $P_F(E)$ is the anisotropic perimeter of $E$, and study the connection with the second eigenvalue of the anisotropic $p$-Laplacian. Finally, we study the twisted anisotropic $q$-Cheeger constant with a volume constraint.