On the second anisotropic Cheeger constant and related questions
Gianpaolo Piscitelli
TL;DR
The paper studies the second eigenfunction of the anisotropic $p$-Laplacian as $p\to 1^+$ and introduces the second anisotropic Cheeger constant $h_{2,F}(\Omega)$, linking it to the asymptotics of $\lambda_{2,F}(p,\Omega)$. It establishes a Cheeger-type lower bound $\lambda_{2,F}(p,\Omega)\ge (h_{2,F}(\Omega)/p)^p$ and proves the sharp limit $\lim_{p\to1^+} \lambda_{2,F}(p,\Omega)^{1/p}=h_{2,F}(\Omega)$, using BV-approximation and disjoint nodal-domain arguments. The work also analyzes the related twisted $q$-Cheeger constant $\mathcal{K}_{q,F}(\Omega)$ and shows that constrained minimizers are unions of two disjoint Wulff shapes, with uniqueness depending on $q$. Together, these results connect anisotropic spectral asymptotics with geometric/variational objects in Finsler geometry, offering a framework for higher-order Cheeger-type inequalities and shape optimization in anisotropic media.
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 $Ω$, 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.