Journal of Convex Analysis 33 (2026), No. 1&2, 303--324
Copyright Heldermann Verlag 2026
On the Second Anisotropic Cheeger Constant and Related Questions
Gianpaolo Piscitelli
Dip. di Scienze Economiche, Giuridiche, Informatiche e Motorie, Universita degli Studi di Napoli Parthenope, Nola, Italy
gianpaolo.piscitelli@uniparthenope.it
[Abstract-pdf]
We study the behavior of the second eigenfunction of the anisotropic $p$-Laplace operator \[ -\mathcal Q_{p}u:=-{\rm div} \left(F^{p-1}(\nabla u)F_\xi (\nabla u)\right), \] 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}(\Omega) :=\inf \left\{ \max\left\{\frac{P_{F}(E_{1})}{|E_{1}|},\frac{P_{F}(E_{2})}{|E_{2}|}\right\},\; E_{1},E_{2}\subset \Omega, 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.
Keywords: Nonlinear eigenvalue problems, second anisotropic Cheeger constant, second eigenfunctions of the $p$-Laplacian.
MSC: 47J10, 49Q20, 52A38.
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