preprint
Inserted: 29 aug 2025
Year: 2025
Abstract:
In this paper we study the behavior of the second eigenfunction of the
anisotropic $p$-Laplace operator \[ - Q_{p}u:=-\textrm{div}
\left(F^{p-1}(\nabla u)F_\xi (\nabla u)\right), \] 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.