The Monotonicity of the Principal Frequency of the Anisotropic $p$-Laplacian

Abstract

Let $D>1$ be a fixed integer. Given a smooth bounded, convex domain $\Omega \subset \mathbb{R}^D$ and $H:\mathbb{R}^D\rightarrow [0,\infty )$ a convex, even, and $1$-homogeneous function of class $C^{3,\alpha }(\mathbb{R}^D\setminus \lbrace 0\rbrace )$ for which the Hessian matrix $D^2(H^p)$ is positive definite in $\mathbb{R}^D\setminus \lbrace 0\rbrace $ for any $p\in (1,\infty )$, we study the monotonicity of the principal frequency of the anisotropic $p$-Laplacian (constructed using the function $H$) on $\Omega $ with respect to $p\in (1,\infty )$. As an application, we find a new variational characterization for the principal frequency on domains $\Omega $ having a sufficiently small inradius. In the particular case where $H$ is the Euclidean norm in $\mathbb{R}^D$, we recover some recent results obtained by the first two authors in [3, 4].