Affine isoperimetric inequalities for the first eigenvalue of the $m$-th order Affine $p$-Laplace Operator

TL;DR

This paper introduces higher-order affine $p$-Laplace operators and establishes new isoperimetric inequalities for their first eigenvalues, extending classical results to more general, asymmetric, and higher-order contexts.

Contribution

It generalizes the affine $p$-Laplace operator to higher orders and proves new affine Talenti and Faber-Krahn inequalities for these operators.

Findings

Established $m$th-order affine Talenti inequality.

Proved $m$th-order affine Faber-Krahn inequality.

Recovered classical inequalities when $m=1$.

Abstract

Recently, Haddad, Jim\'enez, and Montenegro introduced the affine $p$-Laplace operator, $p>1$, and studied associated affine versions of the isoperimetric inequalities for the first eigenvalue of the affine $p$-Laplace operator, including the affine Faber-Krahn inequality and affine Talenti inequality. In this work, we introduce the $m$th-order $p$-Laplace operator $\Delta_{Q,p}^\mathcal{A} f$, which recovers the affine $p$-Laplace operator when $m=1$ and $Q$ is a symmetric interval. Given $n,m \in \mathbb{N}$, a sufficiently smooth convex body $Q \subset \mathbb{R}^m$, a bounded, open set $\Omega \subset \mathbb{R}^n$ and $p >1$, we investigate the eigenvalue problem \[\begin{cases} \Delta_{Q,p}^\mathcal{A} f = \lambda_{1,p}^\mathcal{A}(Q,\Omega) |f|^{p-2} f &\text{ in } \Omega; \\ f=0 & \text{ on } \partial \Omega, \end{cases} \] for $f \in W^{1,p}_0(\Omega)$. Finally, we establish $m$th-order extensions of the affine Talenti inequality and affine Faber-Krahn inequality, which, upon choosing $m=1$, yield new, asymmetric versions of those aforementioned inequalities.