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

TL;DR

This work extends the affine-isoperimetric framework to an $m$-th order setting by introducing the $m$-th order affine $p$-Laplacian $\Delta_{Q,p}^{\mathcal{A}}$ via the energy $\mathcal{E}_{Q,p}$ and the body $L_{Q,p,f}$. It develops a variational theory for the first eigenvalue, proves regularity of minimizers, and derives sharp affine isoperimetric inequalities: a m-th order affine Talenti inequality and a m-th order affine Faber–Krahn inequality, including rigidity results and connections to Cheeger-type sets. The approach relies on the $m$-order convex-body machinery ($\Pi_{Q,p}^{\circ}$, $\Gamma_{Q,p}$, energy representations) and combines coarea, Brunn–Minkowski-type, and centroid inequalities to obtain distributional and geometric inequalities with equality characterizations. The results generalize the known $m=1$ affine inequalities, produce asymmetric variants, and provide a robust framework for regularity and variational characterizations of affine PDEs.

Abstract

Recently, Haddad, Jiménez, 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 $Δ_{Q,p}^\mathcal{A} f$, 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 $Ω\subset \mathbb{R}^n$ and $p >1$, we investigate the eigenvalue problem \[\begin{cases} Δ_{Q,p}^\mathcal{A} f = λ_{1,p}^\mathcal{A}(Q,Ω) |f|^{p-2} f &\text{ in } Ω; \\ f=0 & \text{ on } \partial Ω, \end{cases} \] for $f \in W^{1,p}_0(Ω)$. 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.

Theorems & Definitions (30)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• Lemma 2.2