From affine Poincaré inequalities to affine spectral inequalities
Julián Haddad, Carlos Hugo Jiménez, Marcos Montenegro
TL;DR
This work develops an affine-invariant spectral theory for bounded domains by introducing the affine ball $B^{\mathcal{A}}_p(\Omega)$ and the affine Rayleigh quotient $R^{\mathcal{A}}_p$, linking them to a nonlocal affine $p$-Laplacian $\Delta_p^{\mathcal{A}}$ and the first eigenvalue $\lambda^{\mathcal{A}}_{1,p}(\Omega)$. A central contribution is a sharp reverse inequality $\mathcal{E}_p f \ge C_{n,p}(\Omega) \|f\|_p^{(n-1)/n} \|\nabla f\|_p^{1/n}$ derived via Blaschke–Santaló, which, together with the classical Poincaré inequality, yields affine Poincaré inequalities, compactness, and existence of minimizers. The authors establish the affine $p$-Faber–Krahn inequality showing ellipsoids minimize $\lambda^{\mathcal{A}}_{1,p}$ at fixed volume, prove that minimizers satisfy the affine Euler–Lagrange equation and regularity, and compare affine and classical eigenvalues, including rigidity statements. They also introduce affine Cheeger-type concepts and prove the existence of affine Cheeger sets, highlighting deep ties between convex geometry and spectral theory in the affine setting. The results provide a stronger, affine-invariant counterpart to classical inequalities with broad implications for spectral geometry and PDEs.
Abstract
Given a bounded open subset $Ω$ of $\mathbb R^n$, we establish the weak closure of the affine ball $B^{\mathcal A}_p(Ω) = \{f \in W^{1,p}_0(Ω):\ \mathcal E_p f \leq 1\}$ with respect to the affine functional $\mathcal E_pf$ introduced by Lutwak, Yang and Zhang in [43] as well as its compactness in $L^p(Ω)$ for any $p \geq 1$. These points use strongly the celebrated Blaschke-Santaló inequality. As counterpart, we develop the basic theory of $p$-Rayleigh quotients in bounded domains, in the affine case, for $p\geq 1$. More specifically, we establish $p$-affine versions of the Poincaré inequality and some of their consequences. We introduce the affine invariant $p$-Laplace operator $Δ_p^{\mathcal A} f$ defining the Euler-Lagrange equation of the minimization problem of the $p$-affine Rayleigh quotient. We also study its first eigenvalue $λ^{\mathcal A}_{1,p}(Ω)$ which satisfies the corresponding affine Faber-Krahn inequality, this is that $λ^{\mathcal A}_{1,p}(Ω)$ is minimized (among sets of equal volume) only when $Ω$ is an ellipsoid. This point depends fundamentally on PDEs regularity analysis aimed at the operator $Δ_p^{\mathcal A} f$. We also present some comparisons between affine and classical eigenvalues, including a result of rigidity through the characterization of equality cases for $p \geq 1$. All affine inequalities obtained are stronger and directly imply the classical ones.