From affine Poincar\'e inequalities to affine spectral inequalities

Given a bounded open subset $\mathrm{\Omega }$ of ${\mathbb{R}}^n$, we establish the weak closure of the affine ball $B^{\mathcal A}_p(\Omega) = {f \in W^{1,p}0(\Omega):\ \mathcal E_p f \leq 1}$ with respect to the affine functional ${\mathcal{E}}_{p}f$ introduced by Lutwak, Yang and Zhang in [43] as well as its compactness in $L^p(\mathrm{\Omega })$ for any $p\ge 1$. These points use strongly the celebrated Blaschke-Santal'{o} inequality. As counterpart, we develop the basic theory of $p$-Rayleigh quotients in bounded domains, in the affine case, for $p\ge 1$. More specifically, we establish $p$-affine versions of the Poincar'e inequality and some of their consequences. We introduce the affine invariant $p$-Laplace operator ${\mathrm{\Delta }}_p^{\mathcal{A}}f$ defining the Euler-Lagrange equation of the minimization problem of the $p$-affine Rayleigh quotient. We also study its first eigenvalue $\lambda^{\mathcal A}{1,p}(\Omega)$ which satisfies the corresponding affine Faber-Krahn inequality, this is that ${\lambda }_{1,p}^{\mathcal{A}}(\mathrm{\Omega })$ is minimized (among sets of equal volume) only when $\mathrm{\Omega }$ is an ellipsoid. This point depends fundamentally on PDEs regularity analysis aimed at the operator ${\mathrm{\Delta }}_p^{\mathcal{A}}f$. We also present some comparisons between affine and classical eigenvalues, including a result of rigidity through the characterization of equality cases for $p\ge 1$. All affine inequalities obtained are stronger and directly imply the classical ones.