From affine Poincar\'e inequalities to affine spectral inequalities

Given a bounded open subset $\Omega$ 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 $\mathcal E_pf$ introduced by Lutwak, Yang and Zhang in [43] as well as its compactness in $L^p(\Omega)$ for any $p \geq 1$. 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 $p\geq 1$. 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 $\Delta_p^{\mathcal A} f$ 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 $\lambda^{\mathcal A}_{1,p}(\Omega)$ is minimized (among sets of equal volume) only when $\Omega$ is an ellipsoid. This point depends fundamentally on PDEs regularity analysis aimed at the operator $\Delta_p^{\mathcal A} f$. We also present some comparisons between affine and classical eigenvalues, including a result of rigidity through the characterization of equality cases for $p \geq 1$. All affine inequalities obtained are stronger and directly imply the classical ones.