From affine Poincaré inequalities to affine spectral inequalities

Given a bounded open subset Ω of Rn, we establish the weak closure of the affine ball BpA(Ω)={f∈W01,p(Ω):Epf≤1} with respect to the affine functional Epf introduced by Lutwak, Yang and Zhang in [46] as well as its compactness in Lp(Ω) for any p≥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≥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 ΔpAf defining the Euler-Lagrange equation of the minimization problem of the p-affine Rayleigh quotient. We also study its first eigenvalue λ1,pA(Ω) which satisfies the corresponding affine Faber-Krahn inequality, that is, λ1,pA(Ω) is minimized (among sets of equal volume) only when Ω is an ellipsoid. This point depends fundamentally on the PDEs regularity analysis aimed at the operator ΔpAf. We also present some comparisons between affine and classical eigenvalues, including a result of rigidity through the characterization of equality cases for p≥1. All affine inequalities obtained are stronger and directly imply the classical ones.