From affine Poincaré inequalities to affine spectral inequalities

Abstract Given a bounded open subset Ω of R n , we establish the weak closure of the affine ball B p A ( Ω ) = { f ∈ W 0 1 , p ( Ω ) : E p f ≤ 1 } with respect to the affine functional E p f introduced by Lutwak, Yang and Zhang in [46] as well as its compactness in L p ( Ω ) for any p ≥ 1 . These points use strongly the celebrated Blaschke-Santalo 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 Poincare inequality and some of their consequences. We introduce the affine invariant p-Laplace operator Δ p A f defining the Euler-Lagrange equation of the minimization problem of the p-affine Rayleigh quotient. We also study its first eigenvalue λ 1 , p A ( Ω ) which satisfies the corresponding affine Faber-Krahn inequality, that is, λ 1 , p A ( 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 Δ p 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 ≥ 1 . All affine inequalities obtained are stronger and directly imply the classical ones.