3 years ago

From affine Poincar\'e inequalities to affine spectral inequalities. (arXiv:2003.07391v2 [math.AP] UPDATED)

Julián Haddad, Carlos Hugo Jiménez, Marcos Montenegro
Given a bounded open subset of , we establish the weak closure of the affine ball with respect to the affine functional introduced by Lutwak, Yang and Zhang in [43] as well as its compactness in for any . This part relies strongly on the celebrated Blaschke-Santal\'{o} inequality. As counterpart, we develop the basic theory of -Rayleigh quotients in bounded domains, in the affine case, for . More specifically, we establish -affine versions of the Poincar\'{e} inequality and some of their consequences. We introduce the affine invariant -Laplace operator defining the Euler-Lagrange equation of the minimization problem of the -affine Rayleigh quotient. We also study its first eigenvalue which satisfies the corresponding affine Faber-Krahn inequality, this is that is minimized (among sets of equal volume) only when is an ellipsoid. This point depends fundamentally on PDEs regularity analysis aimed at the operator . We also present some comparisons between affine and classical eigenvalues and characterize the cases of equality for . Lastly, for and convex we find a sufficient condition of spectral type for the domain to be in John's position. All affine inequalities obtained are stronger and directly imply the classical ones.

Publisher URL: http://arxiv.org/abs/2003.07391

DOI: arXiv:2003.07391v2

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