Interpolation between Brezis–Vázquez and Poincaré inequalities on nonnegatively curved spaces: sharpness and rigidities

A. Kristály, Anikó Szakál

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

This paper is devoted to investigate an interpolation inequality between the Brezis–Vázquez and Poincaré inequalities (shortly, BPV inequality) on nonnegatively curved spaces. As a model case, we first prove that the BPV inequality holds on any Minkowski space, by fully characterizing the existence and shape of its extremals. We then prove that if a complete Finsler manifold with nonnegative Ricci curvature supports the BPV inequality, then its flag curvature is identically zero. In particular, we deduce that a Berwald space of nonnegative Ricci curvature supports the BPV inequality if and only if it is isometric to a Minkowski space. Our arguments explore fine properties of Bessel functions, comparison principles, and anisotropic symmetrization on Minkowski spaces. As an application, we characterize the existence of nonzero solutions for a quasilinear PDE involving the Finsler–Laplace operator and a Hardy-type singularity on Minkowski spaces where the sharp BPV inequality plays a crucial role. The results are also new in the Riemannian/Euclidean setting.

Original languageEnglish
JournalJournal of Differential Equations
DOIs
Publication statusAccepted/In press - Jan 1 2018

    Fingerprint

Keywords

  • Bessel functions
  • Brezis–Vázquez inequality
  • Extremals
  • Finsler manifold
  • Poincaré inequality
  • Sharpness

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this