Kristaly, A. & Szakal, A. (2019) Journal of Differential Equations [Matematică, Q1]

Publicat: 05 Noiembrie 2020

Kristaly, A. & Szakal, A. (2019) Interpolation between Brezis-Vazquez and Poincare inequalities on nonnegatively curved spaces: sharpness and rigidities. Journal of Differential Equations, 266(10), 6621-6646

DOI: https://doi.org/10.1016/j.jde.2018.11.013

✓ Publisher: Elsevier
✓ Web of Science Core Collection: Science Citation Index Expanded
✓ Categories: Mathematics
✓ Article Influence Score (AIS): 1.505 (2019) / Q1

Abstract: This paper is devoted to investigate an interpolation inequality between the Brezis-Vazquez and Poincare 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.

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