Sobolev inequalities on a weighted Riemannian manifold of positive Bakry–Émery curvature and convex boundary

Saïd Ilias and Abdolhakim Shouman

Vol. 294 (2018), No. 2, 423–451

DOI: 10.2140/pjm.2018.294.423

Abstract

In this paper, we study some nonlinear elliptic equations on a compact $n$-dimensional weighted Riemannian manifold of positive $m$-Bakry–Émery–Ricci curvature and convex boundary. Our main purpose is to find conditions which imply that such elliptic equations admit only constant solutions. As an application, we obtain weighted Sobolev inequalities with explicit constants that extend the inequalities obtained by Ilias [1983; 1996] in the Riemannian setting. In a last part of the article, as applications we derive a new Onofri inequality, a logarithmic Sobolev inequality and estimates for the eigenvalues of a weighted Laplacian and for the trace of the weighted heat kernel.

To the memory of our friend A. El Soufi

Keywords

Sobolev inequality, Onofri inequality, weighted Riemannian manifold, convex boundary, manifold with density, weighted Laplacian, drifting Laplacian, Bakry–Émery–Ricci curvature, Neumann boundary condition, eigenvalues

Mathematical Subject Classification 2010

Primary: 35J60, 53C21, 58J32, 58J50, 58J60

Milestones

Received: 17 April 2017

Revised: 3 November 2017

Accepted: 3 November 2017

Published: 20 February 2018