Compact Sobolev embeddings on non-compact manifolds via orbit expansions of isometry groups

TL;DR

This paper establishes new compact Sobolev embeddings on non-compact manifolds with isometry group conditions, extending classical results to Finsler manifolds and applying them to quasilinear PDEs.

Contribution

It introduces an orbit expansion condition for isometry groups that ensures compact Sobolev embeddings on non-compact manifolds, including Finsler cases, and applies these results to PDE analysis.

Findings

Characterization of coerciveness of isometry groups via orbit expansions.

Proved compact Sobolev embeddings for Riemannian and Finsler manifolds.

Applied embeddings to study quasilinear PDEs on Randers spaces.

Abstract

Given a complete non-compact Riemannian manifold $(M,g)$ with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries $G$ of $(M,g)$ that characterizes the coerciveness of $G$ in the sense of Skrzypczak and Tintarev (Arch. Math., 2013). Furthermore, under these conditions, compact Sobolev-type embeddings \`a la Berestycki-Lions are proved for the full range of admissible parameters (Sobolev, Moser-Trudinger and Morrey). We also consider the case of non-compact Randers-type Finsler manifolds with finite reversibility constant inheriting similar embedding properties as their Riemannian companions; sharpness of such constructions are shown by means of the Funk model. As an application, a quasilinear PDE on Randers spaces is studied by using the above compact embeddings and variational arguments.