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

Farkas, Csaba and Kristály, Alexandru and Mester, Ágnes (2021) Compact Sobolev embeddings on non-compact manifolds via orbit expansions of isometry groups. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. pp. 1-24. ISSN 0944-2669 (print); 1432-0835 (online)

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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.

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