基于isometry群轨道展开的非紧流形上的紧Sobolev嵌入
Compact Sobolev embeddings on non-compact manifolds via orbit expansions of isometry groups
摘要：
给定一个具有一定曲率限制的完备非紧黎曼流形$(M,g)$，我们引入了一个关于$(M,g)$的等距群$G$的展开条件，它刻画了$G$在Skrzypczak和Tintarev（Arch. Math.，2013年）的报告。此外，在这些条件下，紧Sobolev型嵌入la Berestycki-Lions被证明为全范围的容许参数（Sobolev，Moser-Trudinger和Morrey）。我们还考虑了有限可逆常数的非紧Randers型Finsler流形的情况下，继承类似的嵌入性质作为他们的黎曼同伴，这样的结构的锋利度通过Funk模型。作为应用，利用上述紧嵌入和变分变元研究了Randers空间上的拟线性偏微分方程。
英文摘要：
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 à 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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