Interplay of Sobolev Spaces on Compact Manifolds: Embedding Theorems, Inequalities, and Compactness
Mogoi N. Evans *
Department of Mathematics and Statistics, Kaimosi Friends University, Kenya.
Samuel B. Apima
Department of Mathematics and Statistics, Kaimosi Friends University, Kenya.
*Author to whom correspondence should be addressed.
Abstract
This research paper explores various properties of Sobolev spaces on compact manifolds, focusing on embedding theorems, compactness, and inequalities. We establish the compact embedding of Sobolev spaces into continuous and Lebesgue spaces, as well as the continuity and compactness of embeddings between different Sobolev spaces. We also derive inequalities involving the Laplacian and gradients of functions, providing insights into their behavior on manifolds. These results contribute to our understanding of the interplay between function smoothness, continuity, and distribution on compact manifolds.
Keywords: Sobolev spaces, embedding theorems, Arzel\(\acute{a}\)–Ascoli theorem, Rellich-Kondrachov compactness theorem, inequalities, Laplacian, gradients, functional analysis, differential geometry, continuity, compactness, trace theorems, topology