Advancements in the Analysis of Sobolev Spaces and Function Spaces on Manifolds: Theoretical Framework and Applications
Mogoi N. Evans *
Department of Mathematics and Statistics, Kaimosi Friends University, Kenya.
Samuel B. Apima
Department of Mathematics and Statistics, Kaimosi Friends University, Kenya.
Amos Otieno Wanjara
Department of Mathematics and Statistics, Kaimosi Friends University, Kenya.
*Author to whom correspondence should be addressed.
Abstract
This research paper delves into Sobolev spaces and function spaces on smooth manifolds, revealing fundamental theorems such as existence, embeddings, and compactness properties. Noteworthy results include the Poincare inequality elucidating function behavior on compact manifolds and compactness properties of Sobolev spaces on Riemannian manifolds. The study establishes trace theorems for functions on the boundary and interpolation results between Sobolev spaces. Isoperimetric inequalities and stability under weak convergence contribute to a holistic understanding of geometric and analytical aspects of Sobolev spaces. The research concludes by exploring invariance under diffeomorphisms and compactness in dual spaces, providing a unified framework for analyzing function spaces on manifolds.
Keywords: Sobolev spaces, manifolds;, function spaces, Poincare inequality, compact embeddings