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Research Article Open access CC BY 4.0

Advancements in the Analysis of Sobolev Spaces and Function Spaces on Manifolds: Theoretical Framework and Applications

Mogoi N. Evans, Samuel B. Apima, Amos Otieno Wanjara

Advances in Research · pp. 148–153 · Published 10 Apr 2024

10.9734/air/2024/v25i31060

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.

Sobolev spaces manifolds; function spaces Poincare inequality compact embeddings

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