Date of Completion
Analysis | Geometry and Topology | Mathematics
Sobolev inequalities, named after Sergei Lvovich Sobolev, relate norms in Sobolev spaces and give insight to how Sobolev spaces are embedded within each other. This thesis begins with an overview of Lebesgue and Sobolev spaces, leading into an introduction to Sobolev inequalities. Soon thereafter, we consider the behavior of Sobolev inequalities on Riemannian manifolds. We discuss how Sobolev inequalities are used to construct isoperimetric inequalities and bound volume growth, and how Sobolev inequalities imply families of other Sobolev inequalities. We then delve into the usefulness of Sobolev inequalities in determining the geometry of a manifold, such as how they can be used to bound a manifold’s number of ends. We conclude with examples of real-world applications Sobolev inequalities.
Reever, John, "Sobolev Inequalities and Riemannian Manifolds" (2021). Honors Scholar Theses. 824.