Date of Completion

5-4-2018

Embargo Period

5-3-2020

Major Advisor

Guozhen Lu

Associate Advisor

Xiaodong Yan

Associate Advisor

Maria Gordina

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Campus Access

Abstract

The goal of this thesis is to investigate best constants, extremal functions, and stability for different geometric and functional inequalities such as the Trudinger-Moser, Caffarelli-Kohn-Nirenberg and Hardy-Sobolev inequalities.

It is well known that the geometric and functional inequalities are basic tools in analysis, calculus of variations, PDEs and geometry. Sharp constants and extremal functions play an essential role because they contain rich geometric, analytical and probabilistic information.

In Chapter 2, we first established several sharp weighted Trudinger-Moser inequalities in Euclidean space $\mathbb R^n$. Moreover, we proved the optimizers of these inequalities must be radially symmetric. Our proofs are based on a quasi-conformal mapping type transform.

In Chapter 3, we applied the same quasi-conformal mapping type transform to study the first-order Caffarelli-Kohn-Nirenberg inequalities in Euclidean space $\mathbb R^n$ in the borderline case $p=n$. Additionally, we gave the explicit form of the best constants and extremal functions for particular cases.

In Chapter 4, by demonstrating a compact embedding theorem on weighted Sobolev space and a variational method, we proved the existence of the extremal functions for higher-order Caffarelli-Kohn-Nirenberg inequalities.

Suppose we are given a geometric or functional inequality for which optimizers are known, the stability property tells us for functions almost attain the equality, they must close (in some suitable sense) to the manifold of all extremal functions. We study the stability for Hardy-Sobolev inequalities and weighted Sobolev inequalities in Chapter 5.

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