Hardy-Trudinger-Moser不等式的最佳常数及其极值函数的存在性

由于几何不等式在偏微分方程、几何分析等数学分支中具有广泛应用，成为非线性泛函分析的研究热点。近年来，随着变指标Sobolev空间的发展，超临界Trudinger-Moser不等式逐渐引起关注。但超临界Hardy型Trudinger-Moser不等式目前还没有相关研究。本项目主要研究三个问题：第一，拟采用爆破分析的方法建立高维Hardy-Trudinger-Moser不等式的极值函数存在性；第二，精确刻画超临界Hardy-Trudinger-Moser不等式的最佳常数，并探讨极值函数的存在性；第三，建立二阶四维超临界Hardy-Adams不等式，并通过容量估计等方法建立极值函数的存在性。. 本项目的研究将揭示超临界Hardy型Trudinger-Moser不等式的一般规律，为进一步探索任意偶数维超临界Hardy-Adams不等式提供方法，促进几何不等式之间的联系与发展。

Because geometric inequalities have extensive application in the branches of mathematics such as partial differential equations and geometric analysis, it has become a hot topic in nonlinear functional analysis. In recent years, with the development of variable exponent Sobolev space, supercritical Trudinger-Moser inequality has attracted wide attention of experts and scholars. However, there is no relevant research about supercritical Hardy-Trudinger-Moser inequality and Hardy-Adams inequality with higher order derivatives. This project will focus on three questions. First, we will analyze the existence of extremal functions of high-dimensional Hardy-Trudinger-Moser inequality by applying blow-up analysis method. After that, we will establish the best constants of supercritical Hardy-Trudinger-Moser inequalities and the existence of extremal functions by using concentration-compactness principle. At last, we will prove the supercritical Hardy-Adams inequality with second-order on four dimensional ball. Furthermore, we will establish the existence of extremal functions by capacity estimation. . The research of this project will reveal the general rule for supercritical Hardy-Trudinger-Moser inequalities. We will provide a new method to explore any even dimensional supercritical Hardy-Adams inequalities with higher order. In this way, this project will promote the development and relationship between different geometric inequalities.