Adams迹不等式及其极值函数存在性的相关问题研究

Adams迹不等式是非常重要的一类迹不等式, 偏微分方程和几何分析中的很多重要问题，如，流形上带边 Yamabe 问题、Bieberbach猜想以及平面等谱域的紧性问题等都与这类不等式密切相关。本项目将重点关注Adams迹不等式及其极值函数存在性，以及相关非线性Neumann边值问题的量化性质。由于现有的截断函数不具有足够的光滑性，并且没有相关Liouville方程解的分类，这给Adams迹不等式及其极值函数的研究带来很大困难。本项目拟构造具有足够光滑性的截断函数，并对相关Liouville方程的解进行分类，进而利用爆破分析、调和延拓和变分法等工具研究单位球和半空间上的Adams迹不等式及其极值函数存在性，以及相关非线性Neumann边值问题的量化性质。本项目的研究将有助于揭示Adams迹不等式及其极值函数存在性的一般规律，并为深入研究Adams迹泛函的临界点等相关问题提供一定的理论基础。

Adams trace inequalities are a kind of very important trace inequalities, many important problems in PDE’s and geometric analysis, such as,Yamabe problem on manifolds with boundary, the Bieberbach conjecture and the compactness of isospectral planar domains are closely related to this kind of inequalities. This project will focus on the Adams trace inequalities and the existence of extremal functions, as well as the quantization properties of the nonlinear Neumann boundary value problem. Since the existing truncation functions do not have enough smoothness, and also there are no relevant results about the classification of solutions to the related Liouville equation, this brings great difficulty to the study of Adams trace inequalities and the existence extremals. We will try to construct a kind of truncation functions with sufficient smoothness, classify the solutions of the relevant Liouville equation, and study the Adams trace inequality and the existence of extremals on the unit ball and half spaces, as well as the quantization properties of the relevant nonlinear Neumann boundary value problem by the blow-up analysis, harmonic extension and variational method. The research of this project will reveal the general rules of the existence of extremals for Adams Trace inequalities, and provide the essential theoretical basis for the further study of problems of critical point related with the Adams Trace functional.