上半空间的最佳积分不等式及其应用

最佳Hardy-Littlewood-Sobolev(简称HLS)不等式和Sobolev不等式等积分不等式是分析学和几何学中的基本估计工具，也是物理学中流体力学等领域的重要研究工具。本项目致力于研究上半空间与延拓积分算子相关的对数HLS型不等式、Carleman型不等式等，及与加权散度算子相关的Sobolev(迹)不等式、Moser-Trudinger型不等式等，并讨论其最佳常数和极值函数。作为应用，研究带边流形上相应的曲率方程解的存在性等。将通过分析算子特性，使用基本不等式、逼近技巧等建立积分不等式；利用重排技巧、变分原理证明不等式对应的极值函数的存在性，并使用移动球面法讨论其分类；对于微分算子，借助其与延拓积分算子的联系，利用积分算子的整体性，克服微分算子的奇异、退化带来的困难，给出相关估计。本项目有助于丰富偏微分方程、微分几何和物理学理论，并为其他学科相关问题的研究提供理论依据。

The sharp integral inequalities including the sharp Hardy-Littlewood-Sobolev(HLS) inequality and Sobolev inequality play important roles in analysis and geometry, and also in the hydromechanics of physics. This project is devoted to establishing the logarithmic HLS type inequality, the Carleman type inequality, etc., related to the extensional integral operator on the upper half space, as well as the Sobolev (trace) inequality, Moser-Trudinger type inequality, etc., related to the weighted divergent operator on the upper half space. Moreover, the project will pay much attention to the best constants and extremal functions for these inequalities. As applications, the existence of solutions to curvature problems on compact manifolds with boundary will be investigated. Based on the analysis of specific features of the operators, the integral inequalities will be established by employing some basic inequalities and the approximating technique. On the other hand, the existence of extremal functions will be proved by using the arrangement technique and variational methods, and the extremal functions will be classified via the method of moving spheres. Particularly, the relationship between the differential operator and the extensional integral operator will be used to overcome the difficulty of singularity and degeneracy of the differential equation by using the global property of the integral operator. The results will improve the theory of partial differential equations, differential geometry and physics, and to provide the theoretical foundations to others related fields.