Singular Integrals in Geometric Analysis

几何分析中的奇异积分

• 批准号: 261100-2012
• 负责人: Xiao, Jie
• 金额: $ 1.53万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570966
• 关键词: Singular; Integrals; Geometric; Analysis

Geometric Analysis (GA) is a branch of mathematics treating the interface of Differential Geometry (DG) and Partial Differential Equations (PDE). This field includes both the use of PDE-methods in DG and the motivation of DG-results in PDE. In the study of GA, it is naturally and frequently needed to find complete integrability conditions for the existence of metrics with certain curvature conditions. Such a finding often involves various kinds of singular integrals that exist more or less in Harmonic Analysis (HA). The proposed research program will shed light upon this issue through the following three p-Laplace oriented objectives: 1. p-Green's functions and their geometric properties; 2. Minkowski's problem for generalized torsional rigidity; 3. Morrey-Riesz's integrals with applications to some geometric PDE. The novelty of this research program is to utilize new HA tools and techniques (especially, an effective incorporation of Riesz potentials and Morrey capacities plus Choquet integrals) to settle some fundamental problems on singular integrals arising from GA. This research program, creating a synthesis among a host of distinct topics, will not only be useful to researchers in both analysis and geometry based on either capacity or curvature, but also be of interest to both physicists and engineers in certain areas.

几何分析（GA）是处理微分几何（DG）和偏微分方程（PDE）界面的数学分支。该领域既包括PDE方法在DG中的使用，也包括PDE中DG结果的动机。在遗传算法的研究中，通常需要找到具有一定曲率条件的度量存在的完全可积条件。这一发现往往涉及调和分析（HA）中或多或少存在的各种奇异积分。拟议的研究计划将通过以下三个面向p-拉普拉斯的目标来阐明这个问题：