On regularity and singularity of solutions of some nonlinear elliptic equations

一些非线性椭圆方程解的正则性和奇异性

• 批准号: 1362525
• 负责人: Luis Silvestre
• 金额: $ 12.3万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1362525&HistoricalAwards=false
• 关键词: regularity; singularity; solutions; some; nonlinear

This project concerns the analysis and applications of nonlinear partial differential equations, with emphasis on integro-differential equations. Integro-differential equations are equations which involve both integrals and derivatives. They arise from models such as in diffusion with long range interactions in physics, future options in mathematical finance, and population dynamics in social science. They also appear intrinsically in geometry. One main goal of the proposal is to study fine analysis of a particular family of integro-differential equations, which will be applied to understand a number of scientific phenomena in geometry and physics. Another main goal is to develop general mathematical theories on integro-differential equations for their wide usage in future. Other than its applications, the analysis itself for those equations is of independent interest. It not only extends the current theories of partial differential equations, but also gives new insights and creates new methods of establishing them. This proposal focuses on regularity and singularity of solutions of nonlinear elliptic partial differential equations. The PI proposes to develop a unified approach to study existence and compactness of solutions to a family of prescribed fractional order curvature problems in conformal geometry. The investigation of their singular solutions will not only develop further the fractional singular Yamabe problem, but also lead naturally to boundary reaction-diffusion equations which appear as models of dislocations in crystals and soft thin films in micromagnetism. The study on regularity of fully nonlinear integro-differential equations, which usually arise from stochastic control problems with purely jump Levy process, will enrich the existing general regularity theory. The proposed research on Monge-Ampere equations is motivated by Monge-Ampere metrics on affine manifolds with singularities. Its goal is to address the regularity and analyze the behavior of solutions of such equations with singularities, and to understand their connections to differential geometry.

本课题研究非线性偏微分方程的分析与应用，重点研究积分微分方程。积分微分方程是既包含积分又包含导数的方程。它们来自于物理学中具有长期相互作用的扩散模型、数学金融中的未来选择模型和社会科学中的人口动态模型。它们本质上也出现在几何中。该提案的一个主要目标是研究一组特定的积分-微分方程的精细分析，这将被应用于理解几何和物理中的许多科学现象。另一个主要目标是发展积分-微分方程的一般数学理论，使其在未来得到广泛应用。除了它的应用之外，对这些方程的分析本身也具有独立的意义。它不仅扩展了现有的偏微分方程理论，而且给出了新的见解，创造了建立偏微分方程的新方法。本文主要研究非线性椭圆型偏微分方程解的正则性和奇异性。PI提出了一种统一的方法来研究保形几何中一类规定分数阶曲率问题解的存在性和紧性。对它们的奇异解的研究不仅将进一步发展分数阶奇异的Yamabe问题，而且将自然而然地导致边界反应-扩散方程的出现，作为微磁性中晶体和软薄膜位错的模型。完全非线性积分-微分方程的正则性研究将丰富现有的一般正则性理论，这些方程通常是由纯跳跃Levy过程的随机控制问题引起的。本文提出的蒙日-安培方程的研究是由奇异仿射流形上的蒙日-安培度量驱动的。它的目标是解决这些奇异方程的规律性和分析解的行为，并了解它们与微分几何的联系。