On regularity and singularity of solutions of some nonlinear elliptic equations

本计画系关于非线性偏微分方程式之分析与应用，重点在积微分方程式。积分微分方程是同时包含积分和导数的方程。它们产生于物理学中的扩散与长程相互作用模型、数学金融学中的未来选择模型和社会科学中的人口动力学模型。它们在几何学中也是固有的。该提案的一个主要目标是研究一个特定的积分微分方程族的精细分析，这将被应用于理解几何和物理中的一些科学现象。另一个主要目标是发展积分微分方程的一般数学理论，以便在将来得到广泛的应用。除了它的应用，这些方程的分析本身是独立的兴趣。它不仅扩展了偏微分方程的现有理论，而且还提供了新的见解和建立它们的新方法。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分数阶曲率问题解的存在性和紧性。对它们奇异解的研究，不仅将进一步发展分数阶奇异Yamabe问题，而且也将自然地导出作为微磁学中晶体和软薄膜中位错模型的边界反应扩散方程。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，这类方程通常产生于具有纯跳Levy过程的随机控制问题。Monge-Ampere方程的研究是基于仿射流形上的Monge-Ampere度量。其目标是解决此类具有奇异性的方程的解的规律性并分析其行为，并了解它们与微分几何的联系。

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.