Nonlinear Partial Differential Equations and Applications

非线性偏微分方程及其应用研究摘要：秦汉在获得该奖项后，计划继续其在偏微分方程方面的研究。特别是他将研究微分方程解的节点集的作用，或者更一般的函数的水平集。还有解的奇异集和分支集。研究的一个重要部分是研究这些集附近解的渐近行为或这些集本身的渐近行为。微分方程可以是椭圆型、双曲型或混合型。有些问题与数学的其他领域有密切的联系，包括几个复变量和代数几何。PI将继续研究的另一类问题涉及方程中已知函数的水平集对解的存在性和性质的影响。一个特殊的问题是当高斯曲率的零集表现良好时，二维黎曼流形在三维空间中的等距嵌入问题。奇异集问题经常出现在材料科学和控制理论中。奇点集合，顾名思义，就是那些有奇点的集合。精确的定义因问题的产生而异。通常不可能消除奇异集，即所谓的“坏集”。中心任务之一是确定奇异集可以控制的条件和奇异集很小的条件。该方案中所提出的关于奇异集的问题是其最简单的形式。它们与Erickson液晶模型和超导中的金兹堡-朗道方程有关。PI相信对这些数学问题的讨论将有助于科学家在各种应用中处理奇异集。

Nonlinear Partial Differential Equations and Applications Abstract of Proposed ResearchQing HanUnder this award, the principal investigator plans to continue his work in partial differential equations. In particular he will investigate the role of the nodal sets of solutions of differential equations, or more general level sets of functions. Also the singular sets and the branch sets of solutions. An important part of the study is the investigation of the asymptotic behavior of solutions near these sets or the asymptotic behavior of these sets themselves. The differential equations may be of elliptic, hyperbolic or mixed type. Some problems have close connections with other fields in mathematics, including several complex variables and algebraic geometry. Another class of problems that the PI would continue to work on involves the effect of level sets of known functions in the equations on the existence and properties of solutions. A particular problem is the isometric embedding of 2-dim Riemannian manifolds in 3-space when the zero set of Gauss curvature is well behaved.The problems involving singular sets occur in materials science and control theory. Singular sets, as the name suggests, are those sets where singularities occur. Precise definitions vary depending on the problems where they arise. It often is impossible to eliminate the singular sets, the so-called "bad sets". One of the central tasks is to identify the conditions under which the singular sets can be controlled and the conditions under which the singular sets are small. The proposed problems concerning singular sets in the project are in their simplest forms. They are related to the Erickson's model for liquid crystals and the Ginzburg-Landau equation in the superconductivity. The PI believes that the discussion of these mathematical problems will help scientists work with singular sets in various applications.