CAREER: Singular and Global Solutions to Nonlinear Elliptic Equations

物理定律和曲率条件用偏微分方程组(PDE)的语言写成。这些方程的解往往是奇异的(非光滑的)，这限制了解的数值逼近的可靠性，并在其数学分析中提出了重大挑战。在这个项目中，PI将研究在物理和几何中起核心作用的非线性椭圆型偏微分方程解的定性行为，特别强调奇异和整体例子的构造。该项目包括一个教育部分，通过(A)指导博士后研究人员和博士后学生；(B)每季度组织一次周末会议，对学生进行科学交流方面的培训；(C)组织一次针对本科生研究生和高级本科生的冬季讲习班，由专家就与该项目有关的研究主题举办为期一周的短期课程；在技术层面上，该项目的主要目标是(1)构造最小表面方程变量的非线性整体解的新实例，并证明相关的Bernstein型定理；(2)研究完全非线性椭圆型方程的解中出现的奇异结构，例如Monge-Ampere方程和二次Hessian方程，受复杂几何、最优运输和气象学的应用的启发；(3)构造了低维经典变分积分奇异极小的新例子，发现了防止奇点形成的结构条件。研究中的方程具有共同的特征(退化的椭圆性和奇异解的存在)，这些特征限制了标准技术的有效性。为了应对这些挑战，PI将寻求将几个数学领域结合在一起的新方法，并建立复杂的技术工具来执行这些方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Physical laws and curvature conditions are written in the language of partial differential equations (PDE). Solutions to these equations are often singular (non-smooth), which limits the reliability of numerical approximations of solutions and presents significant challenges in their mathematical analysis. In this project, the PI will investigate the qualitative behavior of solutions to nonlinear elliptic PDE that play a central role in physics and geometry, with particular emphasis on the construction of singular and global examples. The project includes an educational component that involves researchers at many career stages through (a) the supervision of postdoctoral researchers and Ph.D. students; (b) the organization of a quarterly weekend conference to train students in scientific communication; (c) the organization of a winter workshop aimed at graduate and advanced undergraduate students, with week-long short courses by experts on research topics related to this project; and (d) the writing of a book based on advanced topics courses given by the PI, with significant input from the students who attended these courses.At a technical level, the main goals of the project are to (1) construct new examples of nonlinear entire solutions to variants of the minimal surface equation, and prove related Bernstein-type theorems; (2) investigate singular structures that appear in solutions to fully nonlinear elliptic equations such as the Monge-Ampere and quadratic Hessian equations, motivated by applications to complex geometry, optimal transport, and meteorology; and (3) construct new examples of singular minimizers of classical variational integrals in low dimensions, and discover structure conditions that prevent the formation of singularities. The equations under investigation share features (degenerate ellipticity and the existence of singular solutions) that limit the usefulness of standard techniques. To address these challenges, the PI will pursue new approaches that unite several areas of mathematics, and build sophisticated technical tools to carry out these approaches.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.