CAREER: Singular and Global Solutions to Nonlinear Elliptic Equations

物理定律和曲率条件是用偏微分方程（PDE）的语言写的。这些方程的解通常是奇异的（非光滑的），这限制了解的数值近似的可靠性，并对其数学分析提出了重大挑战。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偏微分方程解的定性行为，特别强调奇异和全局例子的构建。该项目包括一个教育部分，涉及到许多职业阶段的研究人员，通过(a)博士后研究人员和博士生的监督；(b)每季度组织一次周末会议，培训学生进行科学传播；(c)为研究生和高级本科生举办冬季讲习班，由专家就与本项目有关的研究课题提供为期一周的短期课程；(d)根据PI提供的高级主题课程编写一本书，并听取参加这些课程的学生的重要意见。在技术层面上，该项目的主要目标是：(1)构造最小曲面方程变异体的非线性完整解的新例子，并证明相关的伯恩斯坦型定理；(2)研究出现在完全非线性椭圆方程（如蒙日-安培方程和二次黑森方程）解中的奇异结构，其动机是应用于复杂几何、最优运输和气象学；(3)构造低维经典变分积分的奇异极小值的新例子，并发现防止奇点形成的结构条件。所研究的方程具有共同的特征（简并椭圆性和奇异解的存在性），这些特征限制了标准技术的有效性。为了应对这些挑战，PI将寻求结合多个数学领域的新方法，并建立复杂的技术工具来实施这些方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.