CAREER: Singular and Global Solutions to Nonlinear Elliptic Equations

职业：非线性椭圆方程的奇异和全局解

• 批准号: 2143668
• 负责人: Connor Mooney
• 金额: $ 50.06万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2027-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2143668&HistoricalAwards=false
• 关键词: CAREER; Singular; Global; Solutions; Nonlinear

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.

物理定律和曲率条件是用偏微分方程 (PDE) 语言编写的。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，并在数学分析中提出了重大挑战。在这个项目中，PI 将研究在物理学和几何学中发挥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示例的构造。该项目包括一个教育部分，通过（a）博士后研究人员和博士的监督，涉及许多职业阶段的研究人员。学生; (b) 每季度组织一次周末会议，对学生进行科学传播方面的培训； (c) 为研究生和高年级本科生举办冬季讲习班，由专家就与该项目相关的研究主题举办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) 构建新的示例最小曲面方程变体的非线性整体解，并证明相关的伯恩斯坦型定理； (2) 研究完全非线性椭圆方程解中出现的奇异结构，例如 Monge-Ampere 和二次 Hessian 方程，其动机是在复杂几何、最优传输和气象学中的应用； (3)构造低维经典变分积分的奇异极小化的新例子，并发现阻止奇点形成的结构条件。所研究的方程具有限制标准技术实用性的共同特征（简并椭圆性和奇异解的存在）。为了应对这些挑战，PI 将寻求将多个数学领域结合起来的新方法，并构建复杂的技术工具来实施这些方法。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能力进行评估，被认为值得支持。更广泛的影响审查标准。

项目成果

Homogeneous functions with nowhere-vanishing Hessian determinant

具有不消失的 Hessian 行列式的齐次函数

• DOI: 10.4171/aihpc/78
• 发表时间: 2023-02
• 期刊: Analyse non linéaire
• 作者: Mooney; Connor
• 通讯作者: Connor

Singular structures in solutions to the Monge-Ampère equation with point masses

具有点质量的 Monge-Ampère 方程解中的奇异结构

• DOI: 10.3934/mine.2023083
• 发表时间: 2023-01
• 期刊: Mathematics in Engineering
• 影响因子: 1
• 作者: Mooney, Connor;Rakshit, Arghya
• 通讯作者: Rakshit, Arghya