Regularity vs. Singularity for Elliptic and Parabolic Systems

椭圆和抛物线系统的正则性与奇异性

• 批准号: 1854788
• 负责人: Connor Mooney
• 金额: $ 14.38万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1854788&HistoricalAwards=false
• 关键词: Regularity; vs.; Singularity; Elliptic; Parabolic

This project investigates partial differential equations that arise in several areas of pure and applied mathematics including meteorology, elasticity, and differential and complex geometry. The proposed problems concern the regularity (smoothness properties) of solutions. Key challenges that these problems share are that ellipticity (an agent of regularity) degenerates, and that solutions can be vector-valued. Overcoming these challenges will not only require significant new mathematical ideas, but could also help us better predict weather patterns and assist in the design of cars. The results will be communicated by publication in peer-reviewed journals, and through the writing of expository notes that the PI intends to make publicly available and will be useful to researchers and students alike.The first project tackles regularity questions for the real Monge-Ampere equation, motivated by applications to meteorology and differential geometry. The PI will investigate singularity formation for the semi-geostrophic system, which models large scale atmospheric flows. Certain examples of irregular stationary solutions in a half-plane may be useful models for blowup at boundary points. The second project concerns local regularity for the complex Monge-Ampere equation, which arises in complex geometry. Difficulties include the non-convexity of solutions, and invariance under adding certain quadratic polynomials. The PI has initiated the study of a model equation that captures these difficulties. The third project concerns non-concave uniformly elliptic equations. A challenging problem is to construct singular solutions in low dimensions. An obstruction is that in 3d, linear equations with rough coefficients have no nontrivial one-homogeneous solutions; the PI will instead consider solutions with "spiraling" one-homogenous symmetry. The fourth project concerns minimizers of variational integrals with convex integrand. A conjecture is that scalar-valued minimizers have the same regularity as the Legendre transform of the integrand. The PI proposes to confirm this when the degeneracy set of the integrand satisfies certain geometric conditions, and to construct a counterexample when these conditions aren't met. The last project aims to answer classical regularity and stability questions for parabolic systems, especially in low dimensions. The PI recently constructed examples of singular solutions to linear parabolic systems in the plane, answering a long-standing question. This may open the way to tackling related problems, e.g. to identify conditions on coefficients that prevent singularities in the linear case, and to determine regularity vs. singularity for nonlinear structures of porous medium type.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.

该项目研究纯数学和应用数学多个领域中出现的偏微分方程，包括气象学、弹性以及微分和复几何。提出的问题涉及解的规律性（平滑性）。这些问题共同面临的主要挑战是椭圆性（规律性的代理）退化，并且解决方案可以是向量值的。 克服这些挑战不仅需要重要的新数学思想，而且还可以帮助我们更好地预测天气模式并协助汽车设计。研究结果将通过在同行评审期刊上发表文章以及撰写解释性说明进行交流，PI 打算公开这些说明性说明，这些说明对研究人员和学生都有用。第一个项目解决了真实 Monge-Ampere 的规律性问题方程，受到气象学和微分几何应用的启发。 PI 将研究半地转系统奇点的形成，该系统模拟大规模大气流动。半平面中不规则稳态解的某些示例可能是边界点爆炸的有用模型。第二个项目涉及复杂几何中出现的复杂 Monge-Ampere 方程的局部正则性。困难包括解的非凸性以及添加某些二次多项式时的不变性。 PI 已开始研究解决这些困难的模型方程。第三个项目涉及非凹均匀椭圆方程。一个具有挑战性的问题是在低维中构建奇异解决方案。一个障碍是，在 3d 中，具有粗糙系数的线性方程没有非平凡的齐次解；相反，PI 将考虑具有“螺旋”单齐对称性的解决方案。第四个项目涉及凸被积函数的变分积分的最小化。一个猜想是，标量值极小化器与被积函数的勒让德变换具有相同的规律性。 PI建议当被积函数的简并集满足一定的几何条件时证实这一点，并在不满足这些条件时构造一个反例。最后一个项目旨在回答抛物线系统的经典规律性和稳定性问题，特别是在低维情况下。 PI 最近构建了平面线性抛物线系统奇异解的示例，回答了一个长期存在的问题。这可能会为解决相关问题开辟道路，例如确定防止线性情况下奇异性的系数条件，并确定多孔介质类型非线性结构的规律性与奇异性。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的评估进行评估，被认为值得支持影响审查标准。

项目成果

Entire solutions to equations of minimal surface type in six dimensions

六维极小曲面方程的全解

• DOI: 10.4171/jems/1202
• 发表时间: 2022-01
• 期刊: Journal of the European Mathematical Society
• 影响因子: 2.6
• 作者: Mooney; Connor
• 通讯作者: Connor

Hilbert’s $$19{\text {th}}$$ problem revisited

希尔伯特 $$19{ ext {th}}$$ 问题重温

• DOI: 10.1007/s40574-021-00315-3
• 发表时间: 2022-01
• 期刊: Bollettino dell'Unione Matematica Italiana
• 作者: Mooney; Connor
• 通讯作者: Connor

Solutions to the Monge–Ampère Equation with Polyhedral and Y-Shaped Singularities

具有多面体和 Y 形奇点的蒙日安培方程的解

• DOI: 10.1007/s12220-021-00615-2
• 发表时间: 2021-01
• 期刊: The Journal of Geometric Analysis
• 作者: Mooney; Connor
• 通讯作者: Connor

Minimizers of convex functionals with small degeneracy set

具有小简并集的凸泛函的极小化

• DOI: 10.1007/s00526-020-1723-9
• 发表时间: 2019-03-07
• 期刊: Calculus of Variations and Partial Differential Equations
• 影响因子: 2.1
• 作者: Connor Mooney

Strict 2-convexity of convex solutions to the quadratic Hessian equation

二次 Hessian 方程凸解的严格 2-凸性

• DOI: 10.1090/proc/15454
• 发表时间: 2021-01
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Mooney; Connor
• 通讯作者: Connor