Regularity vs. Singularity for Elliptic and Parabolic Systems

该项目研究了在纯数学和应用数学的几个领域出现的偏微分方程，包括气象学、弹性、微分和复杂几何。所提出的问题涉及解的规律性（平滑性）。这些问题共同面临的关键挑战是椭圆性（正则性的代理）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，还可以帮助我们更好地预测天气模式，并协助设计汽车。研究结果将在同行评议的期刊上发表，并通过撰写说明性笔记进行交流，PI打算将这些笔记公开发布，并将对研究人员和学生都有用。第一个项目解决了真正的蒙日-安培方程的规则问题，受到气象学和微分几何应用的启发。PI将研究半地转系统的奇点形成，该系统模拟大尺度大气流动。半平面上不规则定解的某些例子可能是边界点爆破的有用模型。第二个项目涉及复杂几何中出现的复杂蒙日-安培方程的局部正则性。难点在于解的非凸性，以及在加入某些二次多项式时的不变性。PI已经开始研究一个模型方程来解决这些困难。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如何构造低维的奇异解。一个障碍是在三维中，具有粗糙系数的线性方程没有非平凡的单齐次解；PI将考虑具有“螺旋”单齐次对称的解。第四个项目是关于凸被积的变分积分的最小化。一个猜想是，标量值最小化与被积函数的勒让德变换具有相同的规律性。PI提出当被积函数的退化集满足一定的几何条件时证实这一点，当这些条件不满足时构造一个反例。最后一个项目旨在回答抛物系统的经典正则性和稳定性问题，特别是在低维。PI最近构造了平面上线性抛物系统奇异解的例子，回答了一个长期存在的问题。这可能为解决相关问题开辟了道路，例如，确定线性情况下防止奇点的系数条件，以及确定多孔介质类型的非线性结构的正则性与奇点。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。

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.