Qualitative Properties of Solutions to Nonlinear Elliptic Partial Differential Equations

该项目侧重于经典分析领域的中心问题，如变分法和自由边界问题。 所调查的问题有一些共同的特点，并在不同的领域，如材料科学，流体力学，几何，成本优化等根源，他们都是相关的纯数学和应用数学。提出的问题需要在已知的方法和技术上取得进展，解决这些问题将有利于数学家，也可能有利于更大的科学界。 研究结果将分发给感兴趣的受众，以推动这一理论的发展。 该项目的一个中心部分涉及的稳定性奇异最小化映射的光滑，凸的能量，出现在非线性弹性。PI将研究相关抛物流中的奇异性形成以及一类具有特殊结构的椭圆型方程组解的正则性。 该项目的另一部分是关于非线性椭圆方程的问题。PI将研究一些退化的Monge-Ampere方程和最优运输问题的边界行为，并为一些有趣的模型方程开发Pogorelov型估计。 在自由边界方面，PI将研究不同运营商的两相自由边界问题以及各种障碍物类型的问题，还将分析边界层相变的刚性。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影响审查标准进行评估，被认为值得支持。

This project focuses on central problems in classical fields of analysis such as the calculus of variations and free boundary problems. The problems under investigation share some common features and have roots in different areas like material sciences, fluid mechanics, geometry, cost optimization etc. They are relevant to both pure and applied mathematics. The proposed problems require progress on the known methods and techniques and addressing them would be beneficial for mathematicians and possibly for the larger scientific community as well. The outcome will be disseminated to the interested audience to invigorate the advancement of the theory. A central part of the project deals with the stability of singular minimizing maps of smooth, convex energies that appear in nonlinear elasticity. The PI will investigate the singularity formation in the associated parabolic flow and the regularity of solutions to a class of elliptic systems with special structure. Another part of the project is concerned with problems in nonlinear elliptic equations. The PI shall study the boundary behavior of some degenerate Monge-Ampere equations and optimal transportation problems, and to develop Pogorelov type estimates for some interesting model equation. In free boundaries, the PI will study the two-phase free boundary problem for different operators together with various obstacle-type problems, and will also analyze the rigidity of boundary layer phase transitions.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.