Degenerate Elliptic Problems in Analysis and Geometry

本课题主要研究在不同数学领域中出现的某些退化椭圆型偏微分方程解的性质。所研究的问题包括四阶Monge-Ampere型方程，非局部自由边界和相变问题，以及凹型退化的完全非线性方程。一个反复出现的问题是解或其水平集的光滑性及其在奇点附近的行为。例如，该项目将研究Lipschitz薄自由边界的光滑性及其奇异集的维度，具有退化边界条件的Monge-Ampere方程解的边界行为，以及非局部相变中水平集的刚性。同时，它们促进了不同数学分支的发展，如微分几何、拓扑学、概率和代数几何。特别是，Monge-Ampere方程自然而然地出现在最优交通问题的数学公式中，其中许多问题涉及我们的日常生活：城市交通网络规划、互联网交通优化、人体静脉系统的血管分支和气象流体动力学。这样的方程也经常出现在数学的其他领域，比如黎曼几何或保形几何。非局部自由边界和相变问题出现在多个科学领域，包括流体力学、等离子体物理和半导体理论。薄的自由边界与我们对火焰传播的理解有关，也与不连续表面的传播有关。这个项目的最终目的是获得对有趣的“退化”方程的正确看法，以便能够引入创新的工具和方法来解决它们。

This project focuses on qualitative properties of solutions to certain degenerate elliptic partial differential equations arising in different areas of mathematics. Some of the problems under investigation include fourth-order Monge-Ampere-type equations, nonlocal free boundary and phase transition problems, and degenerate fully nonlinear equations of concave type. One recurrent question is the smoothness of the solutions or their level sets and their behavior near singular points. For example, the project will investigate the smoothness of Lipschitz thin free boundaries and the dimensions of their singular sets, the boundary behavior of solutions of Monge-Ampere equations with degenerate boundary conditions, and the rigidity of level sets in nonlocal phase transitions.Partial differential equations are used to describe a large variety of physical phenomena. At the same time, they contribute to the development of different branches of mathematics such as differential geometry, topology, probability, and algebraic geometry. In particular, Monge-Ampere equations occur naturally in the mathematical formulation of optimal transportation problems, many of them within the realm of our daily lives: traffic network planning in cities, internet traffic optimization, blood vessel branching in the human venous system, and meteorological fluid dynamics. Such equations also frequently turn up in other areas of mathematics, like Riemannian or conformal geometry. Nonlocal free boundary and phase transition problems appear in several areas of science, including fluid mechanics, plasma physics, and semiconductor theory. Thin free boundaries are relevant to our understanding of flame propagation and also in the propagation of surfaces of discontinuities. The final purpose of this project is to gain the correct perspective on interesting "degenerate" equations in order to be able to introduce innovative tools and methodologies to tackle them.