Interfaces, Degenerate Partial Differential Equations, and Convexity

偏微分方程（PDE）是物理过程建模的基本数学对象。该项目旨在了解复合材料中气体扩散、液滴形状和电力传输的一些基本PDE模型的特性。许多这样的过程表现出一个界面，在这个界面上方程变得简并或奇异。在气体通过多孔介质扩散的模型中，界面是将有气体的区域与无气体的区域分开的一组。液滴的边界是界面的另一个例子。该项目将研究这些界面的定性和定量特性，包括凹凸度和平滑度。对于由两种具有不同导电性的材料组成的复合材料，界面是这些材料相遇的地方。首席研究员（PI）将研究其中一种材料具有非常薄的部分时电场的行为。这个项目可能对材料失效有影响，这是工程中的一个重要问题。学生和博士后学者将接受关于这些PDE模型的技术和理论的培训。该项目围绕四个主题展开，由界面、简并性和凹凸性这三个主题联系在一起。多孔介质方程是用于模拟气体扩散的非线性退化抛物方程。PI将研究解的凹性和凸性问题，并找到全局最优正则性估计。其次，PI将研究线性偏微分方程，其系数沿两个几乎接触的界面不连续，传输问题的模型和复合材料。在这种情况下，PI将研究在界面之间的薄区域获得最佳梯度估计的新方法。第三个课题是研究在固定区域内为抛物型但在边界处退化的线性方程。这些方程是多孔介质方程和高斯曲率流的线性化。PI将研究光滑解存在唯一性的最优条件。最后，将研究具有Dirichlet边界条件的椭圆型扭转问题解的凹性问题以及该方程的动态版本，即准静态液滴模型。此外，PI将与本科生开展暑期研究项目，探索这些方程的显式解。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。

Partial differential equations (PDE) are essential mathematical objects for modeling physical processes. This project aims to understand the properties of some fundamental PDE models for the diffusion of gas, the shape of liquid droplets, and electric transmission in composite materials. Many such processes exhibit an interface, where the equation becomes degenerate or singular. In the case of a model of gas diffusion through a porous medium, the interface is the set which separates the region where there is gas from the region where there is no gas. The boundary of a liquid droplet is another example of an interface. The project will investigate qualitative and quantitative properties of these interfaces, including convexity and smoothness. For a composite material, consisting of two materials with different conductivity properties, the interface is where these materials meet. The Principal Investigator (PI) will study the behavior of the electric field when one of the materials has a very thin part. This project has possible implications for material failure, an important question in Engineering. Students and postdoctoral scholars will be trained on the techniques and theory of these PDE models.This project centers on four topics, connected by the themes of interfaces, degeneracies, and convexity/concavity. The porous medium equation is a nonlinear degenerate parabolic equation used to model the diffusion of gas. The PI will investigate questions of concavity and convexity of solutions and finding global optimal regularity estimates. Secondly, the PI will study linear PDE whose coefficients are discontinuous along two almost touching interfaces, a model for transmission problems and composite materials. In this setting, the PI will investigate new approaches to obtaining optimal gradient estimates in the thin region between the interfaces. A third project is to study linear equations which are parabolic on the interior on a fixed domain but are degenerate at the boundary. These equations arise as linearizations of the porous medium equation and the Gauss curvature flow. The PI will investigate optimal conditions for existence and uniqueness of smooth solutions. Finally, questions of concavity of solutions to the elliptic torsion problem with Dirichlet boundary conditions and the dynamic version of this equation, known as the quasi-static droplet model, will be studied. In addition, the PI will carry out summer research projects with undergraduates, exploring explicit solutions to these equations.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.