On the Homogenization of some Free Boundary Problems

关于某些自由边界问题的齐次化建议研究摘要首席研究员Antoine Mellet建议分析某些自由边界问题的一些性质。本文提出的就是研究这些问题的同质化。齐次化的目的是描述介质局部性质的微小变化对解的整体行为的影响。在过去的三十年里，齐次化理论有了广泛的发展，但处理自由边界问题的工作很少。自由边界问题是在对涉及界面的生物和物理现象进行建模时自然产生的。这里提出的工作将使用均匀化来研究界面的形状。该提案的一部分涉及倾斜平面上的一滴液体的运动问题--例如窗户上的水。众所周知，这是一个困难的数学问题，在许多领域都有应用，包括油漆的涂抹和植物叶片上农药喷雾的保留。国际和平研究所研究了火焰传播分析中的这类问题，他的工作导致了对那里观察到的各种问题的理解，包括火焰锋面的传播。

On the Homogenization of some Free Boundary ProblemsAbstract of proposed researchAntoine MelletThe principal investigator proposes to analyze some properties of certain free boundary problems. This proposal is to study the homogenization of these problems. The aim of homogenization is to describe the influence of small variations in the local properties of the medium on the global behavior of the solutions. There has been extensive development in the theory of homogenization over the last thirty years but very little of the work has treated free boundary problems.Free boundary problems arise naturally in the modeling of biological and physical phenomena involving an interface. The work proposed here will use homogenization to study the shapes of interfaces. Part of the proposal treats the problem of the motion of a drop of liquid on an inclined plane - such as water on a window. This is known to be a difficult mathematical problem and has applications in many areas including the spreading of paint and the retention of pesticide sprays on plant leaves. The PI has studied problems of this type in the analysis of flame propagation and his work has led to an understanding of various issues observed there including the propagation of flame fronts.