Collaborative Research: Capillary Interfaces

数学科学：合作研究：毛细管界面结论：本研究解决了毛细管理论数学方程解的行为。利用从物理问题的内在结构推导出的程序，我们将研究这些方程的非线性通用性。这些程序被证明以一种自然的方式与新开发的几何测量理论方法相互作用，这些方法最初是为其他情况下的最小曲面的特殊情况而开发的。在许多有趣的情况下，通过这种方式发现了解决方案的惊人和不寻常的行为。我们的研究将继续朝着已经导致发现对边界数据的不连续依赖、对称破缺、物理条件下的存在性失效、解存在条件下的唯一性失效以及比较关系的不连续反转的方向进行。毛细界面在许多实际情况下起着重要的作用，特别是在重力减小或物理长度尺度非常小的配置中。毛细现象是描述物质在表面界面接触而不混合的情况的基本概念。对毛细方程解的数学性质的精确理解将为解决诸如如何设计轨道航天器上的燃料箱以使燃料易于获得，预测石油或水驻留在地下岩石孔隙空间中的位置，或确定微电子元件表面浸在涂层浴中时部分被覆盖的方式等问题提供有价值的见解。在这些和其他应用中，正在考虑的工作与预测液体的意外行为特别相关。

NSF Award Abstract - DMS-0103937Mathematical Sciences: Collaborative Research: Capillary InterfacesDMS-0103937ConcusThis research addresses behavior of solutions to the mathematical equations of capillarity theory. The equations will be studied in their full nonlinear generality, using procedures deriving from the intrinsic structure of the physical problems. These procedures have turned out to interact in a natural way with newly developed methods of geometric measure theory, which were developed initially for the special case of minimal surfaces in other contexts. In a number of cases of interest, striking and unusual behavior of solutions was uncovered in this way. Our investigations will continue in directions that have already led to discoveries of discontinuous dependence on boundary data, symmetry breaking, failure of existence under physical conditions, failure of uniqueness under conditions for which solutions exist, and discontinuous reversal of comparison relations.Capillary interfaces play an important role in many practical situations, notably in reduced gravity or in configurations for which physical length scales are very small. Capillarity is an essential concept for describing situations in which substances are in contact at surface interfaces and do not mix. A precise understanding of the mathematical properties of solutions of the capillarity equations will provide valuable insight for resolving such matters as how to design fuel tanks on an orbiting spacecraft so that the fuel will be accessibly located, predicting where oil or water resides in underground pore spaces of rocks, or determining the manner in which the faces of microelectronic components will become partially covered when they are dipped in a coating bath. The work under consideration is of particular relevance for predicting unexpected behavior of liquids in these and in other applications.