Collaborative Research: Capillary Interfaces

NSF奖摘要-DMS-0103937数学科学：合作研究：毛细管界面DMS-0103937讨论这项研究解决了毛细管理论的数学方程的解的行为。我们将使用从物理问题的内在结构中得出的程序，研究这些方程的全部非线性概括性。事实证明，这些程序与新开发的几何测度论方法以一种自然的方式相互作用，这些方法最初是为其他环境中极小曲面的特殊情况而开发的。在一些有趣的案例中，通过这种方式发现了解决方案的惊人和不寻常的行为。我们的研究将继续沿着已经导致发现对边界数据的不连续依赖、对称性破坏、物理条件下的存在失败、存在解的条件下的唯一性失败以及比较关系的不连续颠倒的方向进行。毛细管界面在许多实际情况中发挥着重要的作用，特别是在重力降低或物理长度尺度非常小的构型中。毛细作用是描述物质在表面界面接触而不混合的情况的一个基本概念。对毛细管方程解的数学性质的精确了解，将为解决以下问题提供宝贵的见解：如何设计在轨航天器上的燃料箱，使燃料易于找到，预测石油或水在岩石地下孔隙空间中的位置，或确定微电子元件在浸泡在镀膜槽中时，其表面将以何种方式部分覆盖。正在考虑的工作对于预测液体在这些和其他应用中的意外行为具有特别的相关性。

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.