Mathematical Sciences: Classical and Applied Analysis

在这个项目中，将讨论毛细作用的数学理论和相关几何问题中的几个问题。许多工作的动机是在太空中遇到的具体问题，那里没有重力来掩盖表面张力的影响，而地球表面的许多现象都存在这种影响。一些更直接的应用将是精确测量接触角的新方法(以及通过推断接触角是否可以被视为杨、拉普拉斯和高斯所设想的材料的内在属性的问题)。其他重要的问题集中在在无重力的情况下，一般截面的圆柱管中何时可以存在毛细管表面的问题。已知的情况是，管子不能容纳流体，但横截面上容易获得的几何条件将预测这一点，这是未知的。将在描述垂直接触角和切向接触角的奇异情况下的边界行为以及建立支撑表面上的液体体积的一般平衡条件方面进行工作。原则上，液滴可以实现非一致平衡表面构型的连续体。将努力寻找确保这些表面的唯一性和稳定性的一般条件。在研究给定平均曲率的曲面的几何性质时，毛细问题可以作为障碍。近年来，这种方法得到了相当大的成功。该方法的全部潜力还远未实现；将继续为此开展工作。//

Several questions in the mathematical theory of capillarity and related geometrical problems will be addressed in this project. Much of the work is motivated by concrete problems encountered in space where gravity is not present to mask the effects of surface tension as occurs for many phenomena on the earth's surface. Some of the more immediate applications will be to new procedures for precise measurement of contact angle (and by inference to the question of whether contact angle can be regarded as an intrinsic property of materials in the sense envisaged by Young, Laplace and Gauss). Other important issues center on the question of when capillary surfaces can exist in cylindrical tubes of general section in the absence of gravity. There are known cases where tubes will not hold fluids, but readily accessible geometrical conditions on the cross sections which will predict this are not known. Work will be done in describing boundary behavior in the singular cases of perpendicular and tangential contact angles and on the formulation of general equilibrium conditions for liquid volumes resting on support surfaces. Liquid drops can, in principle, achieve a continuum of non-congruent equilibrium surface configurations. Efforts will be made to find general conditions ensuring uniqueness and stability for such surfaces. Capillary problems can be used as barriers in the study of geometrical properties of surfaces of prescribed mean curvature. This approach has been exploited with considerable success in recent years. The full potential of the method is far from realized; work will continue toward this end. //