Mathematical Sciences: Classical and Applied Analysis

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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. //