Collaborative Research: Capillary Interfaces

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NSF Award Abstract - DMS-0103954Mathematical Sciences: Collaborative Research: Capillary InterfacesDMS-0103954FinnThis 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.