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• W3102764058 abstract "The Schwarz function has played an elegant role in understanding and in generating new examples of exact solutions to the Laplacian growth (or Hele- Shaw) problem in the plane. The guiding principle in this connection is the fact that non-physical singularities in the oil domain of the Schwarz function are stationary, and the physical singularities obey simple dynamics. We give an elementary proof that the same holds in any number of dimensions for the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17] (1989). A generalization is also given for the so-called elliptic growth problem by defining a generalized Schwarz potential. New exact solutions are constructed, and we solve inverse problems of describing the driving singularities of a given flow. We demonstrate, by example, how mathbb{C}^n - techniques can be used to locate the singularity set of the Schwarz potential. One of our methods is to prolong available local extension theorems by constructing globalizing families. We make three conjectures in potential theory relating to our investigation." @default.
• W3102764058 created "2020-11-23" @default.
• W3102764058 creator A5060518867 @default.
• W3102764058 date "2011-02-25" @default.
• W3102764058 modified "2023-10-18" @default.
• W3102764058 title "Laplacian growth, elliptic growth, and singularities of the Schwarz potential" @default.
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• W3102764058 doi "https://doi.org/10.1088/1751-8113/44/13/135202" @default.
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