Hele-Shaw flow on weakly hyperbolic surfaces

Details Hele-Shaw flow on weakly hyperbolic surfaces Hele-Shaw flow on weakly hyperbolic surfaces

2004-06-18

arxiv.org/abs/math/0406368v1

Indiana Univ. Math. J. 54 (2005), 1161--1180

Mathematics

Analysis of PDEs

16 pages

Scientific paper

We consider the Hele-Shaw flow that arises from injection of two-dimensional fluid into a point of a curved surface. The resulting fluid domains have and are more or less determined implicitly by a mean value property for harmonic functions. We improve on the results of Hedenmalm and Shimorin \cite{HS} and obtain essentially the same conclusions while imposing a weaker curvature condition on the surface. Incidentally, the curvature condition is the same as the one that appears in a recent paper of Hedenmalm and Perdomo, where the problem of finding smooth area minimizing surfaces for a given curvature form under a natural normalizing condition was considered. Probably there are deep reasons behind this coincidence.

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