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Lemniscates do not survive Laplace growth. (English) Zbl 1236.30006
The Polubarinova-Galin equation models the movement of the boundary between two fluids in the plane when a sink or source is present. (For background material see [B. Gustafsson and A. Vasil’ev, Conformal and potential analysis in Hele-Shaw cells. Advances in Mathematical Fluid Mechanics. Basel: Birkhäuser (2006; Zbl 1122.76002)].) In the present paper it is shown that the only polynomial lemniscates moving under this process of Laplacian growth (in fact, under a somewhat larger class of processes which are invariant under time reversal) are concentric circles.

MSC:
30C20 Conformal mappings of special domains
35R37 Moving boundary problems for PDEs
76D27 Other free boundary flows; Hele-Shaw flows
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