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arXiv · 1212.2337

The Hele-Shaw flow and moduli of holomorphic discs

Julius Ross·David Witt Nystrom
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Abstract

We present a new connection between the Hele-Shaw flow, also known as two-dimensional (2D) Laplacian growth, and the theory of holomorphic discs with boundary contained in a totally real submanifold. Using this we prove short time existence and uniqueness of the Hele-Shaw flow with varying permeability both when starting from a single point and also starting from a smooth Jordan domain. Applying the same ideas we prove that the moduli space of smooth quadrature domains is a smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential problem in the plane.

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Julius Ross, David Witt Nystrom. 2014-01-30. The Hele-Shaw flow and moduli of holomorphic discs. https://doi.org/10.1112/s0010437x15007526

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