The Hele-Shaw flow and moduli of holomorphic discs

Julius Ross, David Witt Nyström
2015 Compositio Mathematica  
We present a new connection between the Hele-Shaw flow, also known as two-dimensional 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 when starting from a smooth Jordan domain. Applying the same ideas, we prove that the moduli space of smooth quadrature domains is a smooth manifold
more » ... smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential problem in the plane.
doi:10.1112/s0010437x15007526 fatcat:66oneawl4zhgjoievdy7skitcu