On a differential equation arising in a Hele Shaw flow moving boundary problem (Q801186)

The following problem arises in a moving boundary problem for Hele Shaw flows. Given f(\(\zeta)\), (analytic and) univalent for \(| \zeta | \leq 1\), find f(\(\zeta\),t), univalent in \(\zeta\) (\(| \zeta | \leq 1)\) and continuously differentiable in t \((-\epsilon <t<\epsilon),\) such that \[ Re[\dot f(\zeta,t)\overline{\zeta f'(\zeta,t)}]=1\quad (| \zeta | =1) \] and \(f(\zeta,0)=f(\zeta)\). We prove that, for \(\epsilon >0\) small, this problem has a unique solution in the form of rational functions (in \(\zeta)\) if f(\(\zeta)\) is rational. As an application we prove that, within the class of domains \(\Omega\) which are conformal images of the unit disc under a polynomial mapping function, \(\Omega\) is locally uniquely determined by its moments \(\int_{\Omega}z^ n dx dy,\) \(n=0,1,2,... \).