Solving the Loewner PDE in complete hyperbolic starlike domains of   $\C^N$
  We prove that any Loewner PDE in a complete hyperbolic starlike domain of $\C^N$ (in particular in bounded convex domains) admits an essentially unique univalent solution with values in $\C^N$. 
Leandro Arosio, Filippo Bracci, Erlend Fornaess Wold
math.CV math.DS