On the Maximal Rank Problem for the Complex Homogeneous Monge-Amp\`ere   Equation
  We give examples of regular boundary data for the Dirichlet problem for the Complex Homogeneous Monge-Amp\`ere Equation over the unit disc, whose solution is completely degenerate on a non-empty open set and thus fails to have maximal rank. 
Julius Ross and David Witt Nystr\"om
math.CV