  We solve the Plateau problem for marginally outer trapped surfaces in general
Cauchy data sets. We employ the Perron method and tools from geometric measure
theory to force and control a blow-up of Jang's equation. Substantial new
geometric insights regarding the lower order properties of marginally outer
trapped surfaces are gained in the process. The techniques developed in this
paper are flexible and can be adapted to other non-variational existence
problems.
