I currently have a similar issue with my 27" display. Many times it will not come on at all, but all USB devices, speakers and the charge cable from the display function. When I play around with unplugging it and switching between outlets I sometimes get it to come on and work again, or just get the second flash before it goes dark.
I've read on the Apple support site that the Genius bar will sometimes replace the power supply or the logic board, but they're replacing both in a lot of cases.
Ifi decide to do the repair myself. I'll post my findings here.