I have an Uno R3 with two shields (Gameduino Dazzler and Adafruit Ultimate GPS Logging) connected through HDMI to a Lilliput 7" HD Camera Monitor.
As long as I power the UNO R3 from the USB port, the setup works fine and I get the data on the Lilliput. It work regardless if I power the UNO Stack from a USB wall-bug or my computer via the USB cable.
But if I supply 12v DC (or even 9v DC) power to the barrel power connector on the UNO R3, the display shows proper data for about 10 seconds and then is blinks and shows a blue screen and then a black screen. It eventually indicates that it lost the HDMI signal.
I have to leave all of the power off for a minute before I can get it running again with the the USB power to the UNO R3. It is as if something gets locked up in the power regulator on the UNO.
I am presently running the Uno R3 via the USB port from a 1A, 5v DC wall-bug. When I power the UNO R3 through the barrel I am using 12v DC with a capability of handling 3A. So I don't think the supply is sagging with any sort of excessive power load.