After picking up my sons Note 10.1 and noticing that it would apparently come on but had nothing on the screen I began googling. Not much info out there. Appears to be a common problem. His tablet seemed to be working fine but nothing displayed on the screen. Read several threads on here with most ending in sending the units back under warranty. Although I did purchase 3 year warranties on both units I bought...I'm just not one to do the right thing. I powered the unit down and began taking the back off like the poster did in this thread. http://forum.xda-developers.com/show....php?t=2617614. Once I got the back off I tried disconnecting the battery and reconnecting it. No dice. The poster then said he disconnected the two ribbons and reconnected them and that worked for him...still no worky for me. One of the ribbons is labeled "LCD". I followed it to the other end. It is covered by a silver "hold down" sticker. I unstuck the sticker and could then clearly see that the ribbon was clearly cocked very slightly in it's socket. If you don't know, these ribbon sockets come open by prying the black piece up and away from the ribbon. Be very careful doing this. I did this on this end of the LCD ribbon, straightened the ribbon and clipped the socket back closed. BAM. Black Screen Gone! Works like a charm.
If you are comfortable digging around in very small electronics give it a try. If not, send it in.
Hope this helps.