I Have seen this in my brother's laptop, some forums say it is a display panel issue, others say it is a video-ram issue, try hooking up annother display, if it does the same, your computers graphics card is bad, also, which apple display, is it a Studio Display (3 Legged Picture frame like thing with powermac G4's and early G5's) or is it a Cinima Display (Newer Aluminum Displays Matching The PowerMac G5 and Mac Pro) or is it a LED Cinima Display( newer LED Backlight with a Mini-Displayport connector for MacBooks and Mac Mini's)?

Try with a different DVI cable and different input computers to see if it is the screen. If it works with a different cable/computer then the display is fine. If the problem remains the display is broke and I think when that happens you need a new one.

Someone correct me if there is a way of repairing a LCD

the dvi cable is built into your monitor, so its hard to replace that... could be a bad pin on it or a bad inverter assembly inside. theres not much you can do to fix it without some specialized apple or lcd repair shop that deals in re cabling the screens

Have you tried attaching your display to another computer to see if the trouble still occurs?

If it doesn't you have a display driver/hardware issue that you need to address.

If it does, you are in some trouble. With a built-in cable, all you might be able to do yourself is open the monitor up and see if the connection is good on the monitor side, and reattach as necessary.