Look up the galaxy S3 teardown guide hear and stop when you get to unplugging the display. Once unplugged get a small brush, sopropyl alcohol and gently scrub the connector on board and connector on flex. If its just corroion or oxidation causing current not to flow that should fix it. If after doing this it still does not work try getting a replacement screen to see if that's the issue or take it to a repair shop to have them test. I hear Radioshack is doing repairs now, you could try taking it to them to troubleshoot. Hopefully it's just the screen and not board level damage because if not you're looking at a replacement board or entire phone.