http://forums.androidcentral.com/sprint-galaxy-s-iii/215595-phone-turning-itself-off-randomly.html
Do you have a case? If so, take it off and make sure it's not pressing on the power button.
Has the phone always done this, or did it just start?
If the phone is brand new and it's doing this, take it back.
If it used to work and this just started, what changed recently? Did you install any apps, or try to root, or change any configuration options?
If the answer to the above is nothing has changed, then try pulling the battery, leaving it out for a few minutes, and see what happens when you restart.
Has the phone always done this, or did it just start?
If the phone is brand new and it's doing this, take it back.
If it used to work and this just started, what changed recently? Did you install any apps, or try to root, or change any configuration options?
If the answer to the above is nothing has changed, then try pulling the battery, leaving it out for a few minutes, and see what happens when you restart.
Finally, if nothing else, try a factory reset. With the phone off, boot into recovery by pressing the power button, the home key, and volume up all at the same time. As soon as you see the Samsung logo, release all three buttons.(if you release the buttons a little too early or too late, the phone will try to boot normally. Just power it down and try again.) Assuming you're not rooted you use the volume control keys to choose the factory reset menu option, then press the home key to select it.
If a factory reset doesn't fix it, then I'm afraid it's a hardware problem. Take it back to where you bought it, or go to Samsung's web site for warranty support.