Compact cameras. They have one thing in common, i.e. the battery compartment door/cover is made flimsy (non-durable, easily damaged).
This part of the camera is important. The door act not only to cover the battery compartment for aesthetic reason, but also to close the circuit of the battery. When the circuit is not closed, you won't be able to switch the camera on. Try it, open the door ajar and press the "on" button. Nothing will happen.
So, it is important that the door is intact. If you accidentally break the door, you should keep that piece with you. Use a tape to stick the door to the camera for it to work. If you lost the door/cover; you must find a compatible cover for the camera to work. Using a cardboard and sticky tape will not revive your camera.
I have no idea for the reason to ensure that the door must be closed to operate the camera. Was it to prevent shocks from the exposed battery? Whatever it was, it seemed important enough for all camera manufacturers to want to implement the same design concept.
My Nikon coolpix S6000 has a clumsy design at the cover. My camera automatically switches off whenever I want to use it (even though the battery indicator showed full bar). The problem seemed to be due to the cover constantly losing contact with the battery and hence breaking the circuit. If anyone has an "automatic switched off camera" problem, it's likely caused by the cover/door to the battery.
Nikon conveniently stated that the cover/door to the camera is not under any warranty. That was their point when I first bought the said camera (S6000 series). Try asking the sales representative or Nikon Service Center. Most probably, the policy is still retained, i.e. no warranty for battery compartment cover/door.
Another problem I faced with this camera is that, now I'm having dead pixels at two corners of my LCD screen. I just hope they don't spread. Worst come to worst, I'll have to shoot without any cue from the LCD, just like old compact camera (but without "view finder"). Haha.