Power Source of outside outlets
One possibility to consider is that it is the outlets themselves that have failed and that there is still live mains wiring in the electrical box behind the outlet. One scheme would be to attempt to see if a non-contact electrical tester probe can detect the live wire in the box. If so you can use a process of elimination to see which circuit breaker removes the live connection from the electrical box. You may need to do this process separately for each of the outlets as they may very well be on separate circuits.

The outlets may have failed for any number of reasons. The first likely candidate is that they were connected to the wiring using the back stab holes on the outlet. These are a known failure point and in your case the temperature cycling that occurs over time may have contributed to the back stab failure. When you replace such outlets it is highly recommended to not use the back stab style of connection and instead connect the wires using the screw terminals on side.

Another failure possibility could be similar to something I personally experienced. In my case an exterior outlet quit working and there was no detectable live wire in the electrical box. Obviously the process of elimination as described above did not work to determine the circuit breaker. It was not until months later that I discovered that the outlet was wired into the load side of a GFCI type outlet located in an obscure part of my garage. That GFCI had tripped for some unknown reason and resetting it restored operation to my external outlet!!