Center Controller reported an aircraft reported their door opened in flight so they diverted to the nearest airport and descended below the published terrain minimum. The Controller lost communication with the aircraft due to an ATC frequency outage.

Narrative

At around XA:15z Aircraft X told me that their passenger door flew open and they needed to find an airport to land at immediately. I notified them of the 2 closest airports in their proximity. I notified my supervisor of the [priority]. Aircraft X chose to land at ZZZ. I cleared the aircraft to ZZZ. I read them the weather and told them the field elevation of the airport. I also notified them of the minimum altitude for IFR flight in their location. The aircraft said that they were in visual conditions and continued descending below the TAV (Terrain Alert Volume) while they were still IFR. I kept calling the distance they were from ZZZ and after they descended past about 4000ft MSL I was unable to contact them. I had other aircraft relay for the aircraft on my current frequency and on the Unicom frequency for the airport. They were unsuccessful in reaching the aircraft. My supervisor called the airport manager and we found out that they landed successfully.The sector I was working X has had a major frequency outage for about a week now leading up to this event. Our XXX.XX main and standby frequencies have been released for maintenance and we have been working off the BUEC (Backup Emergency Communications) frequencies since. because of this we have had great difficulty talking to aircraft in the southeast corner of our airspace below 7000ft. If our frequencies were working normally I would have remained in contact with this aircraft until they landed. The frequency outage literally put someone's life in danger because of our inability to communicate in [priority] situation.