Air carrier Captain reported abnormal noise/vibration was reported to him during cruise. Flight continued to destination with a discovery of a missing main gear door.

Narrative

During cruise at FL350 about 1.5 hrs into the flight; a Flight Attendant called to advise me there was a constant loud noise and vibration felt through the floor at row 13; predominantly on aircraft left. We checked all indications in the flight deck and there was nothing unusual noted. I requested the Flight Attendant record the sound with her phone. She airdropped me the recording; but it wasn't clear enough to decipher what the problem was. We coordinated for me to leave the flight deck and investigate the problem myself. I walked through the cabin; stopping at row 13 and could definitely hear that the air noise was louder than usual and I could feel an abnormal vibration in through the floor. After I returned to the flight deck; I contacted Maintenance Control through Dispatch via SATCOM. We discussed the issue; concluded there were no other abnormal indications and suspected it might be a loose fairing panel or seal. We concurred that continuing to ZZZZ was safe. I made a logbook writeup; discussed the plan with the FO (First Officer) and the FA's (flight attendants) and we all felt comfortable continuing. On the ground at ZZZZ we contacted Maintenance to come inspect the plane. Maintenance informed me and showed me in-person that we were missing most of our left gear door and it appeared to have broken off at some point. The aircraft was taken out of service for further inspection.