So generally, I want to make an app which has video chat functionality for iPhone. But after many searches, I am still not able to find any successful results. Is there any publ
HTTP live streaming is primarily an approach for adaptive streaming from server-to-client. For client-to-server rather go for traditional streaming. There exists an open library for streaming, see this question.
Whilst it is possible to facetime to do two-way chat, it is not certain that you will be able to using public iOS APIs. That said, I have implemented one-way live streaming for iPhone and the difficult part was not the core streaming itself, but encoding of the payload. You will be able to do H264 in hardware and AAC / iLBC in software.
How you want to feed this to the FFMPEG depends on your transport, possibly changing from 'file' H264 frames to 'streaming' H264. Check out the H264 frame types if you implement frame dropping; reconfiguring the H264 encoder on-the-fly is not possible to my knowledge, but restarting with fresh parameters typically does not take more than a second or so.
Did you attempt to play back a live resource while capturing? That is a good starting point. If you come across an open API for H264 encoding, please post it here ;-)