If your mouth became paralyzed following Smart lipo of the neck and chin, most likely the Marginal Mandibular Branch of the Facial Nerve was injured.
This nerve runs close to the skin in the Jowl area and if the doctor attempted to do liposuction of the jowl area the nerve was likely injured.
Most Neuropraxic or temporary peripheral nerve injuries recover in the first six weeks.
However nerves can continue to heal for up to two years.
I would see a neurologist for an evaluation. I recommend DC (Direct Current) muscle Stimulation of the paralyzed muscle to help recovery and in my experience this can help prevent atrophy of the muscle during re-innervation and strengthen the muscle.
Your doctor can obtain a DC stimulator form EMS Labs in Goleta California-the doctor can contact me for the treatment protocol.
The paralyzed muscle is the Depressor Anguli Oris which normally holds the corner of the mouth in the proper position.
75% of partial nerve injuries in the face recover.