Cracking of skin at the angles of the mouth, tho not eczema
, is very difficult to treat because it 'recracks' each time the mouth opens.
Vicious cycle! keep it elastic & supple with absolutely continuous and constant lip balm, vaseline, etc. Until it heals. If he has eczema & it has 'exploded' recently, there may be a secondary bacterial infection, and a trip to the doctor may be worthwhile.