For a more serious split you may be able to glue and clamp the door.
Split wood door hinge.
Use wood putty to fill the crack and paint or varnish the door as needed.
Serious splits in door frames.
Stainless steel and brass lift off hinge materials are best for high.
It may have holes for screw mounting or spots for welding depending on the style of the hinge and the door material it is mounting to.
How to fix a door that cracked where the hinges go.
Wood glue of any type aliphatic glues like titebond seem to be popular in the.
Cracks on doors are everyday problems.
This video shows how to glue a cracked door stile back in place.
Lift off hinges are surface or mortise mounted devices that allow users to quickly detach the mating door while keeping hardware in place.
In particular the side where the hinges fasten onto the door tends to crack or split in certain situations.
At this point you may need to decide if you will be happy with the patched up door frame when you are done.