Domain or domain ( doe main ) means kingdom or realm, literally or figuratively--
The prince wanted everyone in his domain to be ready for military service.
In the domain of the natural sciences, he was famous and respected.
Domain, legally, means control over and use of, as of land--
The land between the tracks and the river is the domain of our client.
Demesne or demesne ( deh main ) means one's own land, especially the land around one's home--
The dog went out every morning to patrol his demesne.
He dreamed of one day living in his own demesne, surrounded by his own land.
Demesne, historically, meant the land around a manor house, worked for the family that owned the estate--
The crops raised on the demesne are not enough to support the manor house.
Now that you know that, you can say--
"Safe in his own suburban demesne, he became famous in the domain of robotics."