PuzzleManiak: It's demoniak

You have a grid of squares, some of which contain digits, and the rest of which are empty. Your job is to fill in digits in the empty squares, in such a way that each connected region of squares all containing the same digit has an area equal to that digit.

(‘Connected region’, for the purposes of this game, does not count diagonally separated squares as adjacent.)

For instance, it follows that no square can contain a zero, and that two adjacent squares can not both contain a one. No region has an area greater than 9 (because then its area would not be a single digit).