A property of the single layer potential & compatibility conditions in 2D?

So I'm reading a journal and I have a surface of a domain these areas are extended with a variable, and another surface of a domain which contains the domain entirely, also extended with a time variable. I also have a dunemerable, dense set of points on represents the fundamental solution of the heat equation, an arbitrary continuous function and the integral

Then it introduces the potential of a simple layer

from the above equation we know that and because is continuous and is equal to zero for a denumerable, everywhere dense set of points for The journal then states that in the whole external region outside of (Basically in the infinite space ) (with no explaination why) I was just wondering why this was the case that outside or do I need to give more details.

Also does anyone know the compatability conditions for the heat equation in two dimensions or somewhere where I can find them on the internet, (its so I can state that these allow there to be a unique solution to the heat equation).

Any help would be greatly appreciated. Thanks!

(Thanks mush for the tips)