This is the third of a series of posts dealing with the regularity theory of elliptic equations. My motivation in writing these is outlined in the first post. The previous post is here.

Let us recall Green’s identity, if are any functions smooth in and is a bounded domain with smooth boundary we have

this identity can be obtained with a couple of integration by parts involving the vector fields and .

Lets rewrite the identity as

thus, at least formally, if somehow we could find for every a function such that

then Green’s identity applied to both and in would give us an *integral representation formula* for harmonic functions