Harmonic functions, equivalence of boundary conditions with phenomena outside domain.

Question

I'm study a article and it contains:

"It is well known that harmonic functions are associated with boundary conditions, which equivalently means that they depend upon the phenomena occurring outside the given domain."

Can someone give a more detailed explanation?

Answer 1

A harmonic function satisfies the Laplace equation, and since you consider harmonic functions on a domain, it is a solution to the Dirichelt problem with boundary data a continuous function f. But the boundary is "outside" of your domain, because a domain is an open and connected set.