I come from an engineering background and would like to learn about Laplace-Dirichlet operator, which I came across in a couple of posts on math.stackexchange.com.
Could someone kindly help me understand/interpret what we mean by the following statement, which was posted here:
Let $A = -\Delta$ be the Laplace-Dirichlet operator, $D(A) = H_0^1(\Omega) \cap H^2(\Omega)$ where $\Omega \subset \mathbb{R}^d$ is a bounded domain. It is known that there exists a Green's function for $A$, i.e. $G \colon L^2(\Omega) \to D(A)$ such that $Av = f \Longleftrightarrow v = Gf$.
I would be thankful to have some references to read and learn more and about this very interesting topic.
In fact, I have a bounded 3D triangulated surface whose vertices have known value of a function $g$. The region inside the surface is a subset of a larger domain whose field is harmonic. The surface is filled with points and I wanted to see if with the Laplace-Dirichlet operator I can relate the boundary points to the points inside the surface, and if so, how this can be found. Thank you in advance.
Update: I know how this can be done with a finite element method and I already implemented it. However, I would like to learn more about the Green's function approach.