I was playing around with numerical solutions of the Laplace equation with mixed boundary conditions: \begin{alignat}{3} \Delta u(x) &= 0, &\quad &x \in \Omega, \\\\ u(x) &= g(x), &\quad& x \in \Gamma_D,\\\\ \partial_n u(x) &= 0, &\quad& x \in \Gamma_N. \end{alignat}
And I came to the conclusion that the solution can be written as $$u(x) =\int_{\Gamma_D} w(x,y) g(y) dy,$$ where $w(x,y) \geq 0$ and $\int_{\Omega_D} w(x,y) dy = 1$. From my understanding $w$ here is just $\partial_n G(x,y)$ where $G$ are the Green's functions.
Is there a reference that discusses the fact that $\partial_n G$ must be non-negative for the Laplace equation or that it must integrate to one on the boundary? I thought about it and my idea was that if this didn't hold, then the min-max principle would be violated since I could pick $g$ in that case such that the minimum or maximum is achieved inside of the domain instead on the boundary.
This also seems to be Laplacian specific as it doesn't work for the biharmonic equation involving $\Delta^2$.