Greens function with non-zero boundary condition

Question

When solving a differential equation using a Greens function, is it possible to solve a problem with non-zero boundarys directly using a Greens function? For example, when solving a problem with non-zero boundarys I've broken the problem into multiple pieces, using the Greens function to solve the inhomogeneous part and then solving the homogeneous part using some other method, taking advantage of linearity. But is it possible to solve without breaking the problem up?

Answer 1

It's possible to recombine the pieces into a single formula for solution of the equation $\Delta u=f$ in $\Omega$, $u=g$ on $\partial\Omega$. Namely, $$ u(x) = \int_\Omega G(x,y)f(y)\,dy + \int_{\partial\Omega} \frac{\partial G(x,y)}{\partial n}g(y)\,dy $$ (With multiplicative constants subject to the normalization of $G$.) See, for example, Russell L. Herman's lecture notes on PDE.

The formula still contains two terms, reflecting the fact that interior sources and boundary sources are different in nature and affect the solution in different ways.