Solving homogeneous differential equation with boundary condition using Green functions

Question

I have an homogeneous equation $Lf=0$ where $L$ is an operator and $f$ a function, with boundary conditions $f(\partial\Omega)=g(\partial\Omega)$ where $g$ is a known function.

In the case when the Green function for the operator $L$ is known, how is it possible to use it to find $f$ ?

Answer 1

Let $h$ be any function in the domain of $L$ satisfying $h\rvert_{\partial\Omega} = g$, then define $u = f - h$. By linearity of $L$, $u$ now satisfies $Lu = L(f-h) = -Lh$ and $u\rvert_{\partial\Omega} = 0$, so we can apply the Green's function to obtain $u$ then add $h$ back to obtain $f$.