I have been studying Green's functions for Laplace/Poisson's equation and have been having some trouble on a few things. In Strauss's book he claims the solution to the Dirichlet problem is:
$$u(\bf x_0)= \iint_{bdy D} u(\bf{x})\frac{\partial G(\bf{x},\bf{x_0})}{\partial n} \,dS\label{1}\tag{1}$$
But in other texts I have seen it defined as
$$u(x) = \int_\Omega G(x,\xi)\Delta u \,dy + \int_{\partial \Omega} u(y) \frac{\partial G(x,\xi)}{\partial n} \,dS\label{2}\tag{2}$$
I must be missing something here, but I cannot pinpoint what exactly. Which definition is the correct one? Are they equivalent? I suspect the first equation is assuming a homogenous PDE.
Green’s Function for Dirichlet problems
greens functionharmonic functionslaplacianpartial differential equationspoisson's equation
Best Answer
Formula \eqref{2} you give includes the solution to the Dirichlet problem for both equations, i.e. for Laplace's and Poisson's equations.
Precisely, the Dirichlet problem for Poisson's equation reads as $$ \begin{cases} \Delta u(x)=f(x) & x\in \Omega\\ u|_{\partial\Omega}=U(x) & x\in \partial\Omega \end{cases}, $$ while the same problem for Laplace’s equation looks identical except for the fact that in this case $f\equiv0$.
If $f(x)\not\equiv 0$ then $\Delta u(x)\not\equiv 0$ and you have the proper Poisson's equation, thus $$ \int_\Omega G(x,\xi)\Delta u \,dy\neq 0, $$ and formula \eqref{2} holds in full. On the other hand, if $f(x)\equiv 0$ then $\Delta u(x)\equiv 0$ thus you have the proper Laplace's equation and $$ \int_\Omega G(x,\xi)\Delta u \,dy= 0, $$ thus formula \eqref{2} reduces to formula \eqref{1}.