For unit open ball $D=\{{x}\in\mathbb{R}:\lvert{x}\rvert<1\}$, I need to find Green function $G$ with Dirichlet Problem $$ \Delta u =0\,\, \text{ in }D,\qquad u=g\,\, \text{ on }\partial D $$ Let $\Phi({x})=-\frac{1}{2\pi}\log\lvert{x}\rvert$ be fundamental solution of Laplacian at ${x}\neq{0}$ and $\phi^{x}$ be the solution of following Dirichlet problem $$ \Delta \phi^{x} =0\,\, \text{ in }D,\qquad u(y)=\Phi(y-x)\,\, \text{ on }\partial D $$ Note that $G(x,y)=\Phi(y-x) -\phi^x (y)$.
My friend ask me this question, but I cannot understand the conditions because of abstruse terminology and my short study at partial differential equation. Please, give me solution and make me understanding this problem. Thank you for reading my question with my short English.