What I'm looking for is an explicit form of the Green's function $G$ for the Laplace operator $\Delta$ in a rectangular (or square) domain $D\subset\mathbb R^2$, e.g. $D=[0,1]\times[0,1]$. Namely, I'm looking for the solution $G:D\times D\rightarrow \mathbb R$ to the following PDE: $$ \begin{cases} \Delta_x G(x,z) = -\delta_z(x) & \text{for }x\in D \\ G(x,z) = 0 & \text{for }x\in\partial D \end{cases} $$ where $\Delta_x$ is the laplacian with respect to the first variable $x\in\mathbb R^2$ and $\delta_z(x)$ is the Dirac delta ($\delta_z(x)=1$ if $x=z$ and $0$ otherwise).

I need it in order to provide a (hopefully simple) example to a problem I am studying, which involves the Green's function. I don't know where to start and I am not very familiar with second order PDEs; any solution or hint is highly appreciated, thanks in advance!


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