Solution of Poisson equation for $f(x,y)=1$

Consider Poisson equation $$ \Delta u=f $$ To be more specific, $$ \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=f(x,y) $$ What would be the analytical(Exact) solution of the above equation for $f(x,y)=1$?

Assume that the domain is $\Omega=(-1,1)\times (-1,1)$ and the boundary conditions are of Dirichlet type, i.e, $u(x,y)=g(x,y)$ on $\partial \Omega$.

$g(x,y)$ is a general function as stated.

The Green's function for $\Omega = (-1,1)^2 \subset \mathbb R^2$ is $$G((x,y),(\zeta,\eta))= \frac 4 {\pi^2} \sum_{k,\ell=1}^\infty \frac{\sin \big ( \frac{k\pi x}2 +\frac{k\pi } 2 \big )\sin \big ( \frac{k\pi \zeta}2 +\frac{k\pi } 2 \big )\sin \big ( \frac{\ell\pi y}2 +\frac{\ell\pi } 2 \big )\sin \big ( \frac{\ell\pi \eta}2 +\frac{\ell\pi } 2 \big )}{k^2+\ell^2} . $$ Hence, $$u(x,y) = - \int_{\partial \Omega}g(\zeta,\eta) \frac{\partial G}{\partial \nu}((x,y),(\zeta,\eta)) \, d S(\zeta,\eta) - \int_\Omega G((x,y),(\zeta,\eta)) \, d \zeta d \eta$$ where $\nu$ is the outward pointing unit normal to $\partial \Omega$ and $$\frac{\partial G}{\partial \nu}((x,y),(\zeta,\eta)) := \nabla_{(\zeta,\eta)} G((x,y),(\zeta,\eta)) \cdot \nu(\zeta,\eta). $$ Via some large (but elementary) calculations you should be able to work out each of the terms involving $G$.