I need to solve $$\nabla^2u(x,y) = constant$$ with $0\leq x \leq a$, $0\leq y \leq b$ and $u(x,y) = 0$ on the rectangle with vertices $(0,0),(a,0),(0,b),(a,b)$
I know the solution is unique since the difference between any two solutions is a solution to $$\nabla^2u(x,y) = 0$$ which has the trivial solution $u = 0$.
I feel like there is a simple way to do this or even guess a solution in this constant source two dimensional case but I'm stuck.
Any help would be appreciated.
Start with $v(x,y)=x(x-a)/2$ which satisfies $\nabla^{2}v=1$ and $$ v(0,y)=0,\;\;\; v(a,y)=0,\\ v(x,0)=x(x-a)/2,\;\;\; v(x,b)=x(x-a)/2. $$ Then solve for $\nabla^{2}w_1=0$ such that $$ w_1(0,y)=0,\;\;\; w_1(a,y)=0,\\ w_1(x,0)=0,\;\;\; w_1(x,b)=x(x-a)/2. $$ It follows that $w_2(x,y)=w_1(x,b-y)$ satisfies $\nabla^{2}w_2=0$ and $$ w_2(0,y)=0,\;\;\; w_1(a,y)=0,\\ w_2(x,0)=x(x-a)/2,\;\;\;w_2(x,b)=0. $$ The desired solution is $u=v-w_1-w_2$.