Suppose $u$ is C^2 solution of Poisson equatino $u_{xx}+u_{yy}=-1$ in the square $S = \{ (x,y) : |x|<1, |y|<1 \}$. If $u$ satisfies homogeneous dirichlet boundary conditions, how can we find the maximum and minimum values of $u$?
I know that for laplace $\Delta u = 0$, the maximum of $u$ and $-u$ (minimum of u) occurs at the boundary. Do we have the same result for Poisson equation? If so, then max/min is zero since we have dirichlet boundary condition: $u = 0$ on the boundary.
No, you can't say where the maximum and minimum occurs when you have a source term $\Delta u(\mathbf{x})=f(\mathbf{x})$. Such a general result (say for all $f$ continuous) would tell you everything about where maximum/minimum of an arbitrary ($C^2$) function $g$ simply by applying the result to $\Delta u=\Delta g$ and imposing $u=g$ on the boundary.
Edit: In this particular case, as your $u_{xx}+u_{yy}=\operatorname{Tr}\operatorname{Hess}u$ has constant negative sign, we can rule out an interior minimum (since interior minimum has a positive semidefinite Hessian). However, you can't say where the maximum occurs (indeed if $u\neq 0$, then from $u\geq u_{min}=0$ the maximum must be interior).