I understand how to prove the maximum principle for $u_{xx}+u_{yy}=0$, but how does this extend to a maximum principle for the equation $u_{xx}+u_{yy}=f$?
I believe this is called Poisson’s equation. Here $u$ is given by a prescribed function $f$ in a bounded domain $D$.
I just don’t see how the same proof generalises (the one about perturbing $u$ to force a positive Laplacian, eg considering $v(x,y)= u(x,y) + \epsilon (x^2 + y^2)$).
Thanks for help.