Assume we have that problem: $$ - \nabla ^ 2 u = f $$ $$ u|_{\partial \Omega} = 0, \Omega = [0, 1] ^2 $$ The question is what are the natural sufficient conditions on $f$ for the classical solution to exist.
Is it enough for $f$ to be continuous on its domain? $f \in C^1(\Omega)$?
I would like to see some reference if possible.
I am only aware about the existence of the weak solution for the case of $f \in L^2(\Omega)$.