  We show that the classical strong maximum principle, concerning positive
supersolutions of linear elliptic equations vanishing on the boundary of the
domain $\Omega $ can be extended, under suitable conditions, to the case in
which the forcing term $f(x)$ is changing sign. In addition, in the case of
solutions, the normal derivative on the boundary may also vanish on the
boundary (definition of flat solutions). This leads to examples in which the
unique continuation property fails. As a first application, we show the
existence of positive solutions for a sublinear semilinear elliptic problem of
indefinite sign. A second application, concerning the positivity of solutions
of the linear heat equation, for some large values of time, with forcing and/or
initial datum changing sign is also given.
