Beyond the classical strong maximum principle: forcing changing sign near the boundary and flat solutions

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