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

TL;DR

This paper extends the classical strong maximum principle for elliptic equations to cases with sign-changing forcing terms and introduces flat solutions where boundary derivatives vanish, leading to new insights on solution uniqueness and positivity.

Contribution

It generalizes the strong maximum principle to include sign-changing forcing and flat solutions, revealing cases where unique continuation fails and providing applications to elliptic and heat equations.

Findings

Existence of positive solutions for indefinite sign sublinear elliptic problems

Failure of unique continuation in certain flat solutions

Positivity of heat equation solutions with sign-changing data for large times

Abstract

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.