Invariant cones for cooperative elliptic systems with diffusion and drift couplings

The focus of the present paper is to identify suitable conditions on the structure of a linear elliptic system of second-order differential inequalities in non-divergence form in order to guarantee the existence of invariant sets in the sense of Weinberger for the system itself. Here, we show that, under suitable algebraic conditions on the structure of couplings, the system can be scalarized to a single fully nonlinear differential inequality whose viscosity solutions have the desired order preserving property. In the case that the invariant set is the negative orthant in dimension n, this yields the propagation of sign from the boundary to the interior, which generally fails when couplings occur in the system at the level of first or second derivatives, and different invariant cones are involved.