Another way to say subsolution: the maximum principle and sums of Green functions - Laugesen, R.S.; Watson, N. A. (Mathematica Scandinavica, 2005)
Consider an elliptic second order differential operator L with no zeroth order term (for example the Laplacian L=-Δ). If Lu ≤ 0 in a domain U, then of course u satisfies the maximum principle on every subdomain V U. We prove a converse, namely that Lu ≤ 0 on U if on every subdomain V, the maximum principle is satisfied by u+v whenever v is a finite linear combination (with positive coefficients) of Green functions with poles outside V. This extends a result of Crandall and Zhang for the Laplacian. We also treat the heat equation, improving Crandall and Wang’s recent result. The general parabolic case remains open.