Another way to say subsolution: The maximum principle and sums of Green functions

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.