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Consider an elliptic second order differential operator $L$ with no zeroth order term (for example the Laplacian $L=-\Delta$). If $Lu \leq 0$ in a domain $U$, then of course $u$ satisfies the maximum principle on every subdomain $V \subset U$. We prove a converse, namely that $Lu \leq 0$ 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 $\overline{V}$. 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.

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