The equivalence of viscosity and distributional subsolutions for convex subequations - a strong Bellman principle

There are two useful ways to extend nonlinear partial differential inequalities of second order beyond the class of C² functions: one uses viscosity theory and the other uses the theory of distributions. This paper considers the convex situation where both extensions can be applied. The main result is that under a natural "second-order completeness" hypothesis, the two sets of extensons are isomorphic, in a sense that is made precise.