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

TL;DR

This paper demonstrates that under certain conditions, viscosity and distributional subsolutions for convex second-order PDEs are equivalent, unifying two major approaches in nonlinear PDE theory.

Contribution

It establishes an isomorphism between viscosity and distributional subsolutions for convex subequations under a natural completeness hypothesis.

Findings

Viscosity and distributional subsolutions are equivalent under second-order completeness.

The result unifies two approaches to nonlinear PDE inequalities.

Provides a theoretical foundation for applying both methods interchangeably.

Abstract

There are two useful ways to extend nonlinear partial differential inequalities of second order: 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 extensions are isomorphic, in a sense that is made precise.