Sequential Compactness Conditions for the Theorem of Sums in the Viscosity Theory of PDEs

Viscosity solutions offer a theory of weak solutions to nonlinear first and second-order elliptic PDEs. A common way to establish solution regularity is the Ishii-Lions doubling of variables method. A key step involves the theorem of sums, which relates information about solutions to the first and second derivatives of test functions at a point. The conclusion of the theorem of sums takes a familiar, simpler form in the smooth case, and one can sometimes use a compactness argument to recover this simpler form in the general setting. We will summarize known conditions and introduce a new condition to guarantee sequential compactness. Notably, these conditions hold in non-Euclidean geometries.