Viscosity Solutions

Abstract

The main results of Chaps. 3–9 and 11 and are all verification theorems. Any function $\phi$ which satisfies the given requirements is necessarily the value function $\varPhi$ of the corresponding problem. These requirements are made as weak as possible in order to include as many cases as possible. For example, except for the singular control case, we do not require the function $\phi$ to be $C^2$ everywhere (only outside $\partial D$), because except for that case, $\varPhi$ will usually not be $C^2$ everywhere. On the other hand, all the above-mentioned verification theorems require $\phi$ to be $C^1$ everywhere, because this is often the case for $\varPhi$. This $C^1$ assumption on $\varPhi$ is usually called the “high contact” – or “smooth fit” – principle.