On the notion of boundary conditions in comparison principles for viscosity solutions

We collect examples of boundary-value problems of Dirichlet and Dirichlet–Neumann type which we found instructive when designing and analysing numerical methods for fully nonlinear elliptic partial differential equations. In particular, our model problem is the Monge–Ampe`re equation, which is treated through its equivalent reformulation as a Hamilton– Jacobi–Bellman equation. Our examples illustrate how the different notions of boundary conditions appearing in the literature may admit different sets of viscosity sub- and supersolutions. We then discuss how these examples relate to the application of comparison principles in the analysis of numerical methods.