On the notion of boundary conditions in comparison principles for viscosity solutions
This work clarifies subtle theoretical issues in boundary conditions for viscosity solutions, which is important for researchers designing numerical methods for fully nonlinear PDEs.
The paper collects examples of boundary-value problems for fully nonlinear elliptic PDEs, focusing on the Monge-Ampère equation, to illustrate how different notions of boundary conditions affect viscosity sub- and supersolutions and the validity of comparison principles.
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-Ampère 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 validity of comparison principles for these different notions of boundary conditions.