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

## TL;DR

This paper examines various boundary conditions in comparison principles for viscosity solutions, using examples from nonlinear elliptic PDEs like the Monge-Ampère equation reformulated as a Hamilton-Jacobi-Bellman problem, highlighting differences in sub- and supersolutions.

## Contribution

It clarifies how different boundary condition notions affect comparison principles and solution sets in viscosity solutions for fully nonlinear elliptic PDEs.

## Key findings

- Different boundary condition notions admit different sets of viscosity solutions
- Examples illustrate the impact on comparison principles
- Reformulation of Monge-Ampère as Hamilton-Jacobi-Bellman aids analysis

## Abstract

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-Amp\`ere 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.