On the notion of boundary conditions in comparison principles for viscosity solutions
Numerical Analysis
2018-12-18 v1
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.
Keywords
Cite
@article{arxiv.1703.07313,
title = {On the notion of boundary conditions in comparison principles for viscosity solutions},
author = {Max Jensen and Iain Smears},
journal= {arXiv preprint arXiv:1703.07313},
year = {2018}
}