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

Max Jensen (1) , Iain Smears (2, 3)

1 Department of Mathematics [Sussex]
2 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

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.

Mots clés: Viscosity boundary conditions; comparison principles; Hamilton–Jacobi–Bellman equations; Monge-Ampère equations; Barles–Souganidis theorem

Domaines: Analyse numérique [math.NA]

Max Jensen, Iain Smears. On the notion of boundary conditions in comparison principles for viscosity solutions. 2017. ⟨hal-01493586⟩