Flux-limited solutions and state constraints for quasi-convex Hamilton-Jacobi equations

TL;DR

This paper establishes the equivalence between Soner's state constraint formulation and flux-limited solutions for quasi-convex Hamilton-Jacobi equations in multidimensional bounded domains, extending previous results to more general settings.

Contribution

It generalizes the equivalence of state constraint and flux-limited solutions from one-dimensional intervals to multidimensional bounded open sets for stationary and evolution cases.

Findings

Proves equivalence in multidimensional settings

Extends results to stationary and evolution cases

Generalizes to $ ext{C}^1$ bounded open sets

Abstract

Imbert and Monneau proved that Soner's formulation of state constraint problems on a bounded interval is equivalent to the so-called flux-limited formulation they introduced recently. In the multidimensional setting, we show this result for a general $\mathcal{C}^{1}$ bounded open set of $\mathbb{R}^{d}$ and obtain the result in the stationary and the evolution type cases.