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

Recently, Imbert and Monneau have introduced the so-called flux-limited formulation of Hamilton-Jacobi equation with state constraint boundary conditions. When the spatial domain is a bounded interval, they proved that the latter formulation is equivalent to the more classical one which was originally introduced by H-M. Soner. In the present paper, we aim to prove the same result for a multidimensional spatial domain. More precisely, we give the proof for a general bounded domain of Rd with a C1 boundary, in both the stationary and evolutive cases. In this setting, we also prove another result given by Imbert and Monneau in dimension one, namely that it is possible to use only a reduced class of test-functions.