Entropy solutions for first-order quasilinear equations related to a bilateral obstacle condition in a bounded domain

This paper is devoted to the existence and the uniqueness of the entropy solution for a general scalar conservation law associated with a forced bilateral obstacle condition in a bounded domain of R-P, p greater than or equal to 1. The method of penalization is used with a view to obtaining an existence result. However, the former only gives uniform L-infinity-estimates and so leads in fact to look for an Entropy Measure-Valued Solution, according to the specific properties of bounded sequences in L-infinity. The uniqueness of this EMVS is proved. Classically, it first ensures the existence of a bounded and measurable function U entropy solution and then the strong convergence in L-q of approximate solutions to U.