The singular limit of a bilateral obstacle problem for a class of degenerate parabolic-hyperbolic operators

We study the limit as ɛ goes to 0+ of the sequence (uɛ)ɛ>0 of solutions to the Dirichlet problem for the weakly degenerate quasilinear parabolic operators Hɛ(t,x,.):u→∂tu+∑i=1p∂xifi(t,x,u)+g(t,x,u)−ɛΔϕ(u), subject to an inner bilateral constraint in an open bounded domain of Rp, 1⩽p<+∞. We first establish the existence of uɛ by coupling the method of penalization with that of artificial viscosity. The uniqueness proof for uɛ is based on the technique of doubling the time variable and on an assumption on the local behavior of f(.,.,ϕ−1(.)). An L∞-estimate for (uɛ)ɛ>0 is used to take the limit with ɛ through to the notion of entropy process solution.