Strong Variational Formulations for Bilateral Obstacle Problems linked to Parabolic Degenerate Equations and Singular Perturbation Properties

We study the limit as goes to 0+ of the sequence (u)>0 of solutions to the Dirichlet problem for the weakly degenerate quasilinear parabolic operators H(t, x,.) : u → ∂tu+ p∑ i=1 ∂xifi(t, x, u) + g(t, x, u) − ∆φ(u), associated with an inner bilateral constraint in an open bounded domain of Rp, 1 ≤ p < +∞. After setting the positive parameter , we first establish the existence of u by coupling the method of penalization with that of artificial viscosity. The uniqueness proof for u 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 (u)>0 is used to take the limit with through to the notion of entropy process solution