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

Abstract

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