A strongly degenerate quasilinear equation: the elliptic case

We prove existence and uniqueness of entropy solutions for the Neumann problem for the quasilinear elliptic equation u − div a(u, Du) = v, where v ∈ L1, a(z, ξ) = ∇ξ f (z, ξ), and f is a convex function of ξ with linear growth as ‖ξ‖ → ∞, satisfying other additional assumptions. In particular, this class includes the case where f (z, ξ) = φ(z)ψ(ξ), φ > 0, ψ being a convex function with linear growth as ‖ξ‖ → ∞. In the second part of this work, using Crandall-Ligget’s iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions for the corresponding parabolic problem with initial data in L1. Mathematics Subject Classification (2000): 35J60 (primary); 47H06, 47H20 (secondary).