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 is an element of L-1, a(z, epsilon) = del(epsilon) f(z, epsilon), and f is a convex function of with linear growth as parallel toepsilonparallel to --> infinity, satisfying other additional assumptions. In particular, this class includes the case where f(z, epsilon) = phi(z)psi(epsilon), phi > 0, psi being a convex function with linear growth as parallel toepsilonparallel to --> infinity. 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 L-1.