On the solutions to 1-Laplacian equation with L1 data

In the present paper we study the behaviour, as p goes to 1, of the renormalized solutions to the problems −div |∇up|p−2∇up = f in Ω, up =0 on ∂Ω, (0.1) where p >1, Ω is a bounded open set of RN (N 2) with Lipschitz boundary and f belongs to L1(Ω). We prove that these renormalized solutions pointwise converge, up to “subsequences,” to a function u.With a suitable definition of solution we also prove that u is a solution to a “limit problem.” Moreover we analyze the situation occurring when more regular data f are considered. © 2009 Elsevier Inc. All rights reserved.