On the solutions to 1-Laplacian equation with L1L1 data

In the present paper we study the behaviour, as p goes to 1, of the renormalized solutions to the problemsequation(0.1){−div(|∇up|p−2∇up)=finΩ,up=0on∂Ω, where p>1p>1, Ω   is a bounded open set of RNRN(N⩾2)(N⩾2) with Lipschitz boundary and f   belongs to L1(Ω)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.