Nonlinear diffusion in transparent media: the resolvent equation

We consider the partial differential equation u−f=div(u^m ∇u/|∇u|) with f nonnegative and bounded and m∈R. We prove existence and uniqueness of solutions for both the Dirichlet problem (with bounded and nonnegative boundary datum) and the homogeneous Neumann problem. Solutions, which a priori belong to a space of truncated bounded variation functions, are shown to have zero jump part with respect to the ℋ^{N−1}-Hausdorff measure. Results and proofs extend to more general nonlinearities.