Nonlinear diffusion in transparent media: the resolvent equation

We consider the partial differential equation $$u-f=\mathrm{d}\mathrm{i}\mathrm{v}\left(u^m\frac{\mathrm{\nabla }u}{\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }}\right)$$ with $f$ nonnegative and bounded and $m\in \mathbb{R}$. 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 ${\mathcal{H}}^{N-1}$ Haussdorff measure. Results and proofs extend to more general nonlinearities.