Nonlinear diffusion in transparent media: The resolvent equation

Abstract

AbstractWe consider the partial differential equationu-f=\operatornamewithlimits{div}\biggl{(}u^{m}\frac{\nabla u}{|\nabla u|}% \biggr{)}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}}-Hausdorff measure. Results and proofs extend to more general nonlinearities.