Nonlinear diffusion in transparent media: the resolvent equation

Lorenzo Giacomelli, Salvador Moll, Francesco Petitta

Published 2017-02-13Version 1

We consider the partial differential equation $$ u-f={\rm div}\left(u^m\frac{\nabla u}{|\nabla u|}\right) $$ 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.