# Nonlinear diffusion in transparent media: the resolvent equation

## TL;DR

This paper studies a nonlinear PDE involving a diffusion operator with a gradient-dependent term, establishing existence, uniqueness, and regularity of solutions under various boundary conditions.

## Contribution

It proves existence and uniqueness of solutions for the PDE with Dirichlet and Neumann boundary conditions, extending results to more general nonlinearities.

## Key findings

- Solutions have zero jump part with respect to Hausdorff measure.
- Existence and uniqueness are established for bounded, nonnegative data.
- Results extend to more general nonlinearities.

## Abstract

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.