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arXiv · 1702.03746

Nonlinear diffusion in transparent media: the resolvent equation

Lorenzo Giacomelli·Salvador Moll·Francesco Petitta
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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.

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Lorenzo Giacomelli, Salvador Moll, Francesco Petitta. 2017-02-13. Nonlinear diffusion in transparent media: the resolvent equation. https://doi.org/10.1515/acv-2017-0002

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