Nonlinear diffusion in transparent media: the resolvent equation

Abstract

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

Cited by in corpus (4)

• Anzellotti's pairing theory and the Gauss--Green theorem
• The Dirichlet problem for singular elliptic equations with general nonlinearities
• 1-Laplacian type problems with strongly singular nonlinearities and gradient terms
• Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and $L^1$ terms