6533b830fe1ef96bd1297110
RESEARCH PRODUCT
Nonlinear diffusion in transparent media: the resolvent equation
Salvador MollLorenzo GiacomelliFrancesco Petittasubject
Dirichlet problemPure mathematicsTotal variation; transparent media; linear growth Lagrangian; comparison principle; Dirichlet problems; Neumann problems35J25 35J60 35B51 35B99Applied Mathematics010102 general mathematicsMathematics::Analysis of PDEsBoundary (topology)01 natural sciences010101 applied mathematicsMathematics - Analysis of PDEsBounded functionBounded variationFOS: MathematicsNeumann boundary conditionUniquenessNabla symbol0101 mathematicsAnalysisAnalysis of PDEs (math.AP)ResolventMathematicsdescription
Abstract We consider the partial differential equation u - f = div ( u m ∇ u | ∇ u | ) u-f=\operatornamewithlimits{div}\biggl{(}u^{m}\frac{\nabla u}{|\nabla u|}% \biggr{)} with f nonnegative and bounded and m ∈ ℝ {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 ℋ N - 1 {{\mathcal{H}}^{N-1}} -Hausdorff measure. Results and proofs extend to more general nonlinearities.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2017-02-13 |