6533b7d3fe1ef96bd1261494
RESEARCH PRODUCT
Local and nonlocal weighted pLaplacian evolution equations with Neumann boundary conditions
José Manuel Mazón RuizFuensanta Andreu VaílloJosé Julián Toledo MeleroJulio D. Rossisubject
Neumann boundary conditionsDiffusion equationGeneral MathematicsOperator (physics)Nonlocal diffusionMathematical analysisMeasure (mathematics)P-laplacianBounded functionNeumann boundary conditionp-LaplacianUniquenessDiffusion (business)Total variation flowMathematicsMathematical physicsdescription
In this paper we study existence and uniqueness of solutions to the local diffusion equation with Neumann boundary conditions and a bounded nonhomogeneous diffusion coefficient g ≥ 0, {ut = div (g|∇u|p-2∇u) in ]0; T[×Ωg|∇u|p-2u·n = 0 on ]0; T[×∂Ω; for 1 ≤ p < ∞. We show that a nonlocal counterpart of this diffusion problem is ut(t; x)= ∫ω J(x-y)g(x+y/2)|u(t; y)-u(t; x)| p-2 (u(t; y)-u(t; x)) dy in ]0; T[× Ω,where the diffusion coefficient has been reinterpreted by means of the values of g at the point x+y/2 in the integral operator. The fact that g ≥ 0 is allowed to vanish in a set of positive measure involves subtle difficulties, specially in the case p = 1.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2011-01-01 |