The total variation flow with nonlinear boundary conditions

Abstract

We study existence, uniqueness and the asymptotic behaviour of the entropy solutions for the Total Variation Flow with nonlinear boundary conditions. To prove the existence we use the nonlinear semigroup theory and for the uniqueness we apply Kruzhkov's method of doubling variables both in space and in time. We show that when the initial data are in L², the entropy solutions are strong solutions. Respect to the asymptotic behaviour, we show that entropy solutions stabilize as t→∞ by converging to a constant function.