On Constrained Equations with Singular Diffusivity

Abstract

This is a continuation of our work [KG], [GGK], where we studied a gradient (flow) system of an energy whose energy density is not C1 so that the diffusivity in the equation is very strong and its effect is even nonlocal. In this paper we consider the case when the values of unknowns are constrained. To be specific we consider a gradient (flow) system of the total variations of mappings with constraint of their values. We also study the behavior of solution when M is the unit circle S1. The equation of the motion of the plateau is presented, which is written in the form of reducing ODE.We identify the form of stationary solutions and prove that the solution becomes a stationary solution in finite time. In the last part of this paper we demonstrate numerical simulations of the S1-valued problem. Although our theoretical approach is restricted to the piecewise constant solutions at this point, our method also applies to calculating the evolution of the solutions in more general class. Our method to solve S1-valued problem employs an angle variable, thus it is different from the method in [MSO] whereM is represented as a zero level set of some functions.