Local solvability of a constrained gradient system of total variation

Abstract

A $1$-harmonic map flow equation, a gradient system of total variation where values of unknowns are constrained in a compact manifold in ${ℝ}^N$, is formulated by the use of subdifferentials of a singular energy—the total variation. An abstract convergence result is established to show that solutions of approximate problem converge to a solution of the limit problem. As an application of our convergence result, a local-in-time solution of $1$-harmonic map flow equation is constructed as a limit of the solutions of $p$-harmonic $\left(p>1\right)$ map flow equation, when the initial data is smooth with small total variation under periodic boundary condition.