Abstract and Applied Analysis 
Volume 2004 (2004), Issue 8, Pages 651-682
doi:10.1155/S1085337504311048

Local solvability of a constrainedgradient system of total variation

Yoshikazu Giga,1 Yohei Kashima,1,2 and Noriaki Yamazaki3

 1Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
2Department of Mathematics, University of Sussex, Brighton BN1 9QH, UK
3Department of Mathematical Science, Common Subject Division, Muroran Institute of Technology, 27-1 Mizumoto-cho, Muroran 050-8585, Japan
Received 9 October 2003

Abstract

A 1-harmonic map flow equation, a gradient system of total variation where values of unknowns are constrained in a compact manifold in ℝN, 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 (p>1) map flow equation, when the initial data is smooth with small total variation under periodic boundary condition.