Abstract and Applied Analysis
Volume 2004 (2004), Issue 8, Pages 651-682

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

Copyright © 2004 Yoshikazu Giga et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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