## Abstract and Applied Analysis

### 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 $\mathbb{R}^N$, 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 $(p>1)$ map flow equation, when the initial data is smooth with small total variation under periodic boundary condition.

#### Article information

Source
Abstr. Appl. Anal., Volume 2004, Number 8 (2004), 651-682.

Dates
First available in Project Euclid: 20 September 2004

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1095684286

Digital Object Identifier
doi:10.1155/S1085337504311048

Mathematical Reviews number (MathSciNet)
MR2096945

Zentralblatt MATH identifier
1068.35054

Subjects
Primary: 35R70: Partial differential equations with multivalued right-hand sides 35K90
Secondary: 58E20 26A45

#### Citation

Giga, Yoshikazu; Kashima, Yohei; Yamazaki, Noriaki. Local solvability of a constrained gradient system of total variation. Abstr. Appl. Anal. 2004 (2004), no. 8, 651--682. doi:10.1155/S1085337504311048. https://projecteuclid.org/euclid.aaa/1095684286