Hokkaido Mathematical Journal

Nonlinear stability of stationary solutions for curvature flow with triple junction

Harald GARCKE, Yoshihito KOHSAKA, and Daniel ŠEVČOVIČ

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Abstract

In this paper we analyze the motion of a network of three planar curves with a speed proportional to the curvature of the arcs, having perpendicular intersections with the outer boundary and a common intersection at a triple junction. As a main result we show that a linear stability criterion due to Ikota and Yanagida [13] is also sufficient for nonlinear stability. We also prove local and global existence of classical smooth solutions as well as various energy estimates. Finally, we prove exponential stabilization of an evolving network starting from the vicinity of a linearly stable stationary network.

Article information

Source
Hokkaido Math. J., Volume 38, Number 4 (2009), 721-769.

Dates
First available in Project Euclid: 18 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1258554242

Digital Object Identifier
doi:10.14492/hokmj/1258554242

Mathematical Reviews number (MathSciNet)
MR2561958

Zentralblatt MATH identifier
1187.35013

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B35: Stability 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

Keywords
curvature flow triple junction higher order estimates for the curvature nonlinear stability of stationary solutions

Citation

GARCKE, Harald; KOHSAKA, Yoshihito; ŠEVČOVIČ, Daniel. Nonlinear stability of stationary solutions for curvature flow with triple junction. Hokkaido Math. J. 38 (2009), no. 4, 721--769. doi:10.14492/hokmj/1258554242. https://projecteuclid.org/euclid.hokmj/1258554242


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