Advances in Differential Equations

Surface diffusion with triple junctions: A stability criterion for stationary solutions

Harald Garcke, Kazuo Ito, and Yoshihito Kohsaka

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Abstract

We study a fourth-order geometric evolution problem on a network of curves in a bounded domain $\Omega$. The flow decreases a weighted total length of the curves and preserves the enclosed volumes. Stationary solutions of the flow are critical points of a partition problem in $\Omega$. In this paper we study the linearized stability of stationary solutions using the $H^{-1}$-gradient flow structure of the problem. Important issues are the development of an appropriate PDE formulation of the geometric problem and Poincaré type estimate on a network of curves.

Article information

Source
Adv. Differential Equations Volume 15, Number 5/6 (2010), 437-472.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355854677

Mathematical Reviews number (MathSciNet)
MR2643231

Zentralblatt MATH identifier
1228.35042

Subjects
Primary: 35B35: Stability 35G30: Boundary value problems for nonlinear higher-order equations 35K55: Nonlinear parabolic equations 35R35: Free boundary problems 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

Citation

Garcke, Harald; Ito, Kazuo; Kohsaka, Yoshihito. Surface diffusion with triple junctions: A stability criterion for stationary solutions. Adv. Differential Equations 15 (2010), no. 5/6, 437--472. https://projecteuclid.org/euclid.ade/1355854677.


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