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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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

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

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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.)


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.

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