Stability criteria for multiphase partitioning problems with volume constraints

N. Alikakos; A. Faliagas

doi:10.3934/dcds.2017028

Article Contents

2017, Volume 37, Issue 2: 663-683. Doi: 10.3934/dcds.2017028

This issue Previous Article Next Article A non-local bistable reaction-diffusion equation with a gap

Stability criteria for multiphase partitioning problems with volume constraints

N. Alikakos and
A. Faliagas^,

Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece

^* Corresponding author: A. Faliagas
^* Corresponding author: A. Faliagas

Received: June 2015

Revised: February 2016

Published: May 2017

The first author was partially supported through the project PDEGE (Partial Differential Equations Motivated by Geometric Evolution), co-financed by the European Union European Social Fund (ESF) and national resources, in the framework of the program Aristeia of the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF).

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Abstract

We study the stability of partitions involving two or more phases in convex domains under the assumption of at most two-phase contact, thus excluding in particular triple junctions. We present a detailed derivation of the second variation formula with particular attention to the boundary terms, and then study the sign of the principal eigenvalue of the Jacobi operator. We thus derive certain stability criteria, and in particular we recapture the Sternberg-Zumbrun result on the instability of the disconnected phases in the more general setting of several phases.

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Mathematics Subject Classification: Primary:35B35, 35B36, 49R05;Secondary:53Z05.

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Figure 1. Three-phase partitioning of a set $\Omega$. Subsets painted with the same color contain material of the same phase. Interfaces ($M_{1},\cdots,M_{4}$) and subsets $(\Omega_{1},\cdots,\Omega_{5})$ are identified by successive indexing. Phases are enumerated in the same way: yellow is 1, green is 2, and cyan is 3. An alternative, more convenient for the calculations, identification scheme of subsets is shown in parentheses: the first index corresponds to a phase and the second enumerates the connected components of the subsets occupied by that phase; for example $\Omega_{12}(\equiv\Omega_{4})$ is the second connected component of phase 1. For connected phases (i.e. those occupying a single connected subset) we omit, for brevity, the second index; e.g. $\Omega_{31}\equiv\Omega_{3}$

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Figure 2. Disconnected 3-phase partitioning. Differently shaded regions correspond to different phases. The notation is as in Figure 1. The phases are from left to right 1, 2, 3, and 1 (see first index of notation in parentheses). $N_{i}$ is the unit normal field of interface $M_{i}$ (in the indicated orientation)

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