Stability criteria for multiphase partitioning problems with volume constraints
N. Alikakos A. Faliagas

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.

keywords: Phase partitioning stability stability criteria area functional second variation
Analysis of a corner layer problem in anisotropic interfaces
N. D. Alikakos P. W. Bates J. W. Cahn P. C. Fife G. Fusco G. B. Tanoglu
We investigate a model of anisotropic diffuse interfaces in ordered FCC crystals introduced recently by Braun et al and Tanoglu et al [3, 18, 19], focusing on parametric conditions which give extreme anisotropy. For a reduced model, we prove existence and stability of plane wave solutions connecting the disordered FCC state with the ordered $Cu_3Au$ state described by solutions to a system of three equations. These plane wave solutions correspond to planar interfaces. Different orientations of the planes in relation to the crystal axes give rise to different surface energies. Guided by previous work based on numerics and formal asymptotics, we reduce this problem in the six dimensional phase space of the system to a two dimensional phase space by taking advantage of the symmetries of the crystal and restricting attention to solutions with corresponding symmetries. For this reduced problem a standing wave solution is constructed that corresponds to a transition that, in the extreme anisotropy limit, is continuous but not differentiable. We also investigate the stability of the constructed solution by studying the eigenvalue problem for the linearized equation. We find that although the transition is stable, there is a growing number $0(\frac{1}{\epsilon})$, of critical eigenvalues, where $\frac{1}{\epsilon}$ » $1$ is a measure of the anisotropy. Specifically we obtain a discrete spectrum with eigenvalues $\lambda_n = \e^{2/3}\mu_n$ with $\mu_n$ ~ $Cn^{2/3}$, as $n \to + \infty$. The scaling characteristics of the critical spectrum suggest a previously unknown microstructural instability.
keywords: interfaces singular perturbation. corner layers stability anisotropy nonlinear boundary value problems crystalline structure

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