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DCDS

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.

DCDS-B

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.

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