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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Figure 1.
Three-phase partitioning of a set
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).
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