Stability in a ball-partition problem

In [4], stability of bubble clusters is characterized in terms of a quadratic form defined on the surfaces, which is then used to prove that the nonstandard double bubble is unstable. Since quadratic forms are often analyzed using eigenvalue methods, it seems natural to investigate the eigenvalue problem which arises from such a quadratic form. We will look at a specific example of a problem related to bubble clusters, in order to see how the quadratic form gives rise to interesting interactions across the singular curve. The example that we will look at is the following. Suppose that we partition a ball into three regions of equal volume. How can this be done so that the sum of the areas of the partitioning surfaces is a minimum? We will not answer this general question, but we will address the stability of the particular partition given by three half-disks sharing the line segment −1 ≤ x ≤ 1, y = 0, z = 0 in their boundaries and meeting at angles of (2/3)π. It is clear that this configuration is stationary, since the contact angles between the free surfaces and the fixed sphere are all π/2, and the behavior along the line segment satisfies the Plateau conditions (see [9]). This problem was chosen so that the various contributions due to curvature that appear in [4] would disappear, in that we could focus on the analysis rather than the geometry. It is important to realize that we cannot look at arbitrary normal perturbations on each surface. With the orientations of the normals that we will use, the condition that f1(p) + f2(p) + f3(p) = 0 must be imposed along the singular curve, so that infinitesimal perturbations will take the singular curve to a single curve rather than causing it to split. Following [4], we let be the set of triples 〈 f1, f2, f3〉 with fi ∈ H(Σi) satisfying the matching condition f1 + f2 + f3 = 0 along the singular curve γ. We have an additional consideration in that we have terms in the derivatives relating to perturbations of bubble surfaces near fixed surfaces. These are handled in a manner similar to that used in [11], although there is a slight difference from that in [11],