The existence of minimizers in the fractional isoperimetric problem with multiple volume constraints is proved, together with a partial regularity result.

Existence and almost everywhere regularity of isoperimetric clusters for fractional perimeters

In this paper we make the final step in finding the optimal way to enclose and separate four planar regions with equal area. In Paolini and Tamagnini (ESAIM COCV 24(3):1303–1331, 2018) the graph-topology of the optimal cluster was found reducing the set of candidates to a one-parameter family of different clusters. With a simple argument we show that the minimal set has a further symmetry and hence is uniquely determined up to isometries.

The quadruple planar bubble enclosing equal areas is symmetric

We study planar $N$-clusters that minimize, under an area constraint, a weighted perimeter $P_\varepsilon$ depending on a small parameter $\varepsilon>0$. Specifically we weight $2-\varepsilon$ the boundary between the interior chambers and $1$ the boundary between an interior chamber and the exterior one. We prove that as $\varepsilon\to 0$ minimizers of $P_\varepsilon$ converge to configurations of disjoint disks that maximize the number of tangencies, each weighted by the harmonic mean of the radii of the two tangent disks. We also obtain some information on the structure of minimizers for small $\varepsilon$.

Sticky-disk limit of planar $N$-bubbles

Abstract The topology of a minimal cluster of four planar regions with equal areas and smallest possible perimeter was found in [9]. Here we describe the computation used to check that the symmetric cluster with the given topology is indeed the unique minimal cluster.

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Minimal cluster computation for four planar regions with the same area

Abstract We use continuous and discrete unification to prove that standard triple bubbles in ℝ2 and 𝕊2 are the minimizers of perimeter, among all clusters (Definition 2.3) enclosing the same triple of areas. Unification defines a unified measurement that allows all configurations, regardless of areas, to compete together. Continuous unification proves that if a unified minimizer were better than expected, it would have to have at least one interior bubble component. Discrete unification proves there can only be one interior bubble and that it must be connected. This leaves only the “daisy” configurations: one interior bubble surrounded by an even number of “petals.” A more careful analysis also eliminates these, leaving only the standard triple bubbles as minimizers. The result on the sphere is new; the result in the plane is due to Wichiramala [11]. The double bubble in the sphere was done by Masters [6].

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Perimeter-Minimizing Triple Bubbles in the Plane and the 2-Sphere

We prove that the optimal way to enclose and separate four planar regions with equal area using the less possible perimeter requires all regions to be connected. Moreover, the topology of such optimal clusters is uniquely determined.

Andrea Tamagnini

Papers

Minimal clusters of four planar regions with the same area

Minimal clusters of four planar regions with the same area

Minimal cluster computation for four planar regions with the same area