Journal ArticleDOI
The quadruple planar bubble enclosing equal areas is symmetric
Abstract:
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.read more
Citations
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A structure‐preserving finite element approximation of surface diffusion for curve networks and surface clusters
TL;DR: A parametric element method based on a suitable variational formulation is proposed that can be shown to satisfy the volume conservation of each enclosed bubble and the unconditional energy-stability, thus preserving the two fundamental geometric structures of the mesh points.
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Sticky-disk limit of planar $N$-bubbles
TL;DR: In this article, the authors studied planar clustering that minimize, under an area constraint, a weighted perimeter, depending on a small parameter (i.e., varepsilon>0).
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A formula for the minimal perimeter of clusters with density
TL;DR: In this paper , it was shown that a limit of an isoperimetric minimizing sequence of clusters with volumes V is always isomorphic for its own volumes (which may be smaller than V).
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Isoperimetric planar clusters with infinitely many regions
TL;DR: In this article , the authors studied infinite isoperimetric clusters and proved that such a cluster exists in the planar case for any choice of the areas $ a_k $ with $ \sum \sqrt a k < \infty $.
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On the Steiner property for planar minimizing clusters. The anisotropic case
TL;DR: In this article , the Steiner property for minimal clusters in the plane with an anisotropic double density was discussed, and it was shown that this property holds under very weak assumptions on the densities.
References
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Book
Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory
TL;DR: A good introduction to geometric measure theory can be found in this article, which bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis, such as existence, regularity, analysis of singularities, characterization and symmetry results for minimizers in geometric variational problems.
Journal ArticleDOI
The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces
TL;DR: In this paper, a complete classification of the local structure of singularities in a wide class of two-dimensional surfaces in R3 collected under the adjective (M, i, a) minimal by Almgren [A3] was provided.
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Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints
TL;DR: In this article, the authors studied the structure of m-dimensional subsets of R which are well behaved with respect to deformations of R and also showed the existence of such sets as solutions to geometric variational problems satisfying various constraints.