David Caraballo, Georgetown University
Title: Multiple junctions in polycrystals
Partitions minimizing surface energy have been a subject of great interest for decades. One particularly important result is Jean Taylor's theorem concerning the impossibility of 4 or more regions meeting at a point in a length minimizer (satisfying area constraints) in the plane.
We consider a much more general problem in n dimensions, concerning minimizers (satisfying volume constraints) of anisotropic surface energies which may depend on the regions which meet and also on the orientations of the interfaces. We will briefly discuss recent results on the existence of such minimizers. We will also discuss some new results concerning the number of regions which can meet locally in a minimizer. In particular, for minimizers we can rule out there being too many regions, or even components of regions, present inside suitably small balls -- and our upper bounds on the number of such components are explicitly given in terms of the ambient space dimension and the anisotropic surface energy density functions.
These questions are important since most materials are polycrystalline and typically have surface energies which are anisotropic, and since multiple junctions affect physical properties of materials.
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George Mason University
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