Optimal Transportation, Branching, and Interpolation Inequalities
speaker: Felix Otto (Max Planck Institute for Mathematics in the Sciences)abstract: Branched microstructures are abundant in materials science: non-superconducting threads in type-I superconductors, magnetic domains uniaxial ferromagnets, martensitic twins in shape memory alloys. Form a variational point of view, these microstructures arise as a competition between a transport term and a interfacial energy term (= perimeter). These microstructures are characterized by a finite energy per (cross-sectional) area. To rigorously establish this, one needs scale invariant interpolation inequalities that estimate a suitable \(L^p\)-norm by a combination of an \(H^{1,1}\) norm (=perimeter) and a negative norm or a transportation distance (Wasserstein distance), and thus are of Gagliardo-Nirenberg type. We show that behind each of the three different types of branched microstructures in superconductors, there is a suitable (previously unknown) interpolation estimate. We sketch the elementary proof of these estimates.
This is joint work with Eleonora Cinti
timetable:
Wed 7 Nov, 10:00 - 10:50, Aula Dini
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