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Seminario di Equazioni Differenziali e Applicazioni: “Regularity and singularities of multiple valued Dirichlet minimizing harmonic maps”

Mercoledì 20 Giugno 2018, ore 12:15 - Aula 2BC30 - Salvatore Stuvard

ARGOMENTI: Seminari

Mercoledì 20 Giugno 2018 alle ore 12:15 in Aula 2BC30, Salvatore Stuvard (University of Texas, Austin) terrà un seminario dal titolo “Regularity and singularities of multiple valued Dirichlet minimizing harmonic maps”.

Abstract
The Dirichlet energy is one of the simplest possible functionals involving first derivatives, and the analysis of the properties of its minimizers is classically presented as the prototypical application of the by-now standard elliptic regularity theory. On the other hand, when the Dirichlet functional is studied on non-linear spaces, the corresponding Euler-Lagrange equations exhibit non-linearities which make it a challenge to study the properties of the solutions. Two emblematic examples of this general principle are the theory of harmonic maps and the theory of Dir-minimizing multiple valued functions. The former is concerned with the minimization of the Dirichlet energy among $W^{1,2}$ functions which are constrained to take values in some prescribed Riemannian manifold, and is connected with various theories from Physics (e.g. liquid crystals, ferromagnetic materials, superconductors). The latter, instead, prescribes that the functions competing in the minimization process attain, at each point in the domain, a fixed number $Q\geq2$ of values, and it is essential in understanding the regularity of area-minimizing surfaces in codimension higher than one.
In this talk, we let the difficulties of both worlds join forces, and study the regularity of minimizers of the Dirichlet energy among $Q$-valued maps taking values in a given Riemannian manifold. In particular, we apply quantitative stratification techniques to deduce rectifiability properties and uniform Minkowski estimates on the set of singular points. Moreover, we show that, unlike what happens in the single-valued harmonic maps framework, the topology of the target may play a role in establishing whether singularities are actually developing or not.
This is joint work with Jonas Hirsch (University of Leipzig) and Daniele Valtorta (University of Zurich).

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