Constrained BV functions on covering spaces for minimal networks and Plateau's type problems

Maurizio Paolini, Stefano Amato, Giovanni Bellettini

Risultato della ricerca: Contributo in rivistaArticolo in rivistapeer review

5 Citazioni (Scopus)

Abstract

We link covering spaces with the theory of functions of bounded variation, in order to study minimal networks in the plane and Plateau's problem without fixing a priori the topology of solutions. We solve the minimization problem in the class of (possibly vector-valued) BV functions defined on a covering space of the complement of an (n -2)-dimensional compact embedded Lipschitz manifold S without boundary. This approach has several similarities with Brakke's "soap films" covering construction. The main novelty of our method stands in the presence of a suitable constraint on the fibers, which couples together the covering sheets. In the case of networks, the constraint is defined using a suitable subset of transpositions of m elements, m being the number of points of S. The model avoids all issues concerning the presence of the boundary S, which is automatically attained. The constraint is lifted in a natural way to Sobolev spaces, allowing also an approach based on Γ-convergence.
Lingua originaleEnglish
pagine (da-a)25-47
Numero di pagine23
RivistaAdvances in Calculus of Variations
Volume10
DOI
Stato di pubblicazionePubblicato - 2017

Keywords

  • Analysis
  • Applied Mathematics
  • Constrained BV functions
  • Coverings
  • Plateau's problem

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