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- Title
Constrained BV functions on covering spaces for minimal networks and Plateau's type problems.
- Authors
Amato, Stefano; Bellettini, Giovanni; Paolini, Maurizio
- 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.
- Subjects
FUNCTIONS of bounded variation; PLATEAU'S problem; COVERING spaces (Topology); LIPSCHITZ spaces; SOBOLEV spaces; HAUSDORFF measures
- Publication
Advances in Calculus of Variations, 2017, Vol 10, Issue 1, p25
- ISSN
1864-8258
- Publication type
Article
- DOI
10.1515/acv-2015-0021