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EnglishWe 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.
Amato, S., Bellettini, G., Paolini, M., Constrained BV functions on covering spaces for minimal networks and Plateau's type problems, <<ADVANCES IN CALCULUS OF VARIATIONS>>, 2017; 10 (1): 25-47. [doi:10.1515/acv-2015-0021] [http://hdl.handle.net/10807/99226]
| Autori: | |
| Titolo: | Constrained BV functions on covering spaces for minimal networks and Plateau's type problems |
| Digital Object Identifier (DOI): | http://dx.doi.org/10.1515/acv-2015-0021 |
| URL: | http://www.reference-global.com/loi/acv |
| Data di pubblicazione: | 2017 |
| 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: | Inglese |
| Rivista: | |
| Citazione: | Amato, S., Bellettini, G., Paolini, M., Constrained BV functions on covering spaces for minimal networks and Plateau's type problems, <<ADVANCES IN CALCULUS OF VARIATIONS>>, 2017; 10 (1): 25-47. [doi:10.1515/acv-2015-0021] [http://hdl.handle.net/10807/99226] |
| Appare nelle tipologie: | Articolo in rivista, Nota a sentenza |
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