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.
|Titolo:||Constrained BV functions on covering spaces for minimal networks and Plateau's type problems|
|Citazione:||Constrained BV functions on covering spaces for minimal networks and Plateau's type problems / Amato, S; Bellettini, Giovanni; Paolini, M.. - In: ADVANCES IN CALCULUS OF VARIATIONS. - ISSN 1864-8258. - 10:1(2017), pp. 25-47.|
|Appare nelle tipologie:||1.1 Articolo in rivista|