Connected perimeter of planar sets

François Dayrens (1) , Simon Masnou (1) , Matteo Novaga (2) , Marco Pozzetta (2)

1 MMCS - Modélisation mathématique, calcul scientifique
2 Dipartimento di Matematica [Pisa]

We introduce a notion of connected perimeter for planar sets defined as the lower semi-continuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss the application of this notion to the existence of solutions to a nonlocal minimization problem.

Domaines: Analyse fonctionnelle [math.FA]

François Dayrens, Simon Masnou, Matteo Novaga, Marco Pozzetta. Connected perimeter of planar sets. Advances in Calculus of Variations, In press. ⟨hal-02162832v2⟩