Calculus of Variations and Geometric Measure Theory
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F. Dayrens - S. Masnou - M. Novaga - M. Pozzetta

Connected perimeter of planar sets

created by novaga on 21 Jun 2019


Submitted Paper

Inserted: 21 jun 2019
Last Updated: 21 jun 2019

Year: 2019


We introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous 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.


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