Connected perimeter of planar sets

Abstract

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.

Connected Coulomb Columns: Analysis and Numerics

“…(1) Minimizers exist for any mass. This part, in Section 3.1, follows closely the arguments in [DMNP19]. (2) A charged droplet of small mass aggregates in a disk.…”

“…We prove the existence of solutions of the problem in (2.3) for all masses m > 0 and all λ > 0. This section follows the proof in [DMNP19], where the nonlocal part of the energy was given by…”

“…where P (•) is the usual perimeter of a set. It was shown recently [DMNP19], that for essentially bounded sets E ⊂ R 2 such that ∂E = ∂ * E modulo sets of zero H 1 -measure the identity P r C (E) = P (E) + 2St(E) holds, where St(E) is the length of the Steiner tree of Ē, i.e.,…”

“…In our setting, due to the restriction to columnar sets and a full Coulombic interaction, we are faced with nonexistence of minimizers for any prescribed area of a two-dimensional slice E when considering a standard perimeter since the extension to three dimensions always has infinite mass. Instead of introducing a background potential, however, we opt for a different avenue, also pursued in [DMNP19]: we replace the standard perimeter with a 'connected perimeter', which can be briefly described by the relaxation of the perimeter of a connected, L 1 -approximating set. For a precise statement on the setting, see Section 2.…”

“…Furthermore, even though we are talking about a minimization problem which is effectively two-dimensional, we will refer to the sets E as charged 'droplets'. This connected perimeter used here was introduced in [DMNP19]. A phase-field variant of the connected perimeter was developed in [DNWW19].…”