In this paper we study infinite isoperimetric clusters. An infinite cluster E in Rd is a sequence of disjoint measurable sets Ek C Rd, called regions of the cluster, k = 1, 2, 3, ... A natural question is the existence of a cluster E with given volumes ak > 0 of the regions Ek, having finite perimeter P(E), which is minimal among all the clusters with regions having the same volumes. We prove that such a cluster exists in the planar case d = 2, for any choice of the areas ak with Z 11ak < co. We also show the existence of a bounded minimizer with the property P(E) = Tl1( partial differential & SIM;E), where partial differential & SIM;E denotes the measure theoretic boundary of the cluster. Finally, we provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.
Isoperimetric planar clusters with infinitely many regions
Novaga M.;Paolini Emanuele;Stepanov E.;Tortorelli V. M.
2023-01-01
Abstract
In this paper we study infinite isoperimetric clusters. An infinite cluster E in Rd is a sequence of disjoint measurable sets Ek C Rd, called regions of the cluster, k = 1, 2, 3, ... A natural question is the existence of a cluster E with given volumes ak > 0 of the regions Ek, having finite perimeter P(E), which is minimal among all the clusters with regions having the same volumes. We prove that such a cluster exists in the planar case d = 2, for any choice of the areas ak with Z 11ak < co. We also show the existence of a bounded minimizer with the property P(E) = Tl1( partial differential & SIM;E), where partial differential & SIM;E denotes the measure theoretic boundary of the cluster. Finally, we provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.