Local and global minimality results for an isoperimetric problem with long-range interactions

Abstract

In this paper we review some recent results concerning the following non-local isoperimetric problem: Open image in new window where m ∈ (-1,1) is given and prescribes the volume of the two phases {u = 1} and {u = -1}. Here PΩ stands for the perimeter relative to Ω (in the sense of Caccioppoli-De Giorgi) and BV(Ω; {-1, 1}) denotes the space of functions of bounded variation taking values in {-1, 1}. We refer to [4] and [33] for a rather complete account on the properties of functions of bounded variations and sets of finite perimeter.