On the first and second variations of a nonlocal isoperimetric problem

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On the global minimizers of a nonlocal isoperimetric problem in two dimensions (2011, background): “3) as well as the second variation of Eγ are derived in [5] under the assumption of regularity of the free boundary.”
On minimizers of interaction functionals with competing attractive and repulsive potentials (2015, background): “Recently, a binary nonlocal isoperimetric functional appeared in connection with the modeling of self-assembly of diblock copolymers [29, 30, 31].”
Uniform energy distribution for an isoperimetric problem with long-range interactions (2008, background): “2) was presented in [10] (with a formal calculation presented in the appendix of [17]).”
Droplet Minimizers of an Isoperimetric Problem with Long‐Range Interactions (2011, methodology): “The first variations of Fγ,m have been computed for regular sets by Muratov [39] in dimension 2 and 3, and then in any dimension by Choksi and Sternberg [16].”