In contrast to previously studied similar problems, the nonlocal term contributes to the perimeter term to leading order for small regularization parameters.

Indeed, below a critical value for the dipolar strength, the limiting functional is a renormalized perimeter and for small, positive regularization parameters the minimizers are balls.
At critical dipolar strength, we identify the next-order Gamma-limit and prove that a continuous pertubation of the problem has non-spherical minimizers for some masses.

Furthermore, for a wide class of nonlocal isoperimetric problems, we establish existence of generalized minimizers by interpreting them as minimizers of suitably relaxed functionals.
http://cvgmt.sns.it/seminar/642/