Calculus of Variations and Geometric Measure Theory
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A nonlocal isoperimetric problem with dipolar repulsion

Simon Thilo

created by gelli on 18 May 2018

6 jun 2018 -- 17:00   [open in google calendar]

Sala Seminari Dipartimento di Matematica di Pisa

Abstract.

We study a functional in which perimeter and regularized dipolar repulsion compete under a volume constraint.

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.

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