Seminar Analysis: Sara Danieri (Friedrich-Alexander University Erlangen-Nürnberg)

Patterns formation for minimizers of a local/non-local interaction functional in general dimension

We study a functional consisting of a perimeter term and a non-local term which are in competition, both in the discrete and continuous setting.In the discrete setting such functional was introduced by Giuliani, Lebowitz, Liebe and Seiringer. For both the continuous and discrete problem, we show that the global minimizers are exact periodic stripes. One striking feature of the functionals is that the minimizers are invariant under a smaller group of symmetries than the functional itself. In the continuous setting, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functional is invariant under permutation of coordinates and the minimizers display a pattern formation which is one dimensional. Such behaviour for a smaller range of exponents in the discrete setting had been already shown,using different techniques. This is a joint work with E. Runa.

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