Cascade of minimizers for a nonlocal isoperimetric problem in thin domains (Q2878991)

The authors extend previous results of \textit{X. Ren} and \textit{L. Truskinovsky} [J. Elasticity 59, No. 1--3, 319--355 (2000; Zbl 0990.74007)] and \textit{X. Ren} and \textit{J. Wei} [SIAM J. Math. Anal. 31, No. 4, 909--924 (2000; Zbl 0973.49007)] on one-dimensional nonlocal isoperimetric minimization problems, to a corresponding higher dimensional mass constrained case. The latter case, in a different setting, had previously been handled in the two-dimensional case by \textit{G. Alberti} et al. [J. Am. Math. Soc. 22, No. 2, 569--605 (2009; Zbl 1206.49046)] and also by \textit{E. N. Spadaro} [Interfaces Free Bound. 11, No. 3, 447--474 (2009; Zbl 1180.35583)], modelling diblock copolymers. The general case presents a formidable degree of technical difficulty. By initially focusing on the two-dimensional setting the authors show that a special, stripped pattern is the minimizer. Throughout the paper, the authors employ geometric measure theory and its relationship with \(\gamma\)-convergence theory. Their new techniques lead to a stability result that relates the one-dimensional case to the thin rectangles case, allowing them to show that stable lamellar patterns are actually Lebesgue integrable minimizers in rectangular domains. In the final part of the paper the authors show how to generalize the two-dimensional results to higher dimensions.