Abstract
The modeling of pattern morphologies via energy minimization involving long and shortrange competitions is well-established and ubiquitous in science (cf. [47, 23, 33] and the references therein). It also provides mathematicians with a wealth of rich variational problems consisting of both local (associated with short-range interactions) and nonlocal (associated with long-range interactions) terms (see for example [29] and the references therein). Here we are concerned with two nonlocal variational problems which describe the simplest morphology class within this general setting, that of periodic phase separation. Many physical systems exhibit a phase separation which, according to experiments, can be roughly described as follows:
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