Calculus of Variations and Geometric Measure Theory
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Low density phases in a uniformly charged liquid

Matteo Novaga (Dip. Mat. Univ. Pisa)

created by magnani on 25 Nov 2014

26 nov 2014 -- 17:30

Sala Seminari (Dipartimento di Matematica)

Abstract.

This talk is concerned with the macroscopic behavior of global energy minimizers in the three-dimensional sharp interface Ohta-Kawasaki model of diblock copolymer melts. We are interested in the large volume behavior of minimizers in the low volume fraction regime, in which one expects the formation of a periodic lattice of small droplets of the minority phase in a sea of the majority phase. Under periodic boundary conditions, we show that the considered energy converges to a function of the limit "homogenized" measure associated with the minority phase, consisting of a local linear term and a non-local quadratic term mediated by the Coulomb kernel. As a consequence, asymptotically the mass of the minority phase in a minimizer spreads evenly across the domain. We also prove that the energy density distributes uniformly across the domain as well, with the energy density approaching that of the minimizers of the volume constrained problem in the whole space. This suggest that in the microscopic limit the minimizers should appear as a uniformly distributed array of droplets which minimize the energy density for the volume constrained whole space problem.

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