CRM CAMP in Nonlinear Analysis

Preuves mathématiques assistées par ordinateur en analyse non linéaire

23 juin 2020 de 10 h 00 à 11 h 00 (heure de Montréal/HNE) Réunion Zoom

Stable periodic patterns in 3D for the Ohta-Kawasaki problem

Séminaire par Jan Bouwe van den Berg (VU Amsterdam, Netherlands)

In this talk we discuss a mathematically rigorous computational method to compare local minimizers of the Ohta-Kawasaki free energy, describing diblock copolymer melts. This energy incorporates a nonlocal term to take into account the bond between the monomers.

Working within an arbitrary space group symmetry, we explore the phase space, computing candidates both with and without experimentally observed symmetries. We validate the phase diagram, identifying regions of parameter space where different spatially periodic structures have the lowest energy. These patterns may be lamellar layers, hexagonally packed cylinders, body-centered or close-packed spheres, as well as double gyroids and 'O70' arrangements. Each computation is validated by a mathematical theorem, where we bound the truncation errors and apply a fixed point argument to establish a computer-assisted proof. The method can be applied more generally to symmetric space-time periodic solution of many partial differential equations.