Area quasi-minimizing partitions with a graphical constraint: relaxation and two-dimensional partial regularity

• Published in: arXiv.org
• Main Authors: Bonacini, Marco, Cristoferi, Riccardo
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 06.10.2022
• Subjects: Block copolymers; Configurations; Constraints; Mathematics - Analysis of PDEs; Minimal surfaces; Regularity; Thin films; Two dimensional models
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2107.13325

We consider a variational model for periodic partitions of the upper half-space into three regions, where two of them have prescribed volume and are subject to the geometrical constraint that their union is the subgraph of a function, whose graph is a free surface. The energy of a configuration is given by the weighted sum of the areas of the interfaces between the different regions, and a general volume-order term. We establish existence of minimizing configurations via relaxation of the energy involved, in any dimension. Moreover, we prove partial regularity results for volume-constrained minimizers in two space dimensions. Thin films of diblock copolymers are a possible application and motivation for considering this problem.