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

TL;DR

This paper studies a variational model for partitioning the upper half-space into three regions with a graphical constraint, establishing existence of minimizers in any dimension and partial regularity in two dimensions, motivated by diblock copolymer films.

Contribution

It introduces a relaxed variational framework for partitions with graphical constraints and proves existence and regularity results, extending previous models to higher dimensions.

Findings

Existence of minimizers in any dimension.

Partial regularity of minimizers in two dimensions.

Application relevance to diblock copolymer thin films.

Abstract

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.