Calculus of Variations and Geometric Measure Theory

M. Bonacini - R. Cristoferi

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

created by cristoferi on 14 Sep 2021
modified by bonacini on 06 Oct 2022


Published Paper

Inserted: 14 sep 2021
Last Updated: 6 oct 2022

Journal: Journal of Nonlinear Science
Volume: 32
Number: 93
Year: 2022
Doi: 10.1007/s00332-022-09852-3

ArXiv: 2107.13325 PDF


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.