Existence of varifold minimizers for the multiphase Canham-Helfrich functional

TL;DR

This paper proves the existence of varifold minimizers for the multiphase Canham-Helfrich functional, modeling heterogeneous biological membranes with variable properties, without assuming symmetry, and discusses their regularity and bounds.

Contribution

It introduces a new variational framework using oriented curvature varifolds for multiphase membranes, establishing existence, regularity, and bounds without symmetry assumptions.

Findings

Existence of minimizers under area and volume constraints.

Lower semicontinuity of the functional in the varifold setting.

Bounds on the diameter of minimizers.

Abstract

We address the minimization of the Canham-Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers is here assumed. Correspondingly, the problem is reformulated and solved in the weaker frame of oriented curvature varifolds. We present a lower semicontinuity result and prove existence of single- and multiphase minimizers under area and enclosed-volume constrains. Additionally, we discuss regularity of minimizers and establish lower and upper diameter bounds.