Variational models for the interaction of surfactants with curvature -- existence and regularity of minimizers in the case of flexible curves

TL;DR

This paper proves the existence and regularity of minimizers for a geometric variational model describing surfactant interactions with curved interfaces, incorporating Helfrich, Frank, and coupling energies in a one-dimensional curve setting.

Contribution

It introduces a new variational model for surfactant-interface interactions and establishes analytical existence and regularity results for minimizers in the case of flexible curves.

Findings

Existence of minimizers for the proposed energy functional.

Regularity results ensuring smoothness of minimizers.

Analytical framework applicable to one-dimensional curves.

Abstract

Existence and regularity of minimizers for a geometric variational problem is shown. The variational integral models an energy contribution of the interface between two immiscible fluids in the presence of surfactants and includes a Helfrich type contribution, a Frank type contribution and a coupling term between the orientation of the surfactants and the curvature of the interface. Analytical results are proven in a one--dimensional situation for curves.